From ddbdf5973d86242044dec9803dc80ab4a5897228 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar Date: Thu, 18 Jun 2020 22:17:31 -0700 Subject: [PATCH 001/152] Create Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 82 +++++++++++++++++++++++++++++++++++++ 1 file changed, 82 insertions(+) create mode 100644 _posts/Durr_Hoyer_Post_1.md diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md new file mode 100644 index 0000000..97713e7 --- /dev/null +++ b/_posts/Durr_Hoyer_Post_1.md @@ -0,0 +1,82 @@ +--- +title: "In Progress Durr Hoyer Algorithm" +author "mridulsar" +categories: + - Blog +tags: + - qsharp + - algorithm + - open source +--- +# Durr Hoyer + +Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. +## Background +Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. +## Motivation +The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. +## The Algorithm +A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: + +Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: + +![alt text](QMSA.JPG) +![alt text](wrapper.JPG) + + +It is easy to undestand the conceptual implementation in python. It is important to note steps: +## (a..c) + + + +### (a) +Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. + +For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. + +We do know that the register must be initiated as j divided by the square root of N (length of flattened table) along with a qubit for each integer and a redudant qubit, specifically being the randomly chosen y. + +This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. + +In order to mark the indeces as the algorithm states we must run the intialized qubits through grovers algorithm once. + + +### (b) +Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different prepration of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational effeciency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] + +It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". + +![alt text](implementation.JPG) + +A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: + +1. T is just a Hadamard Gate +2. S_A is just a CNOT Gate +3. S_0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back + +Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! + +### (c) + +Part (c) may be unclear simply because it has been commented out for testing purposes. Right now I am working on testing the algorithms and each gate. + + +# Conclusion + +So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. + + +### My current challenges in this project: + +1. Creating a C# test script/any test script for my circuit + +2. Validating generalizations I made of 'Implementation Considerations' + +3. Containerizing this project + +[(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf +[(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf +[3]: Essentialy, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algoirthm will run for the alloted while loop in the pythong code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. + + + From a4047e8952b190e0525f3eb74ddc2049979b33e5 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar Date: Thu, 18 Jun 2020 22:42:56 -0700 Subject: [PATCH 002/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 54 ++++++++++++++++++++++++++++++------- 1 file changed, 44 insertions(+), 10 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 97713e7..1d67153 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -11,23 +11,57 @@ tags: # Durr Hoyer Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. +------------------ + + ## Background Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. +------------------ + ## Motivation The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. +------------------ + ## The Algorithm A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: ![alt text](QMSA.JPG) -![alt text](wrapper.JPG) - +------------------ + +### Psuedocode + +``` + t = table.flatten() + N = len(t) + y = randomuniform[1..N] #N is a integer + time_limit = (22.5*sqrt(N)=1.4*log(N)^2) + qq = preprocess.simulate(t,y') + + while time.clock() < time_limit: + for i in range(N): + while t[i] < t[y]: + j = N-i + if N is even + Grovers Algorithm for one step across [0...j] + q=Algorithm with T[0...j] defined operator + y'=q[0] + else + Grovers Algorithm for one step across [0...j] + t=Algorithm with QFT instead of T[0...j] + y'=q[0] + if t[y'] < t[y]: + y = y' + return + return y +``` +------------------ It is easy to undestand the conceptual implementation in python. It is important to note steps: -## (a..c) - +## (a..b) +------------------ ### (a) Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. @@ -40,6 +74,7 @@ This still needs to be properely implemented as I have been wrestling with which In order to mark the indeces as the algorithm states we must run the intialized qubits through grovers algorithm once. +------------------ ### (b) Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different prepration of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational effeciency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] @@ -56,15 +91,13 @@ A thorough read of this will supply any calirfication to the python code posted Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! -### (c) - -Part (c) may be unclear simply because it has been commented out for testing purposes. Right now I am working on testing the algorithms and each gate. - +------------------ # Conclusion So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. +------------------ ### My current challenges in this project: @@ -74,9 +107,10 @@ So far I have been greatly enjoying throwing myself into this new and exciting w 3. Containerizing this project +------------------ + [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf [(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf -[3]: Essentialy, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algoirthm will run for the alloted while loop in the pythong code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. - +[3]: Essentialy, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algoirthm will run for the alloted while loop in the python code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. 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a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -1,6 +1,6 @@ --- title: "In Progress Durr Hoyer Algorithm" -author "mridulsar" +date: categories: - Blog tags: @@ -8,27 +8,28 @@ tags: - algorithm - open source --- -# Durr Hoyer +## Durr Hoyer Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. ------------------- + ## Background Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. ------------------- + ## Motivation The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. ------------------- + ## The Algorithm A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: -![alt text](QMSA.JPG) ------------------- + +![QMSA](/assets/images//DurrHoyer-QMSA.JPG "QMSA") + ### Psuedocode @@ -81,7 +82,7 @@ Part b must be understood through another paper 'Tight bounds on quantum searchi It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". -![alt text](implementation.JPG) +![Implementation](/assets/images/DurryHoyer-implementation.JPG "Implementation") A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: @@ -99,14 +100,12 @@ So far I have been greatly enjoying throwing myself into this new and exciting w ------------------ -### My current challenges in this project: +## My current challenges in this project: 1. Creating a C# test script/any test script for my circuit 2. Validating generalizations I made of 'Implementation Considerations' -3. Containerizing this project - ------------------ [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf From 73376a237255cb481de6c2b6d6f5382cd19b2dcf Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Thu, 18 Jun 2020 23:06:43 -0700 Subject: [PATCH 005/152] update --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 9c71feb..828b5e6 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -82,7 +82,7 @@ Part b must be understood through another paper 'Tight bounds on quantum searchi It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". -![Implementation](/assets/images/DurryHoyer-implementation.JPG "Implementation") +![Implementation](/assets/images/DurryHoyer-Implementation.JPG "Implementation") A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: From 57fa80c99a14c78fc06a6de648c94be4f63cf042 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Thu, 18 Jun 2020 23:08:42 -0700 Subject: [PATCH 006/152] update --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 828b5e6..c951a9f 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -82,7 +82,7 @@ Part b must be understood through another paper 'Tight bounds on quantum searchi It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". -![Implementation](/assets/images/DurryHoyer-Implementation.JPG "Implementation") +![Implementation](/assets/images//DurryHoyer-Implementation.JPG "Implementation") A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: From 8f47221df78177af7606f20b8524f48a7ace9621 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Thu, 18 Jun 2020 23:10:19 -0700 Subject: [PATCH 007/152] typo --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index c951a9f..85633e7 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -82,7 +82,7 @@ Part b must be understood through another paper 'Tight bounds on quantum searchi It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". -![Implementation](/assets/images//DurryHoyer-Implementation.JPG "Implementation") +![Implementation](/assets/images//DurrHoyer-Implementation.JPG "Implementation") A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: From 6a4e1a934ba4539cefb83b0bae94c7542649484f Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Thu, 18 Jun 2020 23:28:45 -0700 Subject: [PATCH 008/152] update psuedocode --- _posts/Durr_Hoyer_Post_1.md | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 85633e7..eea32a8 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -43,14 +43,12 @@ Durr and Hoyer propose a computationally effecient routine to find the minimum o while time.clock() < time_limit: for i in range(N): while t[i] < t[y]: - j = N-i + j = N-j if N is even - Grovers Algorithm for one step across [0...j] - q=Algorithm with T[0...j] defined operator + q=Algorithm with Hadamard gates[0...j] y'=q[0] else - Grovers Algorithm for one step across [0...j] - t=Algorithm with QFT instead of T[0...j] + t=Algorithm with QFT instead of H[0...j] y'=q[0] if t[y'] < t[y]: y = y' @@ -73,7 +71,7 @@ We do know that the register must be initiated as j divided by the square root o This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. -In order to mark the indeces as the algorithm states we must run the intialized qubits through grovers algorithm once. +In order to mark the indices we simply mark all indices that come before i. ------------------ From 5c02e7a12970a1c0f8e96a3644df6d7f40ac9292 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 01:05:26 -0700 Subject: [PATCH 009/152] headers --- _posts/Durr_Hoyer_Post_1.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index eea32a8..a4719b9 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -8,21 +8,21 @@ tags: - algorithm - open source --- -## Durr Hoyer +### Durr Hoyer Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. -## Background +### Background Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. -## Motivation +### Motivation The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. -## The Algorithm +### The Algorithm A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: @@ -31,7 +31,7 @@ Durr and Hoyer propose a computationally effecient routine to find the minimum o ![QMSA](/assets/images//DurrHoyer-QMSA.JPG "QMSA") -### Psuedocode +#### Psuedocode ``` t = table.flatten() @@ -58,11 +58,11 @@ Durr and Hoyer propose a computationally effecient routine to find the minimum o ------------------ It is easy to undestand the conceptual implementation in python. It is important to note steps: -## (a..b) +### (a,b) ------------------ -### (a) +#### (a) Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. @@ -75,7 +75,7 @@ In order to mark the indices we simply mark all indices that come before i. ------------------ -### (b) +#### (b) Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different prepration of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational effeciency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". @@ -92,7 +92,7 @@ Generalizations are a mathematicians greatest friend, but a downfall if not impl ------------------ -# Conclusion +### Conclusion So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. From f4a09b97f41b9eedd5dec51634a31bf576a9a85a Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 12:05:18 -0700 Subject: [PATCH 010/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index a4719b9..fcb1bbc 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -15,11 +15,11 @@ Hello all! My name is Mridul Sarkar. I am an undergraduate at University of Cali ### Background -Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. +Recently I began working on quantum computing with an idea to treat nodes of neural networks as qubits. From here I took a deep dive into the research papers and found myself amazed by the research being produced. I began with a Coursera course from St. Petersburge University on quantum computing with detail on Duestchs, Shors, and Grovers algorithms. From here I tinkered with Q# and developed basic algorithms that I learnt from Coursera. I found Durr Hoyer and saw it as an ensemble of Duestchs, Shors, and Grovers Algorithm; a perfect stepping stone coming from my background. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time while open to feedback. I will likely be posting about this algorithm more as it continues developing and I learn more. ### Motivation -The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. +The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. With this comes some complexity in the innnworkings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python shell. The python shell will be provided below. The circuit's gates will be analyzed but there is no code because it is still buggy :laughing: . ### The Algorithm @@ -38,7 +38,7 @@ Durr and Hoyer propose a computationally effecient routine to find the minimum o N = len(t) y = randomuniform[1..N] #N is a integer time_limit = (22.5*sqrt(N)=1.4*log(N)^2) - qq = preprocess.simulate(t,y') + q = preprocess.simulate(t,y) while time.clock() < time_limit: for i in range(N): @@ -63,11 +63,11 @@ It is easy to undestand the conceptual implementation in python. It is important ------------------ #### (a) -Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. +Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. -We do know that the register must be initiated as j divided by the square root of N (length of flattened table) along with a qubit for each integer and a redudant qubit, specifically being the randomly chosen y. +We do know that the register must be initiated as i divided by the square root of N (length of flattened table) along with a qubit for each integer and a redudant qubit, specifically being the randomly chosen y. This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. @@ -82,7 +82,7 @@ It is intuituve to see that in the case for finding a minimum of an unsorted tab ![Implementation](/assets/images//DurrHoyer-Implementation.JPG "Implementation") -A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: +A thorough read of this will supply any calirfication to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing affecionados I have reached the following conclusions: 1. T is just a Hadamard Gate 2. S_A is just a CNOT Gate @@ -100,7 +100,7 @@ So far I have been greatly enjoying throwing myself into this new and exciting w ## My current challenges in this project: -1. Creating a C# test script/any test script for my circuit +1. Test script for my circuits 2. Validating generalizations I made of 'Implementation Considerations' From 927451b01fcd1227a60bd89dbbc22d79cb8604e9 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 14:38:52 -0700 Subject: [PATCH 011/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index fcb1bbc..24f2a01 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -67,7 +67,9 @@ Part a is probably the most interesting part of this algorithm from my perspecti For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. -We do know that the register must be initiated as i divided by the square root of N (length of flattened table) along with a qubit for each integer and a redudant qubit, specifically being the randomly chosen y. +The register can be intiated as follows: + +[latex_imp](/assets/images//latex_information.jpg)('latex information') This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. 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z)%OJ-M!kh?{^jPoe8J`B_t^+%vJt>HbLH80UHBY2>AB=n+jkLOhD2sV(t%*xRx}Sa zxCY2px^^#YoA4t}$rIUB-&}`8!%X%ZrEpA>N*V2}O2dJK&bDY1{6Ys`NeGFEuaBr&o^({4|J{Z&K nasDW)?>wtv)QM$d$JFJ*Q#r%B6ZzlK0sb3G!T+nl@_YP0p1A*9 literal 0 HcmV?d00001 From 4be8b095a570401a9613b06314c213788776bc20 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 15:04:46 -0700 Subject: [PATCH 013/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 24f2a01..55e0fa0 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -69,7 +69,7 @@ For now, the more prevelant question is what sort of intializiation we can use t The register can be intiated as follows: -[latex_imp](/assets/images//latex_information.jpg)('latex information') +[latex_imp](/assets/images//latex_information.JPG)('latex information') This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. From 0053d22798d3f5a7c8c9860a650e57c36bd603be Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 15:05:28 -0700 Subject: [PATCH 014/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 55e0fa0..2e1c16c 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -69,7 +69,7 @@ For now, the more prevelant question is what sort of intializiation we can use t The register can be intiated as follows: -[latex_imp](/assets/images//latex_information.JPG)('latex information') +![latex_imp](/assets/images//latex_information.JPG)('latex information') This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. 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z6Nr~=fmT*x8+Ot1!si5nL0kioT^s1ye302h_RHniy2)&ykRxRT0zU#r;(u#0`@y<( z%(jwRhO}!*-gH=*m3J_3e(M6i>m2-~a(T@FW<~K5${Wk8eM&2sWlN=nA$Srq8EBz( zMq?>fbLXiT!8sr)DoHf2i1v4yeZ04JH!)n#=SVH03nMI1&E;=6ZhDRfJTU_UHK}Rf zy&ByWt{{8cqaT+&-NcO2Tz~wLs~HY(*%3#a@$!#Ko-YaLP&Vi^$RcdX$^xW^AiDAV zB&y|la5&mGNN0kCjXj!5ZQ&+#`zfoE&DZ*7v? ze(|0qIBy|m3G%Va7}-eelZ@wz=%Ued=yYe&W42YXkzF1PoDk%e9e;DqPv=yd!P*1) zf=x0h1iAAS*_b|J5S_rBAKieQ_86cH`=5w1aEf-8hCtvSOp4CA@Scfy%-2>cD>O|LGswiU7FHmf3=Ut@~0q0iU8P!|H@r#Sgvu);0HPk_-Fr;2$y}Cysz$6{Pfc_J= z?tz`{Q$y7z@&28!*sY_(XQtfRnC6+0ndEqxPY6Z_mUG{pmDABc9%pUv>t52K)9tj~ zQaNqlO*~AmbAK#Av^Nq-IaN@@4;+FpIN?Awe?jm*tEb<1etkyywC<`rN&_27YCl|=w?n&*V| zJx}3nkB1<8Rlb32M2zydkM|P9?}iJWe6j}vsOedM@Phb?Eo)o4x4gDnZARvKqkvva ztQoDMia8m=XL2-ba5MMv4pdj8(Lf$!XW=b&`s(gTHCx4s3#c7F)#kuvY{o8^<_Gs@ z%E}N3{VMB@REpftd^4 G0{_{sH{4eM From a42263bc168512ea336a7ea8a0b9ce82f4d2e2fe Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 15:13:19 -0700 Subject: [PATCH 016/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 2e1c16c..f84e46d 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -87,7 +87,7 @@ It is intuituve to see that in the case for finding a minimum of an unsorted tab A thorough read of this will supply any calirfication to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing affecionados I have reached the following conclusions: 1. T is just a Hadamard Gate -2. S_A is just a CNOT Gate +2. S_A is a variation of the CNOT Gate 3. S_0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! From 82761c4e94d129cf1b4bef427488d658b71ff682 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 15:19:29 -0700 Subject: [PATCH 017/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index f84e46d..4e35585 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -102,10 +102,12 @@ So far I have been greatly enjoying throwing myself into this new and exciting w ## My current challenges in this project: -1. Test script for my circuits +1. Getting detailed error codes from Q# 2. Validating generalizations I made of 'Implementation Considerations' +3. Developing a cool quantum information algorithm to map integers to qubits. + ------------------ [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf From e4524c1bf03f7a1290ee36c2f7ddfec7386a2743 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 16:03:54 -0700 Subject: [PATCH 018/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 4e35585..adc6a92 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -97,7 +97,7 @@ Generalizations are a mathematicians greatest friend, but a downfall if not impl ### Conclusion So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. - +https://github.com/mridulsar/DurrHoyerLibrary ------------------ ## My current challenges in this project: From b800777e02b508514bf88716881419ea7af0bd12 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 16:04:56 -0700 Subject: [PATCH 019/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index adc6a92..94f7975 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -96,7 +96,7 @@ Generalizations are a mathematicians greatest friend, but a downfall if not impl ### Conclusion -So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. +So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. https://github.com/mridulsar/DurrHoyerLibrary ------------------ From df30119b5c0c85d8a8e9c28017d8e889d238fc43 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 19:55:20 -0700 Subject: [PATCH 020/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 94f7975..11cb7da 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -71,6 +71,8 @@ The register can be intiated as follows: ![latex_imp](/assets/images//latex_information.JPG)('latex information') +Obviously this is unachievable on current quantum comptuer architecutre. We onlyhave a qubit, the above example implies we have harness the tritbit and so on till n bit. In reality we will register all qubits as zero, engtangling them with another qubit registered as 1 representative of y. + This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. In order to mark the indices we simply mark all indices that come before i. From ba1e594a6651278fc1879df1d2d7459b52baebf5 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 22:04:34 -0700 Subject: [PATCH 021/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 11cb7da..1f8865b 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -71,8 +71,8 @@ The register can be intiated as follows: ![latex_imp](/assets/images//latex_information.JPG)('latex information') -Obviously this is unachievable on current quantum comptuer architecutre. We onlyhave a qubit, the above example implies we have harness the tritbit and so on till n bit. In reality we will register all qubits as zero, engtangling them with another qubit registered as 1 representative of y. - +Obviously this is unachievable on current quantum comptuer architecutre. We onlyhave a qubit, the above example implies we have harness the tritbit and so on till n bit. In reality we will register all qubits as unique binary strings which will equate to their uique basis states, qubits. + This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. In order to mark the indices we simply mark all indices that come before i. @@ -104,11 +104,9 @@ https://github.com/mridulsar/DurrHoyerLibrary ## My current challenges in this project: -1. Getting detailed error codes from Q# - -2. Validating generalizations I made of 'Implementation Considerations' +1. Validating generalizations I made of 'Implementation Considerations' -3. Developing a cool quantum information algorithm to map integers to qubits. +2. Developing a cool quantum information algorithm to map integers to qubits. ------------------ From 2781f88e68fb55af2ca3af84d54c61ee4cc10815 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Fri, 19 Jun 2020 23:56:47 -0700 Subject: [PATCH 022/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 35 ++++++++++++++++++++++++++++++++--- 1 file changed, 32 insertions(+), 3 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 1f8865b..4e960a3 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -1,5 +1,5 @@ --- -title: "In Progress Durr Hoyer Algorithm" +title: "Durr Hoyer Algorithm (In Progress)" date: categories: - Blog @@ -69,11 +69,25 @@ For now, the more prevelant question is what sort of intializiation we can use t The register can be intiated as follows: -![latex_imp](/assets/images//latex_information.JPG)('latex information') +![latex_imp](/assets/images//latex_information.JPG) Obviously this is unachievable on current quantum comptuer architecutre. We onlyhave a qubit, the above example implies we have harness the tritbit and so on till n bit. In reality we will register all qubits as unique binary strings which will equate to their uique basis states, qubits. -This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. +Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: + +''' +namespace preprocess { + + operation proper_information(N : Int, y : Int) : Qubit[] + { + using (aux = Qubit[N]) + { + PrepareUniformSuperposition(N,aux); + ApplytoAll(i*aux[y],aux); + } + } +} +''' In order to mark the indices we simply mark all indices that come before i. @@ -92,6 +106,21 @@ A thorough read of this will supply any calirfication to the python code posted 2. S_A is a variation of the CNOT Gate 3. S_0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back +Here is a small snippet of my current circuit: + +''' + operation Algorithm_Even(q_bitz : Qubit[] ) : Unit + { + H(q_bitz); + CNOT(q_bitz); // Place holder for ApplyConditionalPhase() + H(q_bitz); + ApplyConditionalPhase_0(q_bitz); + + DumpMachine(q_bitz); + + } +''' + Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! ------------------ From 06a8babf4fab8af3140d73f2dbc616b3c214c7df Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:03:19 -0700 Subject: [PATCH 023/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 27 ++------------------------- 1 file changed, 2 insertions(+), 25 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 4e960a3..8b4106a 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -75,19 +75,7 @@ Obviously this is unachievable on current quantum comptuer architecutre. We only Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: -''' -namespace preprocess { - - operation proper_information(N : Int, y : Int) : Qubit[] - { - using (aux = Qubit[N]) - { - PrepareUniformSuperposition(N,aux); - ApplytoAll(i*aux[y],aux); - } - } -} -''' +![intialize](/assests/images//intalize.jpg) In order to mark the indices we simply mark all indices that come before i. @@ -108,18 +96,7 @@ A thorough read of this will supply any calirfication to the python code posted Here is a small snippet of my current circuit: -''' - operation Algorithm_Even(q_bitz : Qubit[] ) : Unit - { - H(q_bitz); - CNOT(q_bitz); // Place holder for ApplyConditionalPhase() - H(q_bitz); - ApplyConditionalPhase_0(q_bitz); - - DumpMachine(q_bitz); - - } -''' +![circuit](/assets/pages//Algorithm_Even.JPG) Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! 