deutsch_jozsa.rst

Deutsch-Jozsa Algorithm

Overview

The Deutsch-Jozsa algorithm can determine whether a function mapping all bitstrings to a single bit is constant or balanced, provided that it is one of the two. A constant function always maps to either 1 or 0, and a balanced function maps to 1 for half of the inputs and maps to 0 for the other half. Unlike any deterministic classical algorithm, the Deutsch-Jozsa Algorithm can solve this problem with a single iteration, regardless of the input size. It was one of the first known quantum algorithms that showed an exponential speedup, albeit against a deterministic (non-probabilistic) classical compuetwr, and with access to a blackbox function that can evaluate inputs to the chosen function.

Algorithm and Details

This algorithm takes as input n qubits in state $\ket{x}$, an ancillary qubit in state $\ket{q}$, and additionally a quantum circuit U_w that performs the following:

$$U_w: \ket{x}\ket{q}\to\ket{x}\ket{f(x)\oplus q}$$

In the case of the Deutsch-Jozsa algorithm, the function f is some function mapping from bitstrings to bits:

f : {0, 1}ⁿ → {0, 1}

and is assumed to either be textit{constant} or textit{balanced}. Constant means that on all inputs f takes on the same value, and balanced means that on half of the inputs f takes on one value, and on the other half f takes on a different value. (Here the value is restricted to {0, 1})

We can then describe the algorithm as follows:

Input:

n + 1 qubits

Algorithm:

1. Prepare the textit{ancilla} ($\ket{q}$ above) in the $\ket{1}$ state by performing an X gate.
2. Perform the n + 1-fold Hadamard gate H^{⊗ n + 1} on the n + 1 qubits.
3. Apply the circuit U_w.
4. Apply the n-fold Hadamard gate H^⊗ n on the data qubits, $\ket{x}$.
5. Measure $\ket{x}$. If the result is all zeroes, then the function is constant. Otherwise, it is balanced.

Implementation Notes

The oracle in the Deutsch-Jozsa module is not implemented in such a way that calling Deutsch_Jozsa.is_constant() will yield an exponential speedup over classical implementations. To construct the quantum algorithm that is executing on the QPU we use a Quil defgate, which specifies the circuit U_w as its action on the data qubits $\ket{x}$. This matrix is exponentially large, and thus even generating the program will take exponential time.

Source Code Docs

Here you can find documentation for the different submodules in deutsch-jozsa.

grove.deutsch_jozsa.deutsch_jozsa.py

grove.deutsch_jozsa.deutsch_jozsa