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z2O@Yw0ZlY@#jgs!ZYRZj05I!)q;wpkI@C1J1m$^JB&^2#tZJxW3|Wo!PK^u&7{G5= zE=QVL8hu6NnUUO~=TRHdXLpA;Tg;!)<`?EAGAL)RyN`3T{YbriJ?>oW?e`xK#{Mq$ z9}S6|6hNriSfhK+#QC)QLft0kfbnKBlcUN{?%A~6LT_CF_h8i%RAMvoS>dff9rwgo zoq0Jh-|KzyHY*EVm0NN-j01k8f9b9J-;t*@Bpxr}2iu7&-OTa<*JYms2pF?1F7lC3 aN;~%8oRk3nZ5H*vH!%BuN_o8d>Hh|NywxcH literal 0 HcmV?d00001 From bedcfa5dcaf186a301db402f15563a66e3648a88 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:04:48 -0700 Subject: [PATCH 025/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 8b4106a..215db5e 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -75,7 +75,7 @@ Obviously this is unachievable on current quantum comptuer architecutre. We only Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: -![intialize](/assests/images//intalize.jpg) +![intialize](/assests/images//intialize.jpg) In order to mark the indices we simply mark all indices that come before i. @@ -96,7 +96,7 @@ A thorough read of this will supply any calirfication to the python code posted Here is a small snippet of my current circuit: -![circuit](/assets/pages//Algorithm_Even.JPG) +![circuit](/assets/images//Algorithm_Even.JPG) Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! From 07b8633cddbafeca72ea0b4e753eb9eb51d95752 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:06:40 -0700 Subject: [PATCH 026/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 215db5e..694cd02 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -75,7 +75,7 @@ Obviously this is unachievable on current quantum comptuer architecutre. We only Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: -![intialize](/assests/images//intialize.jpg) +![intialize](/assests/images//intialize.JPG) In order to mark the indices we simply mark all indices that come before i. @@ -91,8 +91,8 @@ It is intuituve to see that in the case for finding a minimum of an unsorted tab A thorough read of this will supply any calirfication to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing affecionados I have reached the following conclusions: 1. T is just a Hadamard Gate -2. S_A is a variation of the CNOT Gate -3. S_0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back +2. Conditional Phase Shift on A is a variation of the CNOT Gate +3. Conditional Phase Shift on 0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back Here is a small snippet of my current circuit: From 207daf87c7ef8f9fc7d9b7d043d30a3d4f425fd8 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:07:06 -0700 Subject: [PATCH 027/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 694cd02..0e562a9 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -75,7 +75,7 @@ Obviously this is unachievable on current quantum comptuer architecutre. We only Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: -![intialize](/assests/images//intialize.JPG) +![intialize](/assets/images//intialize.JPG) In order to mark the indices we simply mark all indices that come before i. From 174b9f4996e26889162252feb73cb50d3ccbd40c Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:18:53 -0700 Subject: [PATCH 028/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 32 +++++++++++++++----------------- 1 file changed, 15 insertions(+), 17 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 0e562a9..3f4aa78 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -10,22 +10,22 @@ tags: --- ### Durr Hoyer -Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. +Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. ### Background -Recently I began working on quantum computing with an idea to treat nodes of neural networks as qubits. From here I took a deep dive into the research papers and found myself amazed by the research being produced. I began with a Coursera course from St. Petersburge University on quantum computing with detail on Duestchs, Shors, and Grovers algorithms. From here I tinkered with Q# and developed basic algorithms that I learnt from Coursera. I found Durr Hoyer and saw it as an ensemble of Duestchs, Shors, and Grovers Algorithm; a perfect stepping stone coming from my background. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time while open to feedback. I will likely be posting about this algorithm more as it continues developing and I learn more. +Recently I began working on quantum computing with an idea to treat nodes of neural networks as qubits. From here I took a deep dive into the research papers and found myself amazed by the research being produced. I began with a Coursera course from St. Petersburg University on quantum computing with detail on Deutschs, Shors, and Grovers algorithms. From here I tinkered with Q# and developed basic algorithms that I learnt from Coursera. I found Durr Hoyer and saw it as an ensemble of Duestchs, Shors, and Grovers Algorithm; a perfect stepping stone coming from my background. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time while open to feedback. I will likely be posting about this algorithm more as it continues developing and I learn more. ### Motivation -The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. With this comes some complexity in the innnworkings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python shell. The python shell will be provided below. The circuit's gates will be analyzed but there is no code because it is still buggy :laughing: . +The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer propose an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python shell. The python shell will be provided below. The circuit's gates will be analyzed and some code is given. ### The Algorithm -A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: +A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: -Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: +Durr and Hoyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: ![QMSA](/assets/images//DurrHoyer-QMSA.JPG "QMSA") @@ -57,21 +57,21 @@ Durr and Hoyer propose a computationally effecient routine to find the minimum o ``` ------------------ -It is easy to undestand the conceptual implementation in python. It is important to note steps: +It is easy to understand the conceptual implementation in python. It is important to note steps: ### (a,b) ------------------ #### (a) -Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. +Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an interesting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. -For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. +For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. My most immediate thought is to entangle all the qubits and superimpose them The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. -The register can be intiated as follows: +The register can be initiated as follows: ![latex_imp](/assets/images//latex_information.JPG) -Obviously this is unachievable on current quantum comptuer architecutre. We onlyhave a qubit, the above example implies we have harness the tritbit and so on till n bit. In reality we will register all qubits as unique binary strings which will equate to their uique basis states, qubits. +Obviously this is unachievable on current quantum computer architecture. We only have a qubit, the above example implies we have harnessed the tritbit and so on till n bit. In reality we will register all qubits as unique binary strings which will equate to their qubits. Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: @@ -82,13 +82,13 @@ In order to mark the indices we simply mark all indices that come before i. ------------------ #### (b) -Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different prepration of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational effeciency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] +Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different preparation of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational efficiency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] -It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". +It is intuitive to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". ![Implementation](/assets/images//DurrHoyer-Implementation.JPG "Implementation") -A thorough read of this will supply any calirfication to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing affecionados I have reached the following conclusions: +A thorough read of this will supply any clarification to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing aficionados I have reached the following conclusions: 1. T is just a Hadamard Gate 2. Conditional Phase Shift on A is a variation of the CNOT Gate @@ -112,12 +112,10 @@ https://github.com/mridulsar/DurrHoyerLibrary 1. Validating generalizations I made of 'Implementation Considerations' -2. Developing a cool quantum information algorithm to map integers to qubits. +2. Developing alleviation for Big-O in a fluid manner. ------------------ [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf [(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf -[3]: Essentialy, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algoirthm will run for the alloted while loop in the python code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. - - +[3]: Essentially, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algorithm will run for the allotted while loop in the python code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. 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tbTQs1uZJc+EP(Gq06h$UHX*qYe^wqbL{AGy)x(e7ptk$Xk;2L6{{oxy5}W`4 literal 0 HcmV?d00001 From 8dfbd37cf31e9bccd1e3e5de91789a72c9e9fcd1 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:23:01 -0700 Subject: [PATCH 030/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 3f4aa78..f2ddb54 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -63,6 +63,8 @@ It is easy to understand the conceptual implementation in python. It is importan ------------------ #### (a) +![a](/assets/images//a.JPG) + Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an interesting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. My most immediate thought is to entangle all the qubits and superimpose them The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. @@ -82,6 +84,8 @@ In order to mark the indices we simply mark all indices that come before i. ------------------ #### (b) +![b](/assets/images//b.JPG) + Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different preparation of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational efficiency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] It is intuitive to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". From c28e2404e1fd5c4ed40848803b33fba482e63282 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sat, 20 Jun 2020 00:26:41 -0700 Subject: [PATCH 031/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index f2ddb54..c567961 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -62,7 +62,6 @@ It is easy to understand the conceptual implementation in python. It is importan ------------------ -#### (a) ![a](/assets/images//a.JPG) Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an interesting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. @@ -83,7 +82,6 @@ In order to mark the indices we simply mark all indices that come before i. ------------------ -#### (b) ![b](/assets/images//b.JPG) Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different preparation of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational efficiency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] From 468ba87bc97bf505f18db001d46a18f41983228e Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Sun, 21 Jun 2020 22:46:25 -0700 Subject: [PATCH 032/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 16 +++++++++------- 1 file changed, 9 insertions(+), 7 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index c567961..ef10a92 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -1,5 +1,5 @@ --- -title: "Durr Hoyer Algorithm (In Progress)" +title: "A neat quantum algorithm for finding the minimum of an unsorted table" date: categories: - Blog @@ -15,7 +15,7 @@ Hello all! My name is Mridul Sarkar. I am an undergraduate at University of Cali ### Background -Recently I began working on quantum computing with an idea to treat nodes of neural networks as qubits. From here I took a deep dive into the research papers and found myself amazed by the research being produced. I began with a Coursera course from St. Petersburg University on quantum computing with detail on Deutschs, Shors, and Grovers algorithms. From here I tinkered with Q# and developed basic algorithms that I learnt from Coursera. I found Durr Hoyer and saw it as an ensemble of Duestchs, Shors, and Grovers Algorithm; a perfect stepping stone coming from my background. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time while open to feedback. I will likely be posting about this algorithm more as it continues developing and I learn more. +Recently I began working on quantum computing with an idea to treat nodes of neural networks as qubits. From here I took a deep dive into the research papers and found myself amazed by the research being produced. I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. From here I tinkered with Q# and developed basic algorithms that I learnt. I found Durr and Hoyer's paper and saw it as an ensemble of Duestchs, Shors, and Grovers Algorithm; a perfect stepping stone coming from my background. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time while open to feedback. I will likely be posting about this algorithm more as it continues developing and I learn more. ### Motivation @@ -106,15 +106,17 @@ Generalizations are a mathematicians greatest friend, but a downfall if not impl ### Conclusion -So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. -https://github.com/mridulsar/DurrHoyerLibrary +So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. In specific, I mentioned I felt Durr and Hoyer used principles from Duestch's, Grover's, and Shor's algorithm. Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function. Grover's Algorithm is quite literally applied in this algorithm. Lastly, Shor's Algorithm shows off the power of combining classical and quantum systems to achieve outstanding results. I love seeing such fundamental concepts continue to push boundaries. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at +msarkar at ucdavis dot edu. + +https://github.com/mridulsar/DurrHoyerLibrary ------------------ -## My current challenges in this project: +## My current challenges in this project: -1. Validating generalizations I made of 'Implementation Considerations' +1. Validating generalizations I made of 'Implementation Considerations' -2. Developing alleviation for Big-O in a fluid manner. +2. Developing alleviation for Big-O in a fluid manner. ------------------ From e83d7bfbaaa8b559230cbd7992b98d88b78cf146 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Mon, 22 Jun 2020 10:54:24 -0700 Subject: [PATCH 033/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index ef10a92..e07e5fe 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -1,5 +1,5 @@ --- -title: "A neat quantum algorithm for finding the minimum of an unsorted table" +title: "A quantum algorithm for finding the minimum of an unsorted table" date: categories: - Blog @@ -72,7 +72,7 @@ The register can be initiated as follows: ![latex_imp](/assets/images//latex_information.JPG) -Obviously this is unachievable on current quantum computer architecture. We only have a qubit, the above example implies we have harnessed the tritbit and so on till n bit. In reality we will register all qubits as unique binary strings which will equate to their qubits. +Obviously this is unachievable on current quantum computer architecture. We only have a qubit, the above example implies we have harnessed the tritbit and so on till n bit. In reality we will register all integers as unique binary strings which will equate to their qubits. Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: From 0771a9d5733034ed5ba78847130dedda00ae7a73 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Mon, 22 Jun 2020 11:48:22 -0700 Subject: [PATCH 034/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 17 ++++++++--------- 1 file changed, 8 insertions(+), 9 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index e07e5fe..962ed5f 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -25,7 +25,7 @@ The motivation behind this project is to provide open source functionality. The ### The Algorithm A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: -Durr and Hoyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: +Durr and Hoyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follows: ![QMSA](/assets/images//DurrHoyer-QMSA.JPG "QMSA") @@ -43,7 +43,7 @@ Durr and Hoyer propose a computationally efficient routine to find the minimum o while time.clock() < time_limit: for i in range(N): while t[i] < t[y]: - j = N-j + i = N-i if N is even q=Algorithm with Hadamard gates[0...j] y'=q[0] @@ -64,9 +64,9 @@ It is easy to understand the conceptual implementation in python. It is importan ![a](/assets/images//a.JPG) -Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an interesting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. +Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an interesting question. -For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. My most immediate thought is to entangle all the qubits and superimpose them The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. +For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. My most immediate thought is to entangle all the qubits and superimpose them. The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. The register can be initiated as follows: @@ -78,19 +78,19 @@ Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have impl ![intialize](/assets/images//intialize.JPG) -In order to mark the indices we simply mark all indices that come before i. +In order to mark items we simply mark all indices that come before i. ------------------ ![b](/assets/images//b.JPG) -Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different preparation of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational efficiency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] +Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different preparation of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push computational efficiency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] It is intuitive to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". ![Implementation](/assets/images//DurrHoyer-Implementation.JPG "Implementation") -A thorough read of this will supply any clarification to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing aficionados I have reached the following conclusions: +A thorough read of this will supply any clarification to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, and the lenght of the table. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing aficionados I have reached the following conclusions: 1. T is just a Hadamard Gate 2. Conditional Phase Shift on A is a variation of the CNOT Gate @@ -106,8 +106,7 @@ Generalizations are a mathematicians greatest friend, but a downfall if not impl ### Conclusion -So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. In specific, I mentioned I felt Durr and Hoyer used principles from Duestch's, Grover's, and Shor's algorithm. Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function. Grover's Algorithm is quite literally applied in this algorithm. Lastly, Shor's Algorithm shows off the power of combining classical and quantum systems to achieve outstanding results. I love seeing such fundamental concepts continue to push boundaries. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at -msarkar at ucdavis dot edu. +So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. In specific, I mentioned I felt Durr and Hoyer used principles from Duestch's, Grover's, and Shor's algorithm. Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function as 'G' does. Grover's Algorithm is quite literally applied in this algorithm. Lastly, Shor's Algorithm shows off the power of combining classical and quantum systems to achieve outstanding results. I love seeing such fundamental concepts continue to push boundaries. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email. https://github.com/mridulsar/DurrHoyerLibrary ------------------ From c0a897cbf1aab6b43a6a3e23bb3c8382cda26b87 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Mon, 22 Jun 2020 11:53:45 -0700 Subject: [PATCH 035/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 962ed5f..dd7f1f8 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -78,6 +78,8 @@ Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have impl ![intialize](/assets/images//intialize.JPG) +Note: make sure to use LittleEndian() on the register of qubits. I didn't include this so it looked neat. + In order to mark items we simply mark all indices that come before i. ------------------ From 72be26e8302da90d2d91904b795ed8419089effa Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Mon, 22 Jun 2020 12:13:40 -0700 Subject: [PATCH 036/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index dd7f1f8..69d0bdc 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -78,7 +78,7 @@ Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have impl ![intialize](/assets/images//intialize.JPG) -Note: make sure to use LittleEndian() on the register of qubits. I didn't include this so it looked neat. +Note: make sure to use LittleEndian() on the register of qubits. From here you will have to unpack the qubits with let qubits = qubits! in order to set the right data type for the rest of our algorithm. I didn't include this so it looked neat. In order to mark items we simply mark all indices that come before i. From dfbdc3e452e29fc412e41866aff7213d44855f2c Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Mon, 22 Jun 2020 12:14:41 -0700 Subject: [PATCH 037/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 69d0bdc..f8da2dc 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -78,7 +78,7 @@ Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have impl ![intialize](/assets/images//intialize.JPG) -Note: make sure to use LittleEndian() on the register of qubits. From here you will have to unpack the qubits with let qubits = qubits! in order to set the right data type for the rest of our algorithm. I didn't include this so it looked neat. +Note: make sure to use LittleEndian() on the register of qubits. From here you will have to unpack the qubits with let qubits = qubits! in order to set the right data type after using PrepareUniformSuperposition() as it spits out LE datatype. I didn't include this so it looked neat. In order to mark items we simply mark all indices that come before i. From d6bff17e010f480409c8d3a3aed9964f9a6fc5a4 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Mon, 22 Jun 2020 12:41:39 -0700 Subject: [PATCH 038/152] Update Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index f8da2dc..fd9e402 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -78,7 +78,7 @@ Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have impl ![intialize](/assets/images//intialize.JPG) -Note: make sure to use LittleEndian() on the register of qubits. From here you will have to unpack the qubits with let qubits = qubits! in order to set the right data type after using PrepareUniformSuperposition() as it spits out LE datatype. I didn't include this so it looked neat. +Note: make sure to use LittleEndian() on the register of qubits before applying PrepareUniformSuperposition(). From here you will have to use LittleEndianAsBigEndian() to set the right data type after using PrepareUniformSuperposition() as it spits out LE datatype. I didn't include this so it looked neat. In order to mark items we simply mark all indices that come before i. From 61dbf09c21908818cd18ea3d3ed324e389390cf6 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:03:39 -0700 Subject: [PATCH 039/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 21 +++++++++++++++------ 1 file changed, 15 insertions(+), 6 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index fd9e402..c2c2d66 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -10,12 +10,21 @@ tags: --- ### Durr Hoyer -Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. - - - -### Background -Recently I began working on quantum computing with an idea to treat nodes of neural networks as qubits. From here I took a deep dive into the research papers and found myself amazed by the research being produced. I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. From here I tinkered with Q# and developed basic algorithms that I learnt. I found Durr and Hoyer's paper and saw it as an ensemble of Duestchs, Shors, and Grovers Algorithm; a perfect stepping stone coming from my background. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time while open to feedback. I will likely be posting about this algorithm more as it continues developing and I learn more. +Hello all! +My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. +I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. +This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum mechanics has been a rewarding experience. +I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. +I will likely be posting about this algorithm and library more as the library and my understanding continues developing. + +### My Background +I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. +I was instantly pulled in by the elegant algorithms and proposed efficiency. +Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. +I took a deep dive into the research papers and found myself amazed by what was out there. +I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. +From here I tinkered with Q# and developed basic algorithms that I previously learned. +I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. ### Motivation From 55e7bd465bd879e605c052b36f434b0a3dd5a7c9 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:04:14 -0700 Subject: [PATCH 040/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index c2c2d66..47db87a 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -66,8 +66,7 @@ Durr and Hoyer propose a computationally efficient routine to find the minimum o ``` ------------------ -It is easy to understand the conceptual implementation in python. It is important to note steps: -### (a,b) +It is easy to understand the conceptual implementation in python. It is important to note steps a and b: ------------------ From 54ed8dac0537603ff80d7673fac7708c0a50c451 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:04:33 -0700 Subject: [PATCH 041/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 47db87a..121f8ab 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -8,7 +8,7 @@ tags: - algorithm - open source --- -### Durr Hoyer +## Introduction Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. From 773c4fbe55adbdeb2ad1c69172323e4131f0ffb4 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:04:46 -0700 Subject: [PATCH 042/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 121f8ab..8d67bb7 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -4,9 +4,9 @@ date: categories: - Blog tags: - - qsharp + - library - algorithm - - open source + - in-development --- ## Introduction From 173c90da639825fad0cc41d94163dfd09c78684a Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:04:56 -0700 Subject: [PATCH 043/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 8d67bb7..e081728 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -1,6 +1,6 @@ --- title: "A quantum algorithm for finding the minimum of an unsorted table" -date: +date: 2020-06-22 categories: - Blog tags: From ab8a410a267cb30afda1444152c2f9dc29bb447d Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:05:04 -0700 Subject: [PATCH 044/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index e081728..8554c76 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -1,5 +1,5 @@ --- -title: "A quantum algorithm for finding the minimum of an unsorted table" +title: "Building a Q# library to find table minima: Part 1" date: 2020-06-22 categories: - Blog From 6de3c1120b26ba5b0fcae4add63ea7b4b0812af9 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:05:20 -0700 Subject: [PATCH 045/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 8554c76..19e97d2 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -31,7 +31,7 @@ I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, a The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer propose an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python shell. The python shell will be provided below. The circuit's gates will be analyzed and some code is given. -### The Algorithm +## The Dürr–Høyer Algorithm A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: Durr and Hoyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follows: From 2c033c38a2ffefa57d9d5d2b48fb2081b0ace793 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:05:35 -0700 Subject: [PATCH 046/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 1 - 1 file changed, 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 19e97d2..4abfc1a 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -68,7 +68,6 @@ Durr and Hoyer propose a computationally efficient routine to find the minimum o It is easy to understand the conceptual implementation in python. It is important to note steps a and b: ------------------- ![a](/assets/images//a.JPG) From 678204d892f4cd6712fe9bbfd2192896f3558962 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 04:05:49 -0700 Subject: [PATCH 047/152] Update _posts/Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kasier --- _posts/Durr_Hoyer_Post_1.md | 1 - 1 file changed, 1 deletion(-) diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md index 4abfc1a..5b7879f 100644 --- a/_posts/Durr_Hoyer_Post_1.md +++ b/_posts/Durr_Hoyer_Post_1.md @@ -64,7 +64,6 @@ Durr and Hoyer propose a computationally efficient routine to find the minimum o return return y ``` ------------------- It is easy to understand the conceptual implementation in python. It is important to note steps a and b: From 8c407142dfb4769e3873e94049330a4507d01194 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 14:29:51 -0700 Subject: [PATCH 048/152] Add files via upload --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 116 +++++++++++++++++++++++++ 1 file changed, 116 insertions(+) create mode 100644 _posts/2020-06-23-Durr_Hoyer_Post_1.md diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md new file mode 100644 index 0000000..7ddc758 --- /dev/null +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -0,0 +1,116 @@ +--- +title: "In Progress Durr Hoyer Algorithm" +author "mridulsar" +categories: + - Blog +tags: + - qsharp + - algorithm + - open source +--- +# Durr Hoyer + +Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. +------------------ + + +## Background +Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. +------------------ + +## Motivation +The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. +------------------ + +## The Algorithm +A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: + +Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: + +![alt text](QMSA.JPG) +------------------ + +### Psuedocode + +``` + t = table.flatten() + N = len(t) + y = randomuniform[1..N] #N is a integer + time_limit = (22.5*sqrt(N)=1.4*log(N)^2) + qq = preprocess.simulate(t,y') + + while time.clock() < time_limit: + for i in range(N): + while t[i] < t[y]: + j = N-i + if N is even + Grovers Algorithm for one step across [0...j] + q=Algorithm with T[0...j] defined operator + y'=q[0] + else + Grovers Algorithm for one step across [0...j] + t=Algorithm with QFT instead of T[0...j] + y'=q[0] + if t[y'] < t[y]: + y = y' + return + return y +``` +------------------ + +It is easy to undestand the conceptual implementation in python. It is important to note steps: +## (a..b) + +------------------ + +### (a) +Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. + +For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. + +We do know that the register must be initiated as j divided by the square root of N (length of flattened table) along with a qubit for each integer and a redudant qubit, specifically being the randomly chosen y. + +This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. + +In order to mark the indeces as the algorithm states we must run the intialized qubits through grovers algorithm once. + +------------------ + +### (b) +Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different prepration of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational effeciency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] + +It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". + +![alt text](implementation.JPG) + +A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: + +1. T is just a Hadamard Gate +2. S_A is just a CNOT Gate +3. S_0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back + +Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! + +------------------ + +# Conclusion + +So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. + +------------------ + +### My current challenges in this project: + +1. Creating a C# test script/any test script for my circuit + +2. Validating generalizations I made of 'Implementation Considerations' + +3. Containerizing this project + +------------------ + +[(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf +[(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf +[3]: Essentialy, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algoirthm will run for the alloted while loop in the python code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. + + From 6df8d0ea3847fe36ad6c7a5c97f59e95990b404e Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 15:14:00 -0700 Subject: [PATCH 049/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 219 ++++++++++++++++--------- 1 file changed, 141 insertions(+), 78 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 7ddc758..0f82068 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -1,116 +1,179 @@ --- -title: "In Progress Durr Hoyer Algorithm" -author "mridulsar" +title: "Building a Q# library to find table minima: Part 1" +date: 2020-06-22 categories: - Blog tags: - - qsharp + - library - algorithm - - open source + - in-development --- -# Durr Hoyer +## Introduction + +Hello all! +My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. +I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. +This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum mechanics has been a rewarding experience. +I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. +I will likely be posting about this algorithm and library more as the library and my understanding continues developing. +https://github.com/mridulsar/DurrHoyerLibrary + +### My Background +I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. +I was instantly pulled in by the elegant algorithms and proposed efficiency. +Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. +I took a deep dive into the research papers and found myself amazed by what was out there. +I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. +From here I tinkered with Q# and developed basic algorithms that I previously learned. +I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. + + +### Motivation +The motivation behind this project is to provide open source functionality. +The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. +On a classical computer it will take as many time steps as there are items to find a minimum. +This means the Big-O is N. +The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. +For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. +The Dürr-Høyer algorithm would have find the minimum in 3 steps. +We can understand this effecieny as the Big-O. +The Big-O essentially means the run time of the program will increase as the table size grows at a rate of square root of table size. +Dürr and Høyer propose an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. +This success rate is bounded by the Big-O. +If we decide to alleviate this Big-O the probability of finding the minimum increases. +Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. +With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. +In order for this library to be used properly it must meet some guidelines. +At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. +I am programming the algorithm in Q# with a python host script. +The python host script will be provided below. The circuit's gates will be analyzed and some code is given. +https://github.com/mridulsar/DurrHoyerLibrary + + +## The Dürr–Høyer Algorithm +A brief introduction to the Dürr–Høyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: + +Dürr and Høyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follows: + + +1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length +2. Repeat the following steps until T = 22.5 * sqrt(N) + 1.4 * log^2(N). Once T equals the expression with N proceed to step c +2(a) Intialize the memory as a uniform superposition of qubits. Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. +2(b) Apply the quantum exponential searching algorithm [(2)]. +2(c) Measure the first qubit, map this qubit back to the integer it represents. If this integer is less than the integer in the y-th position set y as the position of the integer represented by the measured qubit. + + +It is important to note steps a and b: + + +2(a) Intialize the memory as a uniform superposition of qubits. Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. + +Part a is probably the most interesting part of this algorithm from my perspective. +After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. +The creation of qubits from a table of unique integers is an interesting question. + +For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. +My most immediate thought is to entangle all the qubits and superimpose them. +The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. +If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. + +The register can be initiated as follows: + +![latex_imp](/assets/images//latex_information.JPG) + +Obviously this is unachievable on current quantum computer architecture. +We only have a qubit, the above example implies we have harnessed the tritbit and so on till n bit. +In reality we will register all integers as unique binary strings which will equate to their qubits. + +Luckily PrepareUniformSuperposition() in Q# does this for us easily. +I have implemented it as follows: -Hello all! My name is Mridul Sarkar. I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. I began my journey from math to computers through my passion for ML algorithms. Eventually I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed effeciency. ------------------- +--- + operation CreateQuantumInformation( TableLength : Int, RandomIndex : Int ) : Qubit[] + { + using (register = Qubit[TableLength]) + { + PrepareUniformSuperposition(N,LittleEndian(register)); + let register = LittleEndianasBigEndian(register); + ApplytoAll(i*register[RandomIndex], register!); + } + return aux; + } + +--- -## Background -Recently I began working on an open source Durr Hoyer library. This project has pushed me and is still pushing me outside of my comfort zone. Learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have learnt in that time while looking for feedback. I will likely be posting about this algorithm more as it continues developing. ------------------- +In order to mark items we simply mark all indices that come before i. -## Motivation -The motivation behind this project is to provide open source functionality. The effeciency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer proposes an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, which I will explain in detail later, the probability of finding the minimum increases. ------------------ -## The Algorithm -A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: +2(b) Apply the quantum exponential searching algorithm [(2)]. -Durr and Hoyer propose a computationally effecient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follow: +Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. +It details the use of the quantum exponential searching algorithm. +The Dürr–Høyer Algorithm is supposed to push computational efficiency versus computational accuracy. +The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O depends on the number of times the QESA is applied to our quantum information. +Essentially, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. The Dürr–Høyer algorithm answers to a 'sweet spot' of sorts. +The algorithm will run for the allotted time based T = 22.5 * sqrt(N) + 1.4 * log^2(N) to achieve a 50% success rate. +If we do some math and manipulate the expression with N we can achieve better accuracy with a longer run time. +This proof is done in detail in 'A quantum algorithm to find [(1)]. +An important feature of this open source library will be the user's manipulation of how many times step 2(b) is applied. -![alt text](QMSA.JPG) ------------------- -### Psuedocode - -``` - t = table.flatten() - N = len(t) - y = randomuniform[1..N] #N is a integer - time_limit = (22.5*sqrt(N)=1.4*log(N)^2) - qq = preprocess.simulate(t,y') - - while time.clock() < time_limit: - for i in range(N): - while t[i] < t[y]: - j = N-i - if N is even - Grovers Algorithm for one step across [0...j] - q=Algorithm with T[0...j] defined operator - y'=q[0] - else - Grovers Algorithm for one step across [0...j] - t=Algorithm with QFT instead of T[0...j] - y'=q[0] - if t[y'] < t[y]: - y = y' - return - return y -``` ------------------- +The details of step 2(b) are mentioned under "Implementation Considerations". -It is easy to undestand the conceptual implementation in python. It is important to note steps: -## (a..b) +![Implementation](/assets/images//DurrHoyer-Implementation.JPG "Implementation") ------------------- +It is important to note that if the table has odd elements a different form of the algorithm must be used. +Being new to quantum computing I have found myself wading through darkness at times. +Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. +After research and conversations with quantum computing aficionados I have reached the following conclusions: -### (a) -Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an intersting question. An even more interesting question is if we create a function to map values onto integers and must ponder how to create a state of qubits to capture this classical data. +1. T is just a Hadamard Gate +2. Conditional Phase Shift on A is a variation of the CNOT Gate +3. Conditional Phase Shift on 0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back -For now, the more prevelant question is what sort of intializiation we can use to verify the integrity and effeciency of this algorithm. My most immediate thought is entangle all the qubits and set them as unique basis. The classical value of the integer is not represented here, only the index of the integer is caputred at a quantum level on the table assuming an input dataytpe of BigEndian. If the goal is to test the algorithm's probablistic integrity this plan of action will suffice. +Here is a small snippet of my current circuit: -We do know that the register must be initiated as j divided by the square root of N (length of flattened table) along with a qubit for each integer and a redudant qubit, specifically being the randomly chosen y. +--- -This still needs to be properely implemented as I have been wrestling with which route to take. Currently the place holder for this eventual quantum information algorithm is #preprocess.simulate() Suggestions are welcome and appreciated. + operation ApplyDürrHøyer( register : BigEndian[] ) : Unit + { + ApplyToEach(H,register!); + ApplyConditionalPhase_A(register); + ApplyToEach(H,register!); + ApplyConditionalPhase_0(register); + } + +--- -In order to mark the indeces as the algorithm states we must run the intialized qubits through grovers algorithm once. +Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. +If anyone has any input for the generalizations I have made I will take them with open arms! ------------------ -### (b) -Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different prepration of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push the computational effeciency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] +### Conclusion -It is intuituve to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". +So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. +In specific, I mentioned I felt Dürr and Høyer used principles from Duestch's, Grover's, and Shor's algorithm. +Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function as 'G' does. Grover's Algorithm is quite literally applied in this algorithm. +Lastly, Shor's Algorithm shows off the power of combining classical and quantum systems to achieve outstanding results. +I love seeing such fundamental concepts continue to push boundaries. +I am constantly in search for collaboration among those who are equally as passionate about quantum computing. +If you had any suggestions or questions feel free to send me an email. -![alt text](implementation.JPG) - -A thorough read of this will supply any calirfication to the python code posted above. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation. After research and conversations with quantum computing affecionados I have reached the following conclusions: - -1. T is just a Hadamard Gate -2. S_A is just a CNOT Gate -3. S_0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back - -Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! - ------------------- - -# Conclusion - -So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email at msarkar at ucdavis dot edu. +https://github.com/mridulsar/DurrHoyerLibrary ------------------ -### My current challenges in this project: +## My current challenges in this project: -1. Creating a C# test script/any test script for my circuit +1. Validating generalizations I made of 'Implementation Considerations' -2. Validating generalizations I made of 'Implementation Considerations' - -3. Containerizing this project +2. Developing alleviation for Big-O in a fluid manner. ------------------ [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf [(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf -[3]: Essentialy, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algoirthm will run for the alloted while loop in the python code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. - From 80eb26e64822cb4c3d4ba74efec014b8e3f8b6f8 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 15:17:19 -0700 Subject: [PATCH 050/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 0f82068..1a268c8 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -51,9 +51,7 @@ https://github.com/mridulsar/DurrHoyerLibrary ## The Dürr–Høyer Algorithm -A brief introduction to the Dürr–Høyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: - -Dürr and Høyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follows: +A brief introduction to the Dürr–Høyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' as summarized below. If you would like to read the paper in all of its glory, there is a link here:[(1)]. 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length @@ -103,7 +101,7 @@ I have implemented it as follows: --- -In order to mark items we simply mark all indices that come before i. +In order to mark items we run a for loop and compare all values to each other, storing the values that are smaller than the y-th index of our flattened table. ------------------ From 5c4cb0718f3c8eda5f168657c6dc5ec0ff77e8ce Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 15:21:46 -0700 Subject: [PATCH 051/152] Delete Durr_Hoyer_Post_1.md --- _posts/Durr_Hoyer_Post_1.md | 132 ------------------------------------ 1 file changed, 132 deletions(-) delete mode 100644 _posts/Durr_Hoyer_Post_1.md diff --git a/_posts/Durr_Hoyer_Post_1.md b/_posts/Durr_Hoyer_Post_1.md deleted file mode 100644 index 5b7879f..0000000 --- a/_posts/Durr_Hoyer_Post_1.md +++ /dev/null @@ -1,132 +0,0 @@ ---- -title: "Building a Q# library to find table minima: Part 1" -date: 2020-06-22 -categories: - - Blog -tags: - - library - - algorithm - - in-development ---- -## Introduction - -Hello all! -My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. -I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. -This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum mechanics has been a rewarding experience. -I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. -I will likely be posting about this algorithm and library more as the library and my understanding continues developing. - -### My Background -I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. -I was instantly pulled in by the elegant algorithms and proposed efficiency. -Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. -I took a deep dive into the research papers and found myself amazed by what was out there. -I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. -From here I tinkered with Q# and developed basic algorithms that I previously learned. -I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. - - -### Motivation -The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is Big-O = square root of the number of items in the table. Durr and Hoyer propose an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python shell. The python shell will be provided below. The circuit's gates will be analyzed and some code is given. - - -## The Dürr–Høyer Algorithm -A brief introduction to the Durr and Hoyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' [(1)]: - -Durr and Hoyer propose a computationally efficient routine to find the minimum of a table of distinct unsorted integers. The algorithm is as follows: - - -![QMSA](/assets/images//DurrHoyer-QMSA.JPG "QMSA") - - -#### Psuedocode - -``` - t = table.flatten() - N = len(t) - y = randomuniform[1..N] #N is a integer - time_limit = (22.5*sqrt(N)=1.4*log(N)^2) - q = preprocess.simulate(t,y) - - while time.clock() < time_limit: - for i in range(N): - while t[i] < t[y]: - i = N-i - if N is even - q=Algorithm with Hadamard gates[0...j] - y'=q[0] - else - t=Algorithm with QFT instead of H[0...j] - y'=q[0] - if t[y'] < t[y]: - y = y' - return - return y -``` - -It is easy to understand the conceptual implementation in python. It is important to note steps a and b: - - -![a](/assets/images//a.JPG) - -Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. The creation of qubits from a table of unique integers is an interesting question. - -For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. My most immediate thought is to entangle all the qubits and superimpose them. The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. - -The register can be initiated as follows: - -![latex_imp](/assets/images//latex_information.JPG) - -Obviously this is unachievable on current quantum computer architecture. We only have a qubit, the above example implies we have harnessed the tritbit and so on till n bit. In reality we will register all integers as unique binary strings which will equate to their qubits. - -Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: - -![intialize](/assets/images//intialize.JPG) - -Note: make sure to use LittleEndian() on the register of qubits before applying PrepareUniformSuperposition(). From here you will have to use LittleEndianAsBigEndian() to set the right data type after using PrepareUniformSuperposition() as it spits out LE datatype. I didn't include this so it looked neat. - -In order to mark items we simply mark all indices that come before i. - ------------------- - -![b](/assets/images//b.JPG) - -Part b must be understood through another paper 'Tight bounds on quantum searching' [(2)]. It details the use of the quantum exponential searching algorithm in three cases. Those three cases being: finding one solution, multiple solutions, or an unknown number of solutions. For each case there is different preparation of qubits and algorithms used in unison to find the solution(s). The Durr Hoyer Algorithm is supposed to push computational efficiency versus computational accuracy. The statement earlier mentioning the 50% success rate for finding the minimum is bounded by the Big-O can be explored further now. [3] - -It is intuitive to see that in the case for finding a minimum of an unsorted table is finding one unique solution. The details are mentioned under "Implementation Considerations". - -![Implementation](/assets/images//DurrHoyer-Implementation.JPG "Implementation") - -A thorough read of this will supply any clarification to the python code posted above. 'G' mentioned above is the 'Algorithm' in the python psuedo code. It will take in the random value, the table in quantum information, and the lenght of the table. It is important to note that if the table has odd elements a different form of the algorithm must be used. Being new to quantum computing I have found myself wading through darkness at times. Reading this even now leaves me a bit unsure and skeptical of my own implementation, redoing my calculations. After research and conversations with quantum computing aficionados I have reached the following conclusions: - -1. T is just a Hadamard Gate -2. Conditional Phase Shift on A is a variation of the CNOT Gate -3. Conditional Phase Shift on 0 will will change the sign of the qubit if it is in null space, otherwise it will give the same result back - -Here is a small snippet of my current circuit: - -![circuit](/assets/images//Algorithm_Even.JPG) - -Generalizations are a mathematicians greatest friend, but a downfall if not implemented correctly. If anyone has any input for the generalizations I have made I will take them with open arms! - ------------------- - -### Conclusion - -So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. In specific, I mentioned I felt Durr and Hoyer used principles from Duestch's, Grover's, and Shor's algorithm. Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function as 'G' does. Grover's Algorithm is quite literally applied in this algorithm. Lastly, Shor's Algorithm shows off the power of combining classical and quantum systems to achieve outstanding results. I love seeing such fundamental concepts continue to push boundaries. I am constantly in search for collaboration among those who are equally as passionate about quantum computing. If you had any suggestions or questions feel free to send me an email. - -https://github.com/mridulsar/DurrHoyerLibrary ------------------- - -## My current challenges in this project: - -1. Validating generalizations I made of 'Implementation Considerations' - -2. Developing alleviation for Big-O in a fluid manner. - ------------------- - -[(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf -[(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf -[3]: Essentially, the number of times we call upon the algorithm in step (b) the greater our Big-0 and probabilistic success rate. Durr Hoyer answers to a 'sweet spot' of sorts. The algorithm will run for the allotted while loop in the python code above to achieve a 50% success rate. This proof is done in detail in [(1)]. An important feature of this open source library will be the user's manipulation of how many times step (b) is applied. In addition, future implementation can include manipulation of the algorithm in order to implement the different 3 cases mentioned for the quantum exponential searching algorithm. Though this means the algorithm is technically no longer Durr Hoyer it will be an exciting feature. From 6700ad02d97673ea242af4404aada3611853c713 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 18:07:36 -0700 Subject: [PATCH 052/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 19 +++++++++++-------- 1 file changed, 11 insertions(+), 8 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 1a268c8..3e535c0 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -46,7 +46,7 @@ With this comes some complexity in the inner workings of the library which will In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python host script. -The python host script will be provided below. The circuit's gates will be analyzed and some code is given. +The circuit's gates will be analyzed and some code is given. https://github.com/mridulsar/DurrHoyerLibrary @@ -55,7 +55,7 @@ A brief introduction to the Dürr–Høyer Algorithm can be derived from the Qua 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length -2. Repeat the following steps until T = 22.5 * sqrt(N) + 1.4 * log^2(N). Once T equals the expression with N proceed to step c +2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Once time equals the expression with N proceed to step c 2(a) Intialize the memory as a uniform superposition of qubits. Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. 2(b) Apply the quantum exponential searching algorithm [(2)]. 2(c) Measure the first qubit, map this qubit back to the integer it represents. If this integer is less than the integer in the y-th position set y as the position of the integer represented by the measured qubit. @@ -64,24 +64,27 @@ A brief introduction to the Dürr–Høyer Algorithm can be derived from the Qua It is important to note steps a and b: -2(a) Intialize the memory as a uniform superposition of qubits. Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. +2(a) Intialize the memory as a uniform superposition of qubits. +Each qubit represents an integer in the table. +After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. +Additionally, mark all elements of the table which are less than than the y-th integer of the table. Part a is probably the most interesting part of this algorithm from my perspective. After a brief chat with Dr. Hoyer I realized this aspect of the algorithm is a field of research in itself. -The creation of qubits from a table of unique integers is an interesting question. +The creation of qubits from a table of unique integers is an interesting question. For now, the more prevalent question is what sort of initialization we can use to verify the integrity and efficiency of this algorithm. My most immediate thought is to entangle all the qubits and superimpose them. The classical value of the integer is not represented here, only the index of the integer is captured at a quantum level on the table assuming an input datatype of BigEndian. -If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. +If the goal is to test the algorithm's probabilistic integrity this plan of action will suffice. -The register can be initiated as follows: +The register can be initiated as follows: -![latex_imp](/assets/images//latex_information.JPG) +![latex_imp](/assets/images//latex_information.JPG) Obviously this is unachievable on current quantum computer architecture. We only have a qubit, the above example implies we have harnessed the tritbit and so on till n bit. -In reality we will register all integers as unique binary strings which will equate to their qubits. +In reality we will register all integers as unique binary strings which will equate to their qubits. Luckily PrepareUniformSuperposition() in Q# does this for us easily. I have implemented it as follows: From 4e2ea76c7620b90cfa730834ebb49f9c377576b2 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 18:09:33 -0700 Subject: [PATCH 053/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 3e535c0..9707911 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -35,9 +35,9 @@ On a classical computer it will take as many time steps as there are items to fi This means the Big-O is N. The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. -The Dürr-Høyer algorithm would have find the minimum in 3 steps. +The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this effecieny as the Big-O. -The Big-O essentially means the run time of the program will increase as the table size grows at a rate of square root of table size. +The Big-O essentially means the run time of the program will increase as the table size grows. Dürr and Høyer propose an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. From 5235c7938469a39d01458c4913b52539c8ca0642 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 18:11:05 -0700 Subject: [PATCH 054/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 9707911..7786f9c 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -56,15 +56,19 @@ A brief introduction to the Dürr–Høyer Algorithm can be derived from the Qua 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Once time equals the expression with N proceed to step c -2(a) Intialize the memory as a uniform superposition of qubits. Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. -2(b) Apply the quantum exponential searching algorithm [(2)]. -2(c) Measure the first qubit, map this qubit back to the integer it represents. If this integer is less than the integer in the y-th position set y as the position of the integer represented by the measured qubit. +2(a) Intialize the memory as a uniform superposition of qubits. +Each qubit represents an integer in the table. +After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. +Additionally, mark all elements of the table which are less than than the y-th integer of the table. +2(b) Apply the quantum exponential searching algorithm [(2)]. +2(c) Measure the first qubit, map this qubit back to the integer it represents. +If this integer is less than the integer in the y-th position set y as the position of the integer represented by the measured qubit. -It is important to note steps a and b: +It is important to note steps a and b: -2(a) Intialize the memory as a uniform superposition of qubits. +2(a) Intialize the memory as a uniform superposition of qubits. Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. From 156ca593eb3b22b3ab514ceaf21fd3051e6adf21 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 18:17:52 -0700 Subject: [PATCH 055/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 7786f9c..9422e85 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -61,8 +61,10 @@ Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. 2(b) Apply the quantum exponential searching algorithm [(2)]. -2(c) Measure the first qubit, map this qubit back to the integer it represents. -If this integer is less than the integer in the y-th position set y as the position of the integer represented by the measured qubit. +2(c) Measure the first qubit and map this qubit back to the the intialized state of qubits. +One of which will be uniquely identified by the measured qubit and represent the hypothetical minima's index. +Take this hypothetical minima's index as y'. +If the integer in the y'-th position is less than the integer in the y-th position set y as y'. It is important to note steps a and b: From c945d190e6786a4e71d0d65cee0ee38963f7c407 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 18:20:00 -0700 Subject: [PATCH 056/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 9422e85..5ffbdbd 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -64,8 +64,8 @@ Additionally, mark all elements of the table which are less than than the y-th i 2(c) Measure the first qubit and map this qubit back to the the intialized state of qubits. One of which will be uniquely identified by the measured qubit and represent the hypothetical minima's index. Take this hypothetical minima's index as y'. -If the integer in the y'-th position is less than the integer in the y-th position set y as y'. - +If the integer in the y'-th position is less than the integer in the y-th position set y as y'. +3. Return y. It is important to note steps a and b: From 230ffbb5a3c11811f0dd1ca050b8024da8e2f9d2 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 23 Jun 2020 22:18:37 -0700 Subject: [PATCH 057/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 15 +++++++++++++++ 1 file changed, 15 insertions(+) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 5ffbdbd..f1f7d39 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -18,6 +18,21 @@ I would like to pass on what I have seen and understood in that time, as well as I will likely be posting about this algorithm and library more as the library and my understanding continues developing. https://github.com/mridulsar/DurrHoyerLibrary +## TL;DR + +The Dürr-Høyer algorithm takes an unsorted table of integers and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table +- +Open Source Implementation is going smoothly, understanding the Dürr and Høyer paper [(1)] was and is the most difficult challenge. +- +There is a python host file and two Q# circuits. One circuit applies the Quantum Minimum Searching Algorithm and another circuit intializes our register as described in 'A quantum algorithm for finding the minimum'[(1)] +- +The most intersting part of the algorithm is the initializing of the register and applying the quantum exponential searching algorithm [(2)] to the register. +- +If you would like to contribute to this project or just want to learn more head over to: https://github.com/mridulsar/DurrHoyerLibrary +- + + + ### My Background I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. From 6a3dd032b07e33a076fb0524b94ea45bc34ac0af Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Wed, 24 Jun 2020 13:22:55 -0700 Subject: [PATCH 058/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 37 +++++++++++++++----------- 1 file changed, 21 insertions(+), 16 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index f1f7d39..a6e8f0b 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -8,6 +8,26 @@ tags: - algorithm - in-development --- + +## TL;DR +''' + + + The Dürr-Høyer algorithm takes an unsorted table of integers + and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. + Open Source Implementation of the algorithm is easily done, understanding the Dürr and Høyer paper [(1)] + was and is the most difficult challenge. + There is a python host file and two Q# circuits. + One circuit applies the Quantum Minimum Searching Algorithm and + another circuit intializes our register as described in 'A quantum algorithm for finding the minimum'[(1)] + The most intersting part of the algorithm is the initializing of the register + and applying the quantum exponential searching algorithm [(2)] to the register. + + If you would like to contribute to this project or take a look at some code head over to: + https://github.com/mridulsar/DurrHoyerLibrary + + +''' ## Introduction Hello all! @@ -16,22 +36,7 @@ I wanted to share with the community my experiences in learning about the Dürr This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum mechanics has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. -https://github.com/mridulsar/DurrHoyerLibrary - -## TL;DR - -The Dürr-Høyer algorithm takes an unsorted table of integers and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table -- -Open Source Implementation is going smoothly, understanding the Dürr and Høyer paper [(1)] was and is the most difficult challenge. -- -There is a python host file and two Q# circuits. One circuit applies the Quantum Minimum Searching Algorithm and another circuit intializes our register as described in 'A quantum algorithm for finding the minimum'[(1)] -- -The most intersting part of the algorithm is the initializing of the register and applying the quantum exponential searching algorithm [(2)] to the register. -- -If you would like to contribute to this project or just want to learn more head over to: https://github.com/mridulsar/DurrHoyerLibrary -- - - +https://github.com/mridulsar/DurrHoyerLibrary. ### My Background I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. From d828849b0136613993ff5dcf8497a8d6e50ae89f Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Wed, 24 Jun 2020 13:23:19 -0700 Subject: [PATCH 059/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index a6e8f0b..7bb3a65 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -10,8 +10,7 @@ tags: --- ## TL;DR -''' - +--- The Dürr-Høyer algorithm takes an unsorted table of integers and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. @@ -26,8 +25,8 @@ tags: If you would like to contribute to this project or take a look at some code head over to: https://github.com/mridulsar/DurrHoyerLibrary +--- -''' ## Introduction Hello all! From a6271773814fbf55a0b8a9d6355aaeb6ca97f083 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Wed, 24 Jun 2020 15:44:29 -0700 Subject: [PATCH 060/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 7bb3a65..826c53b 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -80,8 +80,7 @@ Each qubit represents an integer in the table. After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. Additionally, mark all elements of the table which are less than than the y-th integer of the table. 2(b) Apply the quantum exponential searching algorithm [(2)]. -2(c) Measure the first qubit and map this qubit back to the the intialized state of qubits. -One of which will be uniquely identified by the measured qubit and represent the hypothetical minima's index. +2(c) Measure the first qubit through use of Shor's Algorithm. Take this hypothetical minima's index as y'. If the integer in the y'-th position is less than the integer in the y-th position set y as y'. 3. Return y. From 01cb2012c05ce4cc3677271de149d33a3893d707 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 18:57:17 -0700 Subject: [PATCH 061/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 147 ++++++++++--------------- 1 file changed, 56 insertions(+), 91 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 826c53b..73ccd54 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -14,13 +14,13 @@ tags: The Dürr-Høyer algorithm takes an unsorted table of integers and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. - Open Source Implementation of the algorithm is easily done, understanding the Dürr and Høyer paper [(1)] + Open Source Implementation of the algorithm is easily done, understanding the Dürr and Høyer paper was and is the most difficult challenge. There is a python host file and two Q# circuits. One circuit applies the Quantum Minimum Searching Algorithm and - another circuit intializes our register as described in 'A quantum algorithm for finding the minimum'[(1)] + another circuit intializes our register as described in 'A quantum algorithm for finding the minimum' The most intersting part of the algorithm is the initializing of the register - and applying the quantum exponential searching algorithm [(2)] to the register. + and applying the quantum exponential searching algorithm to the register. If you would like to contribute to this project or take a look at some code head over to: https://github.com/mridulsar/DurrHoyerLibrary @@ -57,7 +57,7 @@ For example assume there are 9 items in a table, on a classical comptuer this wo The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this effecieny as the Big-O. The Big-O essentially means the run time of the program will increase as the table size grows. -Dürr and Høyer propose an ensemble of quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. +Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. @@ -74,116 +74,81 @@ A brief introduction to the Dürr–Høyer Algorithm can be derived from the Qua 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length -2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Once time equals the expression with N proceed to step c +2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). The time begins once we start step 2(a), if time equals the expression with N proceed to step c 2(a) Intialize the memory as a uniform superposition of qubits. -Each qubit represents an integer in the table. -After Intializing the memory apply the qubit representing y-th index of the flattened table to your register. -Additionally, mark all elements of the table which are less than than the y-th integer of the table. -2(b) Apply the quantum exponential searching algorithm [(2)]. -2(c) Measure the first qubit through use of Shor's Algorithm. +Each qubit represents an index. +After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. +This marks all T[j] Date: Tue, 30 Jun 2020 19:09:35 -0700 Subject: [PATCH 062/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 56 +++++++++++++++++++++++--- 1 file changed, 51 insertions(+), 5 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 73ccd54..f53a474 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -16,11 +16,12 @@ tags: and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. Open Source Implementation of the algorithm is easily done, understanding the Dürr and Høyer paper was and is the most difficult challenge. - There is a python host file and two Q# circuits. - One circuit applies the Quantum Minimum Searching Algorithm and - another circuit intializes our register as described in 'A quantum algorithm for finding the minimum' - The most intersting part of the algorithm is the initializing of the register - and applying the quantum exponential searching algorithm to the register. + There is a python host file and a Q# file. + The python file follows the Quantum Minimum Searching Algorithm closely. + The Q# file follows the Quantum Exponential Searching Algorithm implementation for finding a unique solution. + This implementation will allow for easy use of QMSA on any problem where a unique solution needs to be found. + The python host file is not necessary though it does give us a template to work on other problems. + QESA can be run indepently and will allocate itself to your table size and random index. If you would like to contribute to this project or take a look at some code head over to: https://github.com/mridulsar/DurrHoyerLibrary @@ -137,6 +138,51 @@ For futher information on how this was derived take a look at 'Tight bounds on q Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits holds O(log(n)). +Here is the python host that will apply the conditions of QMSA while using QESA. + +class DH(object): + + def __init__(self,table): + + self.table = np.ndarray.flatten(table) + + self.N= len(self.table) + + self.y = int(np.random.uniform(0,self.N-1)) + + def QMSA(self,N,y,table): + N = self.N + + y = self.y + + t = self.table + + y_prime = N + + time_limit = (22.5*np.sqrt(N)+1.4*np.log(N)**2) + + while time.clock() < time_limit: + + for i in range(N): + + while t[y] > t[y_prime]: + + #if N // 2 : + + y_prime= Algorithm.simulate(N,y) + + #else: + + #y_prime=Algorithm_Odd.simulate(N,y) + + if t[y_prime] < t[y]: + + i=0 + + y = y_prime + + return y + ------------------ 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z{G)j<>8dO$!1C+MY>N8KqGdMtrNdt_t^^J9lB7>QKqo0!>$tB4BEGhTwKu?Q)xu_7 zhv<@Fk*uPXJdr-p41JuKOP20roAnx}9XZ$ZCoK}oKD3Q(R$h>`J(9H>M)7*D$-IR1 z8*XB2ft=HfLe5?RY9^F0ez45LWS}D4xTd+bZqerbqLG5cVsWZ6x>Oo}TOLsq9N;>Y z!w0sleZx9H#9trmeYbJw?E)!7-XC6a>%gEGr z`qB(UsI2Wf@%+-9zJb9`YiMo{%ZJ)0<4pa!&zh;NI{?8_auC|#J zkN4Muk{1e5-?$OJXQ>SURT5XNI6M%&G|oy|=wqnGs~(m+LAS}_&OCm%5fv4c?b^e> zQu^nc4Z+@DyPeYtX(+Z^0@yI_i`n5!f+}|mtM4UyDhuAHlIRUDf zgM8t_4mx2bc-JrLS^{KfB?j#fjrB9i)rdv<)P73O5-f46PAnRAJIY{rk-PgC z#8y%?4j4o_&3-w~{-cP+K#lSY_QmBxs z7p|*ssoFO_(4u0`Ek=d0BX^|#^K|e2phbO(leqid4}0f@obfeZ1 E0Eo)liU0rr From db33b92ad98c1aca02140901e0c2bad67af97028 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 19:37:59 -0700 Subject: [PATCH 068/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 152 ++++++++++++++++++------- 1 file changed, 109 insertions(+), 43 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index f53a474..ce89107 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -89,37 +89,31 @@ If the integer in the y'-th position is less than the integer in the y-th positi The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. - operation Algorithm(TableLength : Int, RandomIndex : Int) : Unit - { +In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. + using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices { within { - let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside - PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices + + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] t[y_prime]: + while t[y] < t[y_prime]: - #if N // 2 : + if N // 2 : # Implementation of odd and even functionality - y_prime= Algorithm.simulate(N,y) + y_prime= Algorithm.simulate(N,y) - #else: + else: - #y_prime=Algorithm_Odd.simulate(N,y) + y_prime=Algorithm_Odd.simulate(N,y) - if t[y_prime] < t[y]: + if t[y_prime] < t[y]: # Check if we have found our minimum - i=0 + i=0 # reset loop if found - y = y_prime + y = y_prime #store index with minimum - return y + return y ------------------ @@ -194,8 +258,10 @@ In specific, I mentioned I felt Dürr and Høyer used principles from Duestch's, Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function as done in Dürr and Høyer's algorithm to mark all states that satisfy T[j] Date: Tue, 30 Jun 2020 19:41:28 -0700 Subject: [PATCH 069/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 52 +++++++++++++++++++------- 1 file changed, 38 insertions(+), 14 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index ce89107..12aa8d5 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -91,29 +91,40 @@ The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. - using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices - { - within - { - PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices +--- + + using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices + { + within + { + PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices - let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside - Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Tue, 30 Jun 2020 19:42:18 -0700 Subject: [PATCH 070/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 12aa8d5..db85310 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -107,11 +107,11 @@ In order to intialize the register we pepare a uniform superposition of qubits, Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. It is stated as follows: ---- + ![](DurrHoyer-Implementation.JPG) ---- + We utilize the register we were working with earlier and apply a T' transform, this is stated to simply be the Hadmard. From d0a1d5b701dcc815fbdd5bf4f6296af4964a30b9 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 19:43:46 -0700 Subject: [PATCH 071/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index db85310..0b2bfbf 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -109,7 +109,7 @@ Step 2(a) has been stasified. Now we must figure out how to apply the QESA algor -![](DurrHoyer-Implementation.JPG) +![implement](/assets/images/DurrHoyer-Implementation.JPG) From 087cc637e1acde28ea4c9af73f109b1c2cf87b00 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 19:49:30 -0700 Subject: [PATCH 072/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 34 +++++++++++++++++++++----- 1 file changed, 28 insertions(+), 6 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 0b2bfbf..fe57be5 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -173,7 +173,7 @@ Our Q# script will be structured as follows: open Microsoft.Quantum.Math; open Microsoft.Quantum.Core; open Microsoft.Quantum.Convert; - operation Algorithm(TableLength : Int, RandomIndex : Int) : Unit + operation Algorithm_Even(TableLength : Int, RandomIndex : Int) : Unit { using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices { @@ -189,13 +189,34 @@ Our Q# script will be structured as follows: { ApplyToEach(H,Register); // Apply Hadamard to register - //QFTLE(LittleEndian(register)); Implemntation for odd number of table entries - ApplyConditionalPhase_0(LittleEndian(Register)); // Reflect qubits that are 0s ApplyToEachA(H,Register); // Apply Adjunct Hadamard to register - //QFT(BigEndian(register)); Inverse QFT by using BigEndian + ApplyConditionalPhase(LittleEndian(Register)); //Reflect qubits that are 1s + } + } + } + + operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Unit + { + using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices + { + within + { + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside + + PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices + + Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Tue, 30 Jun 2020 19:53:15 -0700 Subject: [PATCH 073/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 45 ++++++++++++++------------ 1 file changed, 25 insertions(+), 20 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index fe57be5..5f092a2 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -198,6 +198,31 @@ Our Q# script will be structured as follows: } } + operation ApplyConditionalPhase_0(register: LittleEndian) : Unit is Adj + Ctl + { + using (aux = Qubit()) + { + (ControlledOnInt(0,Z))(register!,aux); // If qubit is 0 flip it! + } + } + operation ApplyConditionalPhase(register : LittleEndian) : Unit is Adj + Ctl + { + using (aux = Qubit()) + { + (ControlledOnInt(1,Z))(register!,aux); // If qubit is 1 flip it! + } + } + } + + +--- + +For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. +It is important to not the above algorithm only works for a table with even entries. +The algorithm breaks down when applying the Hadamard gate as the Hadamard is layed across the diagnoal of a identity matrix which is equal in dimensions to the number of qubits we have. +With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned identity matrix the transformation is not retained. +To circumvent this we introduce the following implementation, utilizing QFT. + operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Unit { using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices @@ -222,26 +247,6 @@ Our Q# script will be structured as follows: } } } - operation ApplyConditionalPhase_0(register: LittleEndian) : Unit is Adj + Ctl - { - using (aux = Qubit()) - { - (ControlledOnInt(0,Z))(register!,aux); // If qubit is 0 flip it! - } - } - operation ApplyConditionalPhase(register : LittleEndian) : Unit is Adj + Ctl - { - using (aux = Qubit()) - { - (ControlledOnInt(1,Z))(register!,aux); // If qubit is 1 flip it! - } - } - } - - ---- - -For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits holds O(log(n)). From 3f59998d5559fa9cab98335ea350691a47e5a388 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 22:57:37 -0700 Subject: [PATCH 074/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 68 +++++++++++++------------- 1 file changed, 35 insertions(+), 33 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 5f092a2..4764509 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -38,38 +38,6 @@ I would like to pass on what I have seen and understood in that time, as well as I will likely be posting about this algorithm and library more as the library and my understanding continues developing. https://github.com/mridulsar/DurrHoyerLibrary. -### My Background -I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. -I was instantly pulled in by the elegant algorithms and proposed efficiency. -Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. -I took a deep dive into the research papers and found myself amazed by what was out there. -I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. -From here I tinkered with Q# and developed basic algorithms that I previously learned. -I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. - - -### Motivation -The motivation behind this project is to provide open source functionality. -The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. -On a classical computer it will take as many time steps as there are items to find a minimum. -This means the Big-O is N. -The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. -For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. -The Dürr-Høyer algorithm would find the minimum in 3 steps. -We can understand this effecieny as the Big-O. -The Big-O essentially means the run time of the program will increase as the table size grows. -Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. -This success rate is bounded by the Big-O. -If we decide to alleviate this Big-O the probability of finding the minimum increases. -Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. -With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. -In order for this library to be used properly it must meet some guidelines. -At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. -I am programming the algorithm in Q# with a python host script. -The circuit's gates will be analyzed and some code is given. -https://github.com/mridulsar/DurrHoyerLibrary - - ## The Dürr–Høyer Algorithm A brief introduction to the Dürr–Høyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' as summarized below. If you would like to read the paper in all of its glory, there is a link here:[(1)]. @@ -301,6 +269,36 @@ Here is the python host that will apply the conditions of QMSA while using QESA. ------------------ +### Motivation +The motivation behind this project is to provide open source functionality. +The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. +On a classical computer it will take as many time steps as there are items to find a minimum. +This means the Big-O is N. +The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. +For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. +The Dürr-Høyer algorithm would find the minimum in 3 steps. +We can understand this effecieny as the Big-O. +The Big-O essentially means the run time of the program will increase as the table size grows. +Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. +This success rate is bounded by the Big-O. +If we decide to alleviate this Big-O the probability of finding the minimum increases. +Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. +With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. +In order for this library to be used properly it must meet some guidelines. +At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. +I am programming the algorithm in Q# with a python host script. +The circuit's gates will be analyzed and some code is given. +https://github.com/mridulsar/DurrHoyerLibrary + +### My Background +I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. +I was instantly pulled in by the elegant algorithms and proposed efficiency. +Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. +I took a deep dive into the research papers and found myself amazed by what was out there. +I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. +From here I tinkered with Q# and developed basic algorithms that I previously learned. +I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. + ### Conclusion So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. @@ -311,7 +309,11 @@ Grover's Algorithm is quite literally applied in this algorithm, though a genera Lastly, Shor's Algorithm shows off the power of combining classical and quantum systems to achieve outstanding results, which is shown in Dürr and Høyer's algorithm by bounding our time, a classical step in the algorithm. I love seeing such fundamental concepts continue to push boundaries. -This algorithm should be available to use within Q# within the next few weeks once testing is done to ensure usability. +This algorithm should be available to use within Q# within the next few weeks once testing is done to ensure usability. +QESA is really the algorithm of interest, to get a running version of QMSA is just to show that this algorithm does indeed work. +What is more interesting is creating variations of QMSA to fit your needs while utilizing QESA. +It is important to note that currently QESA can only solve for one unique solution. +In the future, QESA will be more robust and multiple solutions can be found. In the mean time you can implement this code for yourself, I would love to see what this community can do with an algorithm like this. If you had any suggestions or questions feel free to send me an email. From 0dae5b8cb68c9760572c408873e3403d8fea73c5 Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 22:59:37 -0700 Subject: [PATCH 075/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 4764509..596f608 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -12,10 +12,7 @@ tags: ## TL;DR --- - The Dürr-Høyer algorithm takes an unsorted table of integers - and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. - Open Source Implementation of the algorithm is easily done, understanding the Dürr and Høyer paper - was and is the most difficult challenge. + The Dürr-Høyer algorithm takes an unsorted table of integers and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. There is a python host file and a Q# file. The python file follows the Quantum Minimum Searching Algorithm closely. The Q# file follows the Quantum Exponential Searching Algorithm implementation for finding a unique solution. From 16eb14c41b60b4cdeb482b7e484b12f2e618d07c Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 23:03:08 -0700 Subject: [PATCH 076/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 596f608..07df842 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -213,7 +213,7 @@ To circumvent this we introduce the following implementation, utilizing QFT. } } -Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits holds O(log(n)). +Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. From 5d1d639e268c2d7ad7e788a94e7ad5be6aa6553a Mon Sep 17 00:00:00 2001 From: Mridul Sarkar <42987995+mridulsar@users.noreply.github.com> Date: Tue, 30 Jun 2020 23:04:02 -0700 Subject: [PATCH 077/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 7 ++----- 1 file changed, 2 insertions(+), 5 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 07df842..86923b3 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -12,14 +12,11 @@ tags: ## TL;DR --- - The Dürr-Høyer algorithm takes an unsorted table of integers and finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. + The Dürr-Høyer algorithm takes an unsorted table of integers and + finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. There is a python host file and a Q# file. The python file follows the Quantum Minimum Searching Algorithm closely. The Q# file follows the Quantum Exponential Searching Algorithm implementation for finding a unique solution. - This implementation will allow for easy use of QMSA on any problem where a unique solution needs to be found. - The python host file is not necessary though it does give us a template to work on other problems. - QESA can be run indepently and will allocate itself to your table size and random index. - If you would like to contribute to this project or take a look at some code head over to: https://github.com/mridulsar/DurrHoyerLibrary From 2e5dc9aedfa5ec29fd9996f3eab51ef781fd0194 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Fri, 10 Jul 2020 23:52:39 -0700 Subject: [PATCH 078/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 16 +++------------- 1 file changed, 3 insertions(+), 13 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 86923b3..95f2a7d 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -9,18 +9,9 @@ tags: - in-development --- -## TL;DR ---- - - The Dürr-Høyer algorithm takes an unsorted table of integers and - finds the minima in Big-O(sqrt(N)) N being the # of elements in the table. - There is a python host file and a Q# file. - The python file follows the Quantum Minimum Searching Algorithm closely. - The Q# file follows the Quantum Exponential Searching Algorithm implementation for finding a unique solution. - If you would like to contribute to this project or take a look at some code head over to: - https://github.com/mridulsar/DurrHoyerLibrary - ---- +> TL;DR +> The Dürr-Høyer algorithm takes an unsorted table of integers of length N and finds the minima in O(sqrt(N)). +> I started a project that implements this algorithm here: [https://github.com/mridulsar/DurrHoyerLibrary](https://github.com/mridulsar/DurrHoyerLibrary), contributions welcome! ## Introduction @@ -325,4 +316,3 @@ https://github.com/mridulsar/DurrHoyerLibrary [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf [(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf - From c0c13436d90239ec49b40fef6c4e154b5544ba90 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Fri, 10 Jul 2020 23:53:13 -0700 Subject: [PATCH 079/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 95f2a7d..1683593 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -24,7 +24,9 @@ I will likely be posting about this algorithm and library more as the library an https://github.com/mridulsar/DurrHoyerLibrary. ## The Dürr–Høyer Algorithm -A brief introduction to the Dürr–Høyer Algorithm can be derived from the Quantum Minimum Searching Algorithm outlined in 'A quantum algorithm for finding the minimum' as summarized below. If you would like to read the paper in all of its glory, there is a link here:[(1)]. +The Dürr–Høyer Algorithm was originally proposed in ''A quantum algorithm for finding the minimum +∗":[(1)], where it was called the 'Quantum Minimum Searching Algorithm'. +I'll summarize the main result below in case you don't have time to read the paper in all of its glory. 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length From 11b3518beab251307f2ae355a923b7a1e2adb275 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 11 Jul 2020 14:25:25 -0700 Subject: [PATCH 080/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 1683593..0bfe41c 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -37,9 +37,9 @@ After intializing the memory grab the y-th qubit and entangle the state of your This marks all T[j] + 3. Return y. The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. From c9448ad3707f6258e6c99a50a7c68f479e55b5de Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 11 Jul 2020 14:25:33 -0700 Subject: [PATCH 081/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 0bfe41c..6aab19f 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -173,7 +173,8 @@ Our Q# script will be structured as follows: --- For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. -It is important to not the above algorithm only works for a table with even entries. +It is important to note the above algorithm only works for a table with an even number of entries. + The algorithm breaks down when applying the Hadamard gate as the Hadamard is layed across the diagnoal of a identity matrix which is equal in dimensions to the number of qubits we have. With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned identity matrix the transformation is not retained. To circumvent this we introduce the following implementation, utilizing QFT. From d07a098d6bbe43df0c8634bb111c451a36083450 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 11 Jul 2020 14:25:46 -0700 Subject: [PATCH 082/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 6aab19f..30ccaa3 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -31,7 +31,8 @@ I'll summarize the main result below in case you don't have time to read the pap 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). The time begins once we start step 2(a), if time equals the expression with N proceed to step c -2(a) Intialize the memory as a uniform superposition of qubits. + (a) Initialize the memory as a uniform superposition of qubits.
+ Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. This marks all T[j] Date: Sun, 12 Jul 2020 10:22:20 -0700 Subject: [PATCH 083/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 30ccaa3..3f16959 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -43,7 +43,8 @@ If the integer in the table at index y' is less than the integer at y, set y = y 3. Return y. -The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. +The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first look at how to initialize the register. Then we will see how the quantum exponential searching algorithm (QESA) can be implemented to find a unique minimum. + In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. From 5ed63307b698972038abe4b050086aaeffc2749a Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 12 Jul 2020 10:22:31 -0700 Subject: [PATCH 084/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 17 +++++++---------- 1 file changed, 7 insertions(+), 10 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 3f16959..cab7c75 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -24,27 +24,24 @@ I will likely be posting about this algorithm and library more as the library an https://github.com/mridulsar/DurrHoyerLibrary. ## The Dürr–Høyer Algorithm -The Dürr–Høyer Algorithm was originally proposed in ''A quantum algorithm for finding the minimum +Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in ''A quantum algorithm for finding the minimum ∗":[(1)], where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main result below in case you don't have time to read the paper in all of its glory. 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length -2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). The time begins once we start step 2(a), if time equals the expression with N proceed to step c - (a) Initialize the memory as a uniform superposition of qubits.
- +2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). The time begins once we start step 2(a), if time equals the expression with N proceed to step c.
+ 2(a) Initialize the memory as a uniform superposition of qubits.
Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. -This marks all T[j] + 2(b) Apply the quantum exponential searching algorithm [(2)], which is a generalized Grover's search.
+ 2(c) Measure the first register, call that outcome y' which is an index into the table. If the integer in the table at index y' is less than the integer at y, set y = y'.
3. Return y. -The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first look at how to initialize the register. Then we will see how the quantum exponential searching algorithm (QESA) can be implemented to find a unique minimum. - +The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. From c3c1e77759fbc019c3ba5516ea25a8f11d3eedd8 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 12 Jul 2020 10:29:29 -0700 Subject: [PATCH 085/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index cab7c75..5a2bdfd 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -59,7 +59,9 @@ In order to intialize the register we pepare a uniform superposition of qubits, } --- -Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. It is stated as follows: +Step 2(a) is now complete, now we must figure out how to apply the QESA algorithm. +It is stated as follows: + From 2943fd163895d522b0d806006fe6adf1ae5434e3 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 12 Jul 2020 10:31:52 -0700 Subject: [PATCH 086/152] Update _posts/2020-06-23-Durr_Hoyer_Post_1.md Co-authored-by: Sarah Kaiser --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 5a2bdfd..597d5a0 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -51,7 +51,9 @@ In order to intialize the register we pepare a uniform superposition of qubits, { within { - PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices + // Create Uniform Superposition of all indices + PrepareUniformSuperposition(TableLength, LittleEndian(Register)); + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside From 3f72f2e6907b8de0d2d7ce4492fe6c5cd160e636 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 12 Jul 2020 10:32:27 -0700 Subject: [PATCH 087/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 23 +++++++++-------------- 1 file changed, 9 insertions(+), 14 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 597d5a0..c6e9882 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -30,7 +30,7 @@ I'll summarize the main result below in case you don't have time to read the pap 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length -2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). The time begins once we start step 2(a), if time equals the expression with N proceed to step c.
+2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c.
2(a) Initialize the memory as a uniform superposition of qubits.
Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. @@ -48,22 +48,16 @@ In order to intialize the register we pepare a uniform superposition of qubits, --- using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices - { - within - { - // Create Uniform Superposition of all indices - PrepareUniformSuperposition(TableLength, LittleEndian(Register)); - + { + PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices - let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside - Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Sun, 12 Jul 2020 11:44:18 -0700 Subject: [PATCH 088/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 32 +++++++++++++------------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index c6e9882..0e1a31b 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -45,7 +45,7 @@ The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. ---- +``` using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices { @@ -55,7 +55,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Sun, 12 Jul 2020 11:51:14 -0700 Subject: [PATCH 089/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 71 +++++++++++--------------- 1 file changed, 30 insertions(+), 41 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 0e1a31b..ba932ff 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -46,15 +46,14 @@ The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. ``` - - using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices - { - PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices + using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices + { + PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices - let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside - Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Sun, 12 Jul 2020 12:12:14 -0700 Subject: [PATCH 090/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 114 +++++++++++++------------ 1 file changed, 59 insertions(+), 55 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index ba932ff..c4c6ac2 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -50,19 +50,16 @@ In order to intialize the register we pepare a uniform superposition of qubits, { PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices - let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex and set it aside + let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex (which is intialized in our python host script using numpy.uniform.random()) and set it aside Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Sun, 12 Jul 2020 12:15:58 -0700 Subject: [PATCH 091/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 30 +++++++++++++++----------- 1 file changed, 18 insertions(+), 12 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index c4c6ac2..15bd9fd 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -46,30 +46,34 @@ The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. ``` - using ((Register) = (Qubit[TableLength])) // intialize register to number of qubits as there are indices + // intialize register to number of qubits as there are indices + using ((Register) = (Qubit[TableLength])) { - PrepareUniformSuperposition(TableLength,LittleEndian(Register)); // Create Uniform Superposition of all indices - - let Marker = Register[RandomIndex]; // Grab the qubit in the RandomIndex (which is intialized in our python host script using numpy.uniform.random()) and set it aside - - Controlled Z(Register,Marker); // Apply Oracle to flip all states that are T[j] Date: Sun, 12 Jul 2020 12:17:03 -0700 Subject: [PATCH 092/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 15bd9fd..84f8069 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -11,7 +11,7 @@ tags: > TL;DR > The Dürr-Høyer algorithm takes an unsorted table of integers of length N and finds the minima in O(sqrt(N)). -> I started a project that implements this algorithm here: [https://github.com/mridulsar/DurrHoyerLibrary](https://github.com/mridulsar/DurrHoyerLibrary), contributions welcome! +> I started a project that implements this algorithm here : [https://github.com/mridulsar/DurrHoyerLibrary](https://github.com/mridulsar/DurrHoyerLibrary), contributions welcome! ## Introduction @@ -24,8 +24,7 @@ I will likely be posting about this algorithm and library more as the library an https://github.com/mridulsar/DurrHoyerLibrary. ## The Dürr–Høyer Algorithm -Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in ''A quantum algorithm for finding the minimum -∗":[(1)], where it was called the 'Quantum Minimum Searching Algorithm'. +Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in ''A quantum algorithm for finding the minimum":[(1)], where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main result below in case you don't have time to read the paper in all of its glory. From 9372b8b47ab4cc42ab62883f2f18a63446bb8bf4 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Tue, 14 Jul 2020 14:45:32 -0700 Subject: [PATCH 093/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 84f8069..c08dea4 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -314,4 +314,4 @@ https://github.com/mridulsar/DurrHoyerLibrary [(1)]:https://arxiv.org/pdf/quant-ph/9607014.pdf [(2)]:https://arxiv.org/pdf/quant-ph/9605034.pdf -[(3)]: Kitaev, A. Yu., “Quantum measurements and theAbelian stabilizer problem”, manuscript quant-ph/9511-026 (1995). +[(3)]: Kitaev, A. Yu., “Quantum measurements and the Abelian stabilizer problem”, manuscript quant-ph/9511-026 (1995). From d20b0586d7c783c4ddef6aaae53ba9806eda057f Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Thu, 16 Jul 2020 13:09:06 -0700 Subject: [PATCH 094/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 1 + 1 file changed, 1 insertion(+) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index c08dea4..590d30d 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -58,6 +58,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, ``` Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. +https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%22H%22%2C%22H%22%2C%22H%22%5D%2C%5B%22%E2%80%A2%22%2C1%2C%22X%22%5D%2C%5B%22H%22%2C%22H%22%5D%2C%5B%22X%22%2C%22X%22%5D%2C%5B%22H%22%2C%22H%22%5D%2C%5B%22X%22%2C%22X%22%5D%5D%2C%22init%22%3A%5B0%2C1%2C%22-%22%5D%7D The QESA algorithm is a generalized Grover's search characterized by H,S_0,H^-1,S_A for an even number of table elements. If we have an odd number of table elements we use QFT,S_0,QFT^-1,S_A where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. H is defined as a Hadamard transform, S_0 flips our qubit if it is 0, H^-1 is inverse of Hadamard- the conjugate transpose, S_A flips out qubit if it is 1. 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<42987995+mertall@users.noreply.github.com> Date: Thu, 16 Jul 2020 16:16:27 -0700 Subject: [PATCH 098/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 590d30d..fc44496 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -58,10 +58,13 @@ In order to intialize the register we pepare a uniform superposition of qubits, ``` Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. -https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%22H%22%2C%22H%22%2C%22H%22%5D%2C%5B%22%E2%80%A2%22%2C1%2C%22X%22%5D%2C%5B%22H%22%2C%22H%22%5D%2C%5B%22X%22%2C%22X%22%5D%2C%5B%22H%22%2C%22H%22%5D%2C%5B%22X%22%2C%22X%22%5D%5D%2C%22init%22%3A%5B0%2C1%2C%22-%22%5D%7D -The QESA algorithm is a generalized Grover's search characterized by H,S_0,H^-1,S_A for an even number of table elements. If we have an odd number of table elements we use QFT,S_0,QFT^-1,S_A where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. +For the following circuits assume our Random Index is 0. +![2 items in a list](/assets/images/circuit for 2 qubits random value 0.JPG) +The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. +![3 items in a list](/assets/images/circuit for 3 qubits random value 0.JPG) +If we have an odd number of table elements we use this circuit, where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. -H is defined as a Hadamard transform, S_0 flips our qubit if it is 0, H^-1 is inverse of Hadamard- the conjugate transpose, S_A flips out qubit if it is 1. +Here is a nice visual of the circuit running through various LittleEndian Orderings We utilize the register we were working with earlier and apply a H transform, this is stated to simply be the Hadmard. 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ziC1M0x4ytOIHyHV7b;}jE*+GIxE(zAPH@2-thnJpH=&0rkZ~(In75hy844%DYAVEQ zkK@X)4+@zBq3^M$$~8a1Mu$Z|22PXd09L7JeGl3L>XbKk+6btF9{6z?NLP@&#_~Ze zs+8fb@OZcED4DflAfz;^LME+jIguJkBbTKRDIbCo5u<2}V&pHCm$evWm`z9*XWXeu z7jL?FFFjsz#{H^*_VbD0TZtkJ^fumNQS^~R%++Zrk^GRvR7SZ;NSd?mNk}@4%}Hnm yj@x8rY=EB`W0tzn?MOvrl(Rr17@nlAfg!oQ=`#~@d3{?$LrVoKVN)YGp8pR~OF+l~ literal 0 HcmV?d00001 From a033d505a393a7bed079c5fd59e66119cb102040 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Thu, 16 Jul 2020 16:36:04 -0700 Subject: [PATCH 104/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index fc44496..d861723 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -61,7 +61,7 @@ Step 2(a) has been stasified. Now we must figure out how to apply the QESA algor For the following circuits assume our Random Index is 0. ![2 items in a list](/assets/images/circuit for 2 qubits random value 0.JPG) The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. -![3 items in a list](/assets/images/circuit for 3 qubits random value 0.JPG) +![3 items in a list](/assets/images/3-item-list.gif) If we have an odd number of table elements we use this circuit, where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. Here is a nice visual of the circuit running through various LittleEndian Orderings From bcdf277d7dbd634dfec4c2231d67f9922faf29cf Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Thu, 16 Jul 2020 16:38:59 -0700 Subject: [PATCH 105/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index d861723..0e3f88b 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -59,7 +59,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. For the following circuits assume our Random Index is 0. -![2 items in a list](/assets/images/circuit for 2 qubits random value 0.JPG) +![2 items in a list](/assets/images/2-item-list.gif) The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. ![3 items in a list](/assets/images/3-item-list.gif) If we have an odd number of table elements we use this circuit, where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. 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zGz5cn!e9+BSS7-S6WocHAQF^-_z}F6!CkTOG$m#X!|B}L|xQY|@b zzMJ?^s}I!6wWkH|M?yo5)oatSJhJYD>uQb^%L^KNl_{^{S&TTl`af zo>f(qmhU%~xAAN*A5Rv#Vh=ANFhYKLCFQfG>PBj3pDG$CE9*Sd Date: Thu, 16 Jul 2020 16:40:29 -0700 Subject: [PATCH 107/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 0e3f88b..dcb22a6 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -59,11 +59,13 @@ In order to intialize the register we pepare a uniform superposition of qubits, Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. For the following circuits assume our Random Index is 0. -![2 items in a list](/assets/images/2-item-list.gif) The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. -![3 items in a list](/assets/images/3-item-list.gif) + +![2 items in a list](/assets/images/2-item-list.gif) If we have an odd number of table elements we use this circuit, where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. +![3 items in a list](/assets/images/3-item-list.gif) + Here is a nice visual of the circuit running through various LittleEndian Orderings We utilize the register we were working with earlier and apply a H transform, this is stated to simply be the Hadmard. From 1e5ed9eef415e60327283d4d6fe1c2da903c032a Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Thu, 16 Jul 2020 16:41:28 -0700 Subject: [PATCH 108/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index dcb22a6..2c9393c 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -66,8 +66,7 @@ If we have an odd number of table elements we use this circuit, where QFT is the ![3 items in a list](/assets/images/3-item-list.gif) -Here is a nice visual of the circuit running through various LittleEndian Orderings - +Lets break down whats going on here. We utilize the register we were working with earlier and apply a H transform, this is stated to simply be the Hadmard. ``` From d83f32181f506dfe2473022ac260d5a232350fa9 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Thu, 16 Jul 2020 16:49:00 -0700 Subject: [PATCH 109/152] Add files via upload --- assets/images/2-item-list.gif | Bin 183673 -> 168836 bytes assets/images/3-item-list.gif | Bin 207383 -> 238627 bytes 2 files changed, 0 insertions(+), 0 deletions(-) diff --git a/assets/images/2-item-list.gif b/assets/images/2-item-list.gif index 5bf0d69e792e17619f7187d1bbe9a4acd8e183f8..a7b570a376b52cdccea57a8654dfd1b630e0e24d 100644 GIT binary patch delta 95158 zcmZshbyQSQyN6*Ini+=f?i7&j?iM6OKpG?!ke&gCt^uS&Qd&BsyQM+8JCrVAyngqt z@4x%5=j^@CTIaug&ff3(o#!DF7hZycqoSrPD)x>S{Rt%zhY9+hxdfSkK;~yZR~9|A z=iV+S-(O`Qk+n!<-G8Y+zwTMySy*L|LXMk@Z#{``rzQ^;NbS)@b2*F{^%dakM_qWe@^~! z`seiY=|7x3pPeE9@$29C3;FB+_;*``e2hdQ{}J(D!X6((@9sjbu7WQv;8#~EH#b%H 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z>l?|#Gh@-hri*a~lL_{?%RcP({$^&$GZ;PF@)di$&%|7Bls1n&Nto7{IYLKI(HWg# zo)yYs>07We`wW9$5y=QUswj{Z(GqHx9TTKU%yBm`W66xS}}e*Id0fFf3JZPjgVHvj8iZC}3hZETEyTiW_MWo-=zbe2K)>cfZC4Baji!NQ81P>1wd{W-6_qY0)Blfmy*B3f*r>s^R zWUk)KVdQ~M#}S(E!_M!b98s|f#@Ho7d8=&tSL7CdmwD8>>ceZ_!>2m5F;%K-KlWDS z$(lQq8a+SiElhQ+-`0cRbe|K0HKzkaCTwVtxiU#HmtOPUhs#5kvhhJ*2PljE7Jpg7(!cCtEX5K(6Qi! z%3rk8Fd0Sj=-zOj$i@m8e^6BG3azK#>c4oTGK5U}d_jO;T8llUBO>+M*gkjHjhF0 Date: Sat, 18 Jul 2020 13:35:36 -0700 Subject: [PATCH 110/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 21 +++++++++++---------- 1 file changed, 11 insertions(+), 10 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 2c9393c..1e52674 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -13,21 +13,21 @@ tags: > The Dürr-Høyer algorithm takes an unsorted table of integers of length N and finds the minima in O(sqrt(N)). > I started a project that implements this algorithm here : [https://github.com/mridulsar/DurrHoyerLibrary](https://github.com/mridulsar/DurrHoyerLibrary), contributions welcome! -## Introduction +### Introduction Hello all! -My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Scientific and Mathematical Computation. +My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. -This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum mechanics has been a rewarding experience. +This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. https://github.com/mridulsar/DurrHoyerLibrary. -## The Dürr–Høyer Algorithm -Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in ''A quantum algorithm for finding the minimum":[(1)], where it was called the 'Quantum Minimum Searching Algorithm'. -I'll summarize the main result below in case you don't have time to read the paper in all of its glory. - +### The Dürr–Høyer Algorithm +Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" [(1)], where it was called the 'Quantum Minimum Searching Algorithm'. +I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. +## QMSA 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c.
2(a) Initialize the memory as a uniform superposition of qubits.
@@ -39,7 +39,7 @@ This marks all T[j] If the integer in the table at index y' is less than the integer at y, set y = y'.
3. Return y. - +## Implementing QMSA The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. @@ -59,6 +59,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. For the following circuits assume our Random Index is 0. +# QESA The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. ![2 items in a list](/assets/images/2-item-list.gif) @@ -108,7 +109,7 @@ The last step is another conditional phase shift, though it is applied if the qu } } ``` - +# QESA Script Our Q# script will be structured as follows: ``` @@ -209,7 +210,7 @@ To circumvent this we introduce the following implementation, utilizing QFT. Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. - +## QMSA Script ``` class DH(object): From 6440f6c976342acd8fe19d70c004045cc50614f6 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 13:36:45 -0700 Subject: [PATCH 111/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 1e52674..7edc6f1 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -13,7 +13,7 @@ tags: > The Dürr-Høyer algorithm takes an unsorted table of integers of length N and finds the minima in O(sqrt(N)). > I started a project that implements this algorithm here : [https://github.com/mridulsar/DurrHoyerLibrary](https://github.com/mridulsar/DurrHoyerLibrary), contributions welcome! -### Introduction +# Introduction Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. @@ -23,7 +23,7 @@ I would like to pass on what I have seen and understood in that time, as well as I will likely be posting about this algorithm and library more as the library and my understanding continues developing. https://github.com/mridulsar/DurrHoyerLibrary. -### The Dürr–Høyer Algorithm +# The Dürr–Høyer Algorithm Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" [(1)], where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. @@ -59,7 +59,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. For the following circuits assume our Random Index is 0. -# QESA +### QESA The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. ![2 items in a list](/assets/images/2-item-list.gif) @@ -109,7 +109,7 @@ The last step is another conditional phase shift, though it is applied if the qu } } ``` -# QESA Script +### QESA Script Our Q# script will be structured as follows: ``` @@ -210,7 +210,7 @@ To circumvent this we introduce the following implementation, utilizing QFT. Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. -## QMSA Script +### QMSA Script ``` class DH(object): @@ -258,7 +258,7 @@ Here is the python host that will apply the conditions of QMSA while using QESA. ``` -### Motivation +# Motivation The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. @@ -279,7 +279,7 @@ I am programming the algorithm in Q# with a python host script. The circuit's gates will be analyzed and some code is given. https://github.com/mridulsar/DurrHoyerLibrary -### My Background +# My Background I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. @@ -288,7 +288,7 @@ I began with a Coursera course from St. Petersburg University on quantum computi From here I tinkered with Q# and developed basic algorithms that I previously learned. I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. -### Conclusion +# Conclusion So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. From 94f37f9e46f478f3f4028def629e0febad59a4de Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 13:37:33 -0700 Subject: [PATCH 112/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 7edc6f1..dc40582 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -13,7 +13,7 @@ tags: > The Dürr-Høyer algorithm takes an unsorted table of integers of length N and finds the minima in O(sqrt(N)). > I started a project that implements this algorithm here : [https://github.com/mridulsar/DurrHoyerLibrary](https://github.com/mridulsar/DurrHoyerLibrary), contributions welcome! -# Introduction +## Introduction Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. @@ -23,11 +23,11 @@ I would like to pass on what I have seen and understood in that time, as well as I will likely be posting about this algorithm and library more as the library and my understanding continues developing. https://github.com/mridulsar/DurrHoyerLibrary. -# The Dürr–Høyer Algorithm +## The Dürr–Høyer Algorithm Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" [(1)], where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. -## QMSA +### QMSA 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c.
2(a) Initialize the memory as a uniform superposition of qubits.
@@ -39,7 +39,7 @@ This marks all T[j] If the integer in the table at index y' is less than the integer at y, set y = y'.
3. Return y. -## Implementing QMSA +### Implementing QMSA The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. @@ -59,7 +59,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, Step 2(a) has been stasified. Now we must figure out how to apply the QESA algorithm. For the following circuits assume our Random Index is 0. -### QESA +#### QESA The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. ![2 items in a list](/assets/images/2-item-list.gif) @@ -109,7 +109,7 @@ The last step is another conditional phase shift, though it is applied if the qu } } ``` -### QESA Script +#### QESA Script Our Q# script will be structured as follows: ``` @@ -210,7 +210,7 @@ To circumvent this we introduce the following implementation, utilizing QFT. Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. -### QMSA Script +#### QMSA Script ``` class DH(object): @@ -258,7 +258,7 @@ Here is the python host that will apply the conditions of QMSA while using QESA. ``` -# Motivation +## Motivation The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. @@ -279,7 +279,7 @@ I am programming the algorithm in Q# with a python host script. The circuit's gates will be analyzed and some code is given. https://github.com/mridulsar/DurrHoyerLibrary -# My Background +## My Background I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. @@ -288,7 +288,7 @@ I began with a Coursera course from St. Petersburg University on quantum computi From here I tinkered with Q# and developed basic algorithms that I previously learned. I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. -# Conclusion +## Conclusion So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. From c050b2a9eba1bfde903be324036a25ab37add14b Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 15:39:25 -0700 Subject: [PATCH 113/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 188 ++++++++++++++----------- 1 file changed, 106 insertions(+), 82 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index dc40582..873988a 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -80,13 +80,15 @@ We then apply a conditional phase shift if the qubit is 0. // Reflect qubits that are 0s ApplyConditionalPhase_0(LittleEndian(Register)); - operation ApplyConditionalPhase_0(register: LittleEndian) : Unit is Adj + Ctl - { - using (aux = Qubit()) - { - (ControlledOnInt(0,Z))(register!,aux); // If qubit is 0 flip it! - } - } + operation ApplyConditionalPhase_0(register: LittleEndian) : Unit is Adj + Ctl + { + using (aux = Qubit()) + { + // If qubit is 0 flip it! + (ControlledOnInt(0,X))(register!,aux); + } + } + ``` From here we apply the inverse of the Hadmard, this is just the conjugate transpose since Hadamard is unitary. This can be implemented be calling the Adjunct of Hadamard. @@ -96,19 +98,11 @@ This can be implemented be calling the Adjunct of Hadamard. ApplyToEachA(H,Register); ``` The last step is another conditional phase shift, though it is applied if the qubit is 1. +This can be done by using the Z gate. +The Z gate takes our qubit in and checks if it is |0> or |1>. +If |0> leave it be. +If |1> map to |0>. -``` - // Reflect qubits that are 1s - ApplyConditionalPhase(LittleEndian(Register)); - - operation ApplyConditionalPhase(register : LittleEndian) : Unit is Adj + Ctl - { - using (aux = Qubit()) - { - (ControlledOnInt(1,Z))(register!,aux); // If qubit is 1 flip it! - } - } -``` #### QESA Script Our Q# script will be structured as follows: @@ -125,50 +119,77 @@ Our Q# script will be structured as follows: open Microsoft.Quantum.Core; open Microsoft.Quantum.Convert; operation Algorithm_Even(TableLength : Int, RandomIndex : Int) : Unit + { + // intialize register to number of qubits as there are indices + using ((Register) = (Qubit[TableLength])) { - // intialize register to number of qubits as there are indices - using 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Now we must figure out how to apply the QESA algor For the following circuits assume our Random Index is 0. #### QESA The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. - +We can visuailze our algorithm with a circuit design. +Below is the QESA circuit. +![QESA circuit](/assests/images/QESA.jpeg) +Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. ![2 items in a list](/assets/images/2-item-list.gif) -If we have an odd number of table elements we use this circuit, where QFT is the approixmate Fourier transform as given by Kitaev [(3)]. -![3 items in a list](/assets/images/3-item-list.gif) Lets break down whats going on here. We utilize the register we were working with earlier and apply a H transform, this is stated to simply be the Hadmard. 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zsRMMMj*owzdu8aacn#cZ&DXit&HQ#-W7+p-?Q9te+<*5>r`eOuvqT3HPvsR@Z9RYL z1bVEeR0;n9&0O&-2~$lDDw=&%-36((Rax@k^QyVE$C*rpt&$VFErhabiI3}wy?sS# zIGc6!fW&xbx^eKetZQhc=N0UXBv*+Zt6MkdQ(w*^LesygO+R#bdFjCP&9T>o78F&U znX^vAUD*u;-b;&X}>jVdDg@mYuhAP6)Kt_jjM1)Lqv|MbgLVUb(a=$nrAr219g=Tk3&#(acuB< z)v`FNa1eTIe@Bml2(gZ?^Hf_4uc0Qb-qelv$;4WdXkJp?MId4IH*IMYG2OVQ#K&y@ z=9PzZv!M Date: Sat, 18 Jul 2020 16:00:01 -0700 Subject: [PATCH 123/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 64965ae..d9a8aa1 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -63,7 +63,7 @@ For the following circuits assume our Random Index is 0. The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. We can visuailze our algorithm with a circuit design. Below is the QESA circuit. -![QESA circuit](/assests/images/QESA.jpeg) +![QESA circuit](/assests/images/QESA.JPG) Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. ![2 items in a list](/assets/images/2-item-list.gif) @@ -119,7 +119,7 @@ Our Q# script will be structured as follows: open Microsoft.Quantum.Math; open Microsoft.Quantum.Core; open Microsoft.Quantum.Convert; - operation Algorithm_Even(TableLength : Int, RandomIndex : Int) : Unit + operation Algorithm_Even(TableLength : Int, RandomIndex : Int) : Int { // intialize register to number of qubits as there are indices using ((Register) = (Qubit[TableLength])) @@ -149,7 +149,7 @@ Our Q# script will be structured as follows: } - operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Unit + operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Int { // intialize register to number of qubits as there are indices using ((Register) = (Qubit[TableLength])) @@ -202,7 +202,7 @@ With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned ident To circumvent this we introduce the following implementation, utilizing QFT. #### QESA special case ``` - operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Unit + operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Int { // intialize register to number of qubits as there are indices using ((Register) = (Qubit[TableLength])) From 4e2b8280b6bff7f1849548525ebbca7ebf04f15a Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 16:01:19 -0700 Subject: [PATCH 124/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index d9a8aa1..bee1ac0 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -62,10 +62,10 @@ For the following circuits assume our Random Index is 0. #### QESA The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. We can visuailze our algorithm with a circuit design. -Below is the QESA circuit. +Below is the QESA circuit. ![QESA circuit](/assests/images/QESA.JPG) -Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. -![2 items in a list](/assets/images/2-item-list.gif) +Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. +![2 items in a list](/assets/images/2-item-list0.gif) Lets break down whats going on here. 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The QESA algorithm is a generalized Grover's search characterized by the following circuit for an even number of table elements. We can visuailze our algorithm with a circuit design. Below is the QESA circuit. + ![QESA circuit](/assests/images/QESA.JPG) -Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. + +Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. + ![2 items in a list](/assets/images/2-item-list0.gif) From 0501ce52e3556389cd101afe709dcf3f9556496b Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 16:06:08 -0700 Subject: [PATCH 128/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index f50c1ed..26dc575 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -64,7 +64,7 @@ The QESA algorithm is a generalized Grover's search characterized by the followi We can visuailze our algorithm with a circuit design. Below is the QESA circuit. -![QESA circuit](/assests/images/QESA.JPG) +![QESA circuit](/assets/images/QESA.JPG) Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. From 65d12e858aa108ead0836db84cdaf9f591ce7b82 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 16:11:32 -0700 Subject: [PATCH 129/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 26dc575..9416f86 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -310,8 +310,9 @@ https://github.com/mridulsar/DurrHoyerLibrary ## My Background I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. -Recently, I started looking into quantum machine learning where you can model nodes of neural networks as qubits. +About a year ago, I started looking into a model that treats nodes of neural networks as qubits. I took a deep dive into the research papers and found myself amazed by what was out there. +Realizing I had hunger to learn more I looked for online courses in quantum computing. I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. From here I tinkered with Q# and developed basic algorithms that I previously learned. I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. @@ -340,9 +341,9 @@ https://github.com/mridulsar/DurrHoyerLibrary ## My current challenges in this project: -1. Implementing generalized Grover's search for multiple solutions +1. Implementing QESA for multiple solutions -2. Developing feature for users to control the time limit that QMSA runs for +2. Fluid QMSA time manipulation ------------------ From 132c7f06479241d85149978844c25d48cc8a70fc Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sat, 18 Jul 2020 19:06:31 -0700 Subject: [PATCH 130/152] Add files via upload From 2f91f713a2aadbbb9f8352dd992827b6b0414114 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 19 Jul 2020 08:21:28 -0700 Subject: [PATCH 131/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 161 ++++++++----------------- 1 file changed, 51 insertions(+), 110 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 9416f86..28cc816 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -15,36 +15,30 @@ tags: ## Introduction -Hello all! -My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. -I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. -This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. -I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. -I will likely be posting about this algorithm and library more as the library and my understanding continues developing. -https://github.com/mridulsar/DurrHoyerLibrary. - -## The Dürr–Høyer Algorithm -Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" [(1)], where it was called the 'Quantum Minimum Searching Algorithm'. -I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. - -### QMSA -1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length -2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c.
- 2(a) Initialize the memory as a uniform superposition of qubits.
-Each qubit represents an index. -After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. -This marks all T[j] - 2(b) Apply the quantum exponential searching algorithm [(2)], which is a generalized Grover's search.
- 2(c) Measure the first register, call that outcome y' which is an index into the table. -If the integer in the table at index y' is less than the integer at y, set y = y'.
- -3. Return y. +Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. +https://github.com/mridulsar/DurrHoyerLibrary. +##The Dürr–Høyer Algorithm + +Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. +## QMSA + + Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length + + Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. + + 2(a) Initialize the memory as a uniform superposition of qubits. + Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. This marks all T[j] or |1>. -If |0> leave it be. -If |1> map to |0>. - -#### QESA Script + +The last step is another conditional phase shift, though it is applied if the qubit is 1. This can be done by using the Z gate. The Z gate takes our qubit in and checks if it is |0> or |1>. If |0> leave it be. If |1> map to |0>. + +### QESA Script + Our Q# script will be structured as follows: -``` namespace QESA { open Microsoft.Quantum.Intrinsic; @@ -195,16 +176,12 @@ Our Q# script will be structured as follows: } -``` -For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. -It is important to note the above algorithm only works for a table with an even number of entries. +For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. It is important to note the above algorithm only works for a table with an even number of entries. + +The algorithm breaks down when applying the Hadamard gate as the Hadamard is layed across the diagnoal of a identity matrix which is equal in dimensions to the number of qubits we have. With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned identity matrix the transformation is not retained. To circumvent this we introduce the following implementation, utilizing QFT. +QESA special case -The algorithm breaks down when applying the Hadamard gate as the Hadamard is layed across the diagnoal of a identity matrix which is equal in dimensions to the number of qubits we have. -With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned identity matrix the transformation is not retained. -To circumvent this we introduce the following implementation, utilizing QFT. -#### QESA special case -``` operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Int { // intialize register to number of qubits as there are indices @@ -234,12 +211,12 @@ To circumvent this we introduce the following implementation, utilizing QFT. } } -``` + Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). -Here is the python host that will apply the conditions of QMSA while using QESA. -#### QMSA Script -``` +Here is the python host that will apply the conditions of QMSA while using QESA. +QMSA Script + class DH(object): def __init__(self,table): @@ -284,56 +261,20 @@ Here is the python host that will apply the conditions of QMSA while using QESA. return y -``` - ## Motivation -The motivation behind this project is to provide open source functionality. -The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. -On a classical computer it will take as many time steps as there are items to find a minimum. -This means the Big-O is N. -The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. -For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. -The Dürr-Høyer algorithm would find the minimum in 3 steps. -We can understand this effecieny as the Big-O. -The Big-O essentially means the run time of the program will increase as the table size grows. -Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. -This success rate is bounded by the Big-O. -If we decide to alleviate this Big-O the probability of finding the minimum increases. -Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. -With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. -In order for this library to be used properly it must meet some guidelines. -At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. -I am programming the algorithm in Q# with a python host script. -The circuit's gates will be analyzed and some code is given. -https://github.com/mridulsar/DurrHoyerLibrary + +The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. This means the Big-O is N. The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this effecieny as the Big-O. The Big-O essentially means the run time of the program will increase as the table size grows. Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python host script. The circuit's gates will be analyzed and some code is given. +https://github.com/mridulsar/DurrHoyerLibrary ## My Background -I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. -I was instantly pulled in by the elegant algorithms and proposed efficiency. -About a year ago, I started looking into a model that treats nodes of neural networks as qubits. -I took a deep dive into the research papers and found myself amazed by what was out there. -Realizing I had hunger to learn more I looked for online courses in quantum computing. -I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. -From here I tinkered with Q# and developed basic algorithms that I previously learned. -I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. + +I began my journey from math to computers through my passion for ML algorithms, where I stumbled upon quantum computing. I was instantly pulled in by the elegant algorithms and proposed efficiency. About a year ago, I started looking into a model that treats nodes of neural networks as qubits. I took a deep dive into the research papers and found myself amazed by what was out there. Realizing I had hunger to learn more I looked for online courses in quantum computing. I began with a Coursera course from St. Petersburg University on quantum computing with detail on quantum algorithm design and quantum computer architecture. From here I tinkered with Q# and developed basic algorithms that I previously learned. I found Dürr and Høyer's paper and saw it as an fusion of Duestch's, Shor's, and Grover's Algorithm; a perfect next step for me. ## Conclusion -So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. -I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. -In specific, I mentioned I felt Dürr and Høyer used principles from Duestch's, Grover's, and Shor's algorithm. -Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function as done in Dürr and Høyer's algorithm to mark all states that satisfy T[j] Date: Sun, 19 Jul 2020 08:22:45 -0700 Subject: [PATCH 132/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 28 +++++++++++++------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 28cc816..b24ecd9 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -38,7 +38,7 @@ Say we have a table with N unsorted items where you want o find the minimul valu The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. - +``` // intialize register to number of qubits as there are indices using ((Register) = (Qubit[TableLength])) { @@ -49,7 +49,7 @@ In order to intialize the register we pepare a uniform superposition of qubits, // Apply Oracle to flip all states that are T[j] or |1>. If |0> leave it be. If |1> map to |0>. ### QESA Script Our Q# script will be structured as follows: - +``` namespace QESA { open Microsoft.Quantum.Intrinsic; open Microsoft.Quantum.Diagnostics; @@ -175,13 +175,13 @@ Our Q# script will be structured as follows: } - +``` For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. It is important to note the above algorithm only works for a table with an even number of entries. The algorithm breaks down when applying the Hadamard gate as the Hadamard is layed across the diagnoal of a identity matrix which is equal in dimensions to the number of qubits we have. With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned identity matrix the transformation is not retained. To circumvent this we introduce the following implementation, utilizing QFT. QESA special case - +``` operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Int { // intialize register to number of qubits as there are indices @@ -211,12 +211,12 @@ QESA special case } } - +``` Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. QMSA Script - +``` class DH(object): def __init__(self,table): @@ -260,7 +260,7 @@ QMSA Script y = y_prime return y - +``` ## Motivation The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. This means the Big-O is N. The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this effecieny as the Big-O. The Big-O essentially means the run time of the program will increase as the table size grows. Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O the probability of finding the minimum increases. Essentialy if we let the algorithm run for longer than the time it takes to guarantee a 50% success rate, we will get a success rate greater than 50 %. With this comes some complexity in the inner workings of the library which will be explained when analyzing the algorithm. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python host script. The circuit's gates will be analyzed and some code is given. From 8fdc6426eb5b13e70d9218f6ec9691f429c8850b Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 19 Jul 2020 08:23:36 -0700 Subject: [PATCH 133/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index b24ecd9..363c03e 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -22,17 +22,17 @@ https://github.com/mridulsar/DurrHoyerLibrary. Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. ## QMSA - Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length + 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length - Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. + Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. - 2(a) Initialize the memory as a uniform superposition of qubits. + 2(a) Initialize the memory as a uniform superposition of qubits. Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. This marks all T[j] Date: Sun, 19 Jul 2020 08:26:55 -0700 Subject: [PATCH 134/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 363c03e..639148b 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -57,10 +57,10 @@ The QESA algorithm is a generalized Grover's search characterized by the followi We can visuailze our algorithm with a circuit design. Below is the QESA circuit. ### QESA circuit +![QESA Circuit](assets/images/QESA.JPG). Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. - -2 items in a list +![QESA](assets/images/2-item-list0.gif). Lets break down whats going on here. We utilize the register we were working with earlier and apply a H transform, this is stated to simply be the Hadmard. ``` From ff70dcdc53ff3085666c40a83a45a33eedc7d53f Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 19 Jul 2020 08:28:16 -0700 Subject: [PATCH 135/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 639148b..59eeb9c 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -57,10 +57,10 @@ The QESA algorithm is a generalized Grover's search characterized by the followi We can visuailze our algorithm with a circuit design. Below is the QESA circuit. ### QESA circuit -![QESA Circuit](assets/images/QESA.JPG). +![QESA Circuit](/assets/images/QESA.JPG). Here is the QESA circuit treated as a gate with the intialization on qubits as mentioned above. -![QESA](assets/images/2-item-list0.gif). +![QESA](/assets/images/2-item-list0.gif). Lets break down whats going on here. We utilize the register we were working with earlier and apply a H transform, this is stated to simply be the Hadmard. ``` From e2988a9b491ba0618fc21175f9122d0b23ec4902 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 19 Jul 2020 08:32:59 -0700 Subject: [PATCH 136/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 59eeb9c..bc84750 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -215,7 +215,7 @@ QESA special case Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. -QMSA Script +## QMSA Script ``` class DH(object): From 963b2b55ce91079b692d134e9e921ccc8f8f8b96 Mon Sep 17 00:00:00 2001 From: mertall <42987995+mertall@users.noreply.github.com> Date: Sun, 19 Jul 2020 08:33:28 -0700 Subject: [PATCH 137/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index bc84750..fbbdb2c 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -180,7 +180,7 @@ Our Q# script will be structured as follows: For futher information on how this was derived take a look at 'Tight bounds on quantum searching' [2]. It is important to note the above algorithm only works for a table with an even number of entries. The algorithm breaks down when applying the Hadamard gate as the Hadamard is layed across the diagnoal of a identity matrix which is equal in dimensions to the number of qubits we have. With a bit of math, if we try to lay a 2x2 matrix along an odd dimensioned identity matrix the transformation is not retained. To circumvent this we introduce the following implementation, utilizing QFT. -QESA special case +### QESA special case ``` operation Algorithm_Odd(TableLength : Int, RandomIndex : Int) : Int { @@ -215,7 +215,7 @@ QESA special case Now we refer back to our QMSA outline to observe that the Algorithm(TableLenght,RandomIndex) is iterated on until we find a suitable y' or we simply hit our time limit. The true stars of this algorithm are the time limit, which guarantees O(sqrt(N)), the generalized Grovers algorithm given in QESA, which provides for easy implementation and has O(1) for each iteration, and lastly, the oracle function which marks our states, which along with our intialization of qubits has O(log(n)). Here is the python host that will apply the conditions of QMSA while using QESA. -## QMSA Script +### QMSA Script ``` class DH(object): From 044fe1a8b3255a77a1d1a5f82da27261ad2db93d Mon Sep 17 00:00:00 2001 From: mertall Date: Thu, 20 Aug 2020 12:56:44 -0700 Subject: [PATCH 138/152] Delete QESA.JPG --- assets/images/QESA.JPG | Bin 15101 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assets/images/QESA.JPG diff --git a/assets/images/QESA.JPG b/assets/images/QESA.JPG deleted file mode 100644 index 42d0edb3d274c5c2df50c6d4f9bfa27c86728a66..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 15101 zcmeHO2{@Ho+g{toHWkX8DMMu18FuFBlsQ5JWhgW4Y;$ItgrW$glCekwM>Y|M|~#{nux^mc8EheY|Tu_j=yvzMu782;GDMK%=Rup$cFy z7&r_40Ky>6fQpBmH2~V$KmY(h28dy@fC%a#paabiJ3tKeVNhQ`E`H-1>;!=3^#tqJ z>r~LO0@MKO1oVq+qqqIpMqnF(Z3MOv*hXL*f&WGXw4LoN-OfXvlU#>?RT6ZNVo179 z{A)iyD)MU|jyecHRMc<%-&6(JjSb|sRmFc}{A~NYjlebn+X!qUu#LcP2#6gJJ0NpF 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zO4Eiog*MM8;^W~9#OdS!k&RF3D4UHLEGB_VMrrKABk$kW)YjJ2)xEE$r)6NEWn`pfY^-f!qHSub zZDyuzZmw-%p$&!Z3I^MirRA=yto|lz>j!(%v9STtwXxB&wbi$^HL$ZYw6im^w>Q~q zfX$tppw7-#aJY@DtG&Cslcy)#+uPm8$NRt1>qel_h=c?r=x+f}B`XV^ld}uA&d-l6 zEQ~EKjw>mNFD;EPD+4Q?e{}`4q5`0)0kV2ufM6X5THihWcXmO4TO5r6JnX8fgz9R* zRoHEVL9e^l=$3#NW8jq@80Z9{o1Z>Sg6Iffjs~&6`=02(yzu$+w&xcjagIowBoYUS z#4aMSl}LQ?^QtG2xXaH5`g!5=$IfTi*WaE|9Bqv-1Mr4SJ1hC&@6Od&uKEq}JES*7 zp5{tb3XB++kEYVB7vpy{lSf@N9&cB0X8 zC>Z`lALoC&SJqYY>k#(Yk&B@cU+uNp><)D|39)`LYz{mle^Ekf>XTmv=e=IK7)Qgm zaa=Cjv98Xqo6D8QuX)*vD~$^Ij^=j^rrHfRKIA)6=UF*Bo*!_-+)I@3YT`p2&%^xq z%Z`h09vpXJ`Ig}6nlRCc(tKFA;nX)@5hs4>G8e`XcoeyIjw&0Pn4W|~oh(k?HFhvH NwNbZ~hnkUG{tvjoq&ffq literal 0 HcmV?d00001 From e19c07dcc95d52aea11793d3d6d7a881b5f7814e Mon Sep 17 00:00:00 2001 From: mertall Date: Thu, 20 Aug 2020 13:09:01 -0700 Subject: [PATCH 141/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 250 +++++-------------------- 1 file changed, 50 insertions(+), 200 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index fbbdb2c..58ce34e 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -16,70 +16,73 @@ tags: ## Introduction Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. -https://github.com/mridulsar/DurrHoyerLibrary. -##The Dürr–Høyer Algorithm +https://github.com/mridulsar/DurrHoyerLibrary. + +## The Dürr–Høyer Algorithm Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. -## QMSA + +## QMSA 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. 2(a) Initialize the memory as a uniform superposition of qubits. - Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle given by Grover. This marks all T[j]) = |->, so don't neet to reset, but the phase kickback will remain. + (ControlledOnInt(0, X))(register!, aux); + } } - } ``` From here we apply the inverse of the Hadmard, this is just the conjugate transpose since Hadamard is unitary. This can be implemented be calling the Adjunct of Hadamard. ``` @@ -88,182 +91,29 @@ From here we apply the inverse of the Hadmard, this is just the conjugate transp ``` The last step is another conditional phase shift, though it is applied if the qubit is 1. This can be done by using the Z gate. The Z gate takes our qubit in and checks if it is |0> or |1>. If |0> leave it be. If |1> map to |0>. -### QESA Script - -Our Q# script will be structured as follows: - ``` - namespace QESA { - open Microsoft.Quantum.Intrinsic; - open Microsoft.Quantum.Diagnostics; - open Microsoft.Quantum.Arrays; - open Microsoft.Quantum.Preparation; - open Microsoft.Quantum.Canon; - open Microsoft.Quantum.Arithmetic; - open Microsoft.Quantum.Math; - open Microsoft.Quantum.Core; - open Microsoft.Quantum.Convert; - operation Algorithm_Even(TableLength : Int, RandomIndex : Int) : Int - { - // intialize register to number of qubits as there are indices - using ((Register) = (Qubit[TableLength])) - { - within - { - // Create Uniform Superposition of all indices - PrepareUniformSuperposition(TableLength,LittleEndian(Register)); - // Grab the qubit in the RandomIndex and set it aside - let Marker = Register[RandomIndex]; - // Apply Oracle to flip all states that are T[j] Date: Thu, 20 Aug 2020 13:11:23 -0700 Subject: [PATCH 142/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 33 +++++++++++++------------- 1 file changed, 17 insertions(+), 16 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 58ce34e..340df52 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -16,24 +16,24 @@ tags: ## Introduction Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. -https://github.com/mridulsar/DurrHoyerLibrary. +https://github.com/mridulsar/DurrHoyerLibrary. ## The Dürr–Høyer Algorithm -Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. +Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. ## QMSA 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length - Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. + Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. 2(a) Initialize the memory as a uniform superposition of qubits. - Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle that marks all T[j] Date: Thu, 20 Aug 2020 13:13:07 -0700 Subject: [PATCH 143/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 15 +++++++-------- 1 file changed, 7 insertions(+), 8 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 340df52..c197467 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -24,16 +24,15 @@ Say we have a table with N unsorted items where you want o find the minimul valu ## QMSA - 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length - - Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. + ### 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length. + + ### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. - 2(a) Initialize the memory as a uniform superposition of qubits. - Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle that marks all T[j] Date: Thu, 20 Aug 2020 13:14:27 -0700 Subject: [PATCH 144/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index c197467..4043134 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -24,15 +24,15 @@ Say we have a table with N unsorted items where you want o find the minimul valu ## QMSA - ### 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length. + #### 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length. - ### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. + #### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. - ### 2(a) Initialize the memory as a uniform superposition of qubits. Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle that marks all T[j] Date: Thu, 20 Aug 2020 13:14:57 -0700 Subject: [PATCH 145/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 4043134..73c23d2 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -26,7 +26,7 @@ Say we have a table with N unsorted items where you want o find the minimul valu #### 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length. - #### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() whcih returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. + #### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() which returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. ####    2(a) Initialize the memory as a uniform superposition of qubits. Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle that marks all T[j] Date: Thu, 20 Aug 2020 13:16:57 -0700 Subject: [PATCH 146/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 73c23d2..e9569b5 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -36,9 +36,9 @@ Say we have a table with N unsorted items where you want o find the minimul valu ### Implementing QMSA -The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. +The steps 2(a) and 2(b) pose the biggest challenge if someone has no experience with Quantum Computing. We will first observe how to initalize the register. Then we will see how the QESA can be implemented to find a unique solution, in this case the minimum. Here is a python file implementing QMSA: https://github.com/mertall/DurrHoyerLibrary/blob/master/library/QMSA.py. -## Initializing our Qubits +### Initializing our Qubits In order to intialize the register we pepare a uniform superposition of qubits, the number of qubits is determined by the number of elements in our table. We then grab our y-th index and entangle it with our register using a Controlled Z. ``` @@ -101,12 +101,10 @@ Now we refer back to our QMSA outline to observe that the Algorithm(TableLength, The full script for QESA can be found here: https://github.com/mertall/DurrHoyerLibrary/blob/master/library/Library.qs. -## Measuring register +### Measuring register At the moment we are measuring our register by using MeasureInteger(), though this is decoding LittleEndian into an integer as is represented by some binary string. We will have to develop a different way to measure our qubits to get probabilistic results on each qubit. -## QMSA - To implement QESA we must use a python host file which can be found here: https://github.com/mertall/DurrHoyerLibrary/blob/master/library/QMSA.py. ## Motivation From 9cb66c7042a28c7d5390c17a82e0f8bff048ee56 Mon Sep 17 00:00:00 2001 From: mertall Date: Thu, 20 Aug 2020 13:17:26 -0700 Subject: [PATCH 147/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index e9569b5..f7debeb 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -103,9 +103,7 @@ The full script for QESA can be found here: https://github.com/mertall/DurrHoyer ### Measuring register -At the moment we are measuring our register by using MeasureInteger(), though this is decoding LittleEndian into an integer as is represented by some binary string. We will have to develop a different way to measure our qubits to get probabilistic results on each qubit. - -To implement QESA we must use a python host file which can be found here: https://github.com/mertall/DurrHoyerLibrary/blob/master/library/QMSA.py. +At the moment we are measuring our register by using MeasureInteger(), though this is decoding LittleEndian into an integer as is represented by some binary string. We will have to develop a different way to measure our qubits to get probabilistic results on each qubit. ## Motivation From b7117c6cd3fbe8b6007412e4b52870eeb232d2e5 Mon Sep 17 00:00:00 2001 From: mertall Date: Thu, 20 Aug 2020 17:33:38 -0700 Subject: [PATCH 148/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index f7debeb..0608820 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -15,7 +15,7 @@ tags: ## Introduction -Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. +Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. https://github.com/mridulsar/DurrHoyerLibrary. ## The Dürr–Høyer Algorithm @@ -109,7 +109,7 @@ At the moment we are measuring our register by using MeasureInteger(), though th The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. This means the Big-O is N. The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. For example assume there are 9 items in a table, on a classical comptuer this would take 9 time steps. The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this effecieny as the Big-O. The Big-O essentially means the run time of the program will increase as the table size grows. Dürr and Høyer propose a quantum algorithms in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, meaning we let the algorithm run for longer, the probability of finding the minimum increases. -In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. I am programming the algorithm in Q# with a python host script. The circuit's gates will be analyzed and some code is given. +In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. https://github.com/mridulsar/DurrHoyerLibrary ## My Background From d8e7cb818e83e7387daa31ec291b50aea590aaa6 Mon Sep 17 00:00:00 2001 From: mertall Date: Fri, 28 Aug 2020 03:28:16 -0700 Subject: [PATCH 149/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 72 +++++++++++++------------- 1 file changed, 36 insertions(+), 36 deletions(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 0608820..25903a0 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -15,32 +15,31 @@ tags: ## Introduction -Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithim and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intution has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. -https://github.com/mridulsar/DurrHoyerLibrary. - +Hello all! My name is Mridul Sarkar, and I am an undergraduate at University of California Davis studying Mathematics and Scientific Computation. I wanted to share with the community my experiences in learning about the Dürr–Høyer algorithm and creating a library for Q# implementing it. This project has pushed me and is still pushing me outside of my comfort zone, but learning Q# while slowly refining my quantum intuition has been a rewarding experience. I would like to pass on what I have seen and understood in that time, as well as ask the community for feedback. I will likely be posting about this algorithm and library more as the library and my understanding continues developing. +https://github.com/mridulsar/DurrHoyerLibrary. + ## The Dürr–Høyer Algorithm -Say we have a table with N unsorted items where you want o find the minimul value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum" (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. +Say we have a table with N unsorted items where you want to find the minimum value stored in this table. The Dürr–Høyer Algorithm helps us solve this problem. It was originally proposed in "A quantum algorithm for finding the minimum'' (1), where it was called the 'Quantum Minimum Searching Algorithm'. I'll summarize the main ideas below in case you don't have time to read the paper in all of its glory. ## QMSA - #### 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length. - - #### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() which returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. - - ####    2(a) Initialize the memory as a uniform superposition of qubits. Each qubit represents an index. After intializing the memory grab the y-th qubit and entangle the state of your register with this y-th qubit according the the Oracle that marks all T[j] Date: Tue, 24 Nov 2020 18:39:19 -0800 Subject: [PATCH 150/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 25903a0..057fdfc 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -25,7 +25,7 @@ Say we have a table with N unsorted items where you want to find the minimum val ## QMSA #### 1. Choose an integer (y) uniformly at random between 0...N-1, where N = flattened table length. -#### 2. Repeat the following steps until time = 22.5 * sqrt(N) + 1.4 * log^2(N). Time can be easily captured in python using time.clock() which returns wall clock time since program was started. The time begins once we start step 2(a), if time equals the expression with N proceed to step c. +#### 2. Repeat the following steps until you find an answer. #### 2(a) Initialize the memory as a uniform superposition of qubits. Each qubit represents an index. After initializing the memory, grab the y-th qubit and entangle the state of your register with this y-th qubit according to the Oracle that marks all T[j] Date: Tue, 24 Nov 2020 18:40:56 -0800 Subject: [PATCH 151/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 057fdfc..832d316 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -107,7 +107,7 @@ At the moment we are measuring our register by using MeasureInteger(), though th ## Motivation -The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. This means the Big-O is N. The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. For example assume there are 9 items in a table, on a classical computer this would take 9 time steps. The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this efficiency as the Big-O. The Big-O essentially means the run time of the program will increase as the table size grows. Dürr and Høyer propose a quantum algorithm in order to find the minimum of an unsorted table with a 50% success rate. This success rate is bounded by the Big-O. If we decide to alleviate this Big-O, meaning we let the algorithm run for longer, the probability of finding the minimum increases. +The motivation behind this project is to provide open source functionality. The efficiency that is proposed by this algorithm is much better than a typical algorithm for finding the minimum of an unsorted table. On a classical computer it will take as many time steps as there are items to find a minimum. This means the Big-O is N. The Dürr-Høyer algorithm takes this problem and solves it in sqrt(N) time steps. For example assume there are 9 items in a table, on a classical computer this would take 9 time steps. The Dürr-Høyer algorithm would find the minimum in 3 steps. We can understand this efficiency as the Big-O. The Big-O essentially means the run time of the program will increase as the table size grows. Dürr and Høyer propose a quantum algorithm in order to find the minimum of an unsorted table with a 50% success rate. In order for this library to be used properly it must meet some guidelines. At the moment I am referencing the amazing template given by Dr. Sarah Kaiser https://github.com/crazy4pi314/qsharp-library-template in order to make this library usable in Q#. https://github.com/mridulsar/DurrHoyerLibrary From 80a477a5c12d1b13fc370d3a3a1c2271763b3cbd Mon Sep 17 00:00:00 2001 From: mertall Date: Tue, 24 Nov 2020 18:45:07 -0800 Subject: [PATCH 152/152] Update 2020-06-23-Durr_Hoyer_Post_1.md --- _posts/2020-06-23-Durr_Hoyer_Post_1.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2020-06-23-Durr_Hoyer_Post_1.md b/_posts/2020-06-23-Durr_Hoyer_Post_1.md index 832d316..3683fdb 100644 --- a/_posts/2020-06-23-Durr_Hoyer_Post_1.md +++ b/_posts/2020-06-23-Durr_Hoyer_Post_1.md @@ -120,7 +120,7 @@ I began my journey from math to computers through my passion for ML algorithms, So far I have been greatly enjoying throwing myself into this new and exciting world of quantum computing. I have seen some awesome principles translate across quantum algorithms that are keeping me engaged. In specific, I mentioned I felt Dürr and Høyer used principles from Duestch's, Grover's, and Shor's algorithm. Duestsch's Algorithm famously simplifies a classical problem into an non intuitive oracle function as done in Dürr and Høyer's algorithm to mark all states that satisfy T[j]

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