lecture videos

source/07-stabilisers.Rmd

@@ -1,5 +1,7 @@
 # Stabilisers {#stabilisers}
 
+<div class="video" title="Stabilizers" data-videoid="HsizVGW2sk0"></div>
+
 > About the structure of the **Pauli group**, which is the group generated by tensor products of the Pauli matrices, including the identity.
 > It has nice algebraic properties which are useful in many areas of quantum information science, in particular quantum error correction and classical simulations of some types of quantum computation.
 > We will discuss how certain subgroups of the Pauli group, and in particular stabilisers and normalisers of these subgroups, slice the Pauli group into interesting cosets that have a group structure of their own.
@@ -631,6 +633,8 @@ As we shall soon see, Pauli stabilisers are not normal subgroups of $\mathcal{P}
 
 ## Pauli normalisers
 
+<div class="video" title="Normalizers (and centralizers)" data-videoid="_8QnQG0jS1Q"></div>
+
 There are two subgroups that pop up once we choose a stabiliser $\mathcal{S}$.
 The subgroup of $\mathcal{P}_n$ consisting of all elements that commute which every element of $\mathcal{S}$ is called the **centraliser of $\mathcal{S}$**, denoted by
 $$

source/13-correcting-quantum-channels.Rmd

@@ -1,5 +1,7 @@
 # Correcting quantum channels {#correcting-quantum-channels}
 
+<div class="video" title="Overview of quantum error correction" data-videoid="WJr7jqpjge0"></div>
+
 > About the one big problem that hinders us from physically implementing everything that we've learnt so far: **decoherence**.
 > But also about how we can start to deal with it via some elementary **error correction**, including the **Shor $[[9,1,3]]$ quantum code**, which generalises the classical **$[3,1,3]$-code**.
 
@@ -292,6 +294,10 @@ iii. there exists a set of orthogonal isometries $\{V_i\}$ and a probability dis
 
 <div class="video" title="Decoherence vs interference" data-videoid="ZPfdKNzjWbo"></div>
 
+<div class="video" title="Inverting quantum channels" data-videoid="dBqLrfFNJko"></div>
+
+<div class="video" title="Inverting quantum channels (revisited)" data-videoid="c4ruTbmVl_Q"></div>
+
 Consider the following interaction between a qubit and its environment:
 $$
   \begin{aligned}
@@ -637,6 +643,10 @@ If a quantum error correction method corrects errors $E_1$ and $E_2$, then it al
 
 ## Towards error correction {#towards-error-correction}
 
+<div class="video" title="Correctable errors" data-videoid="gJYfrnpLNHw"></div>
+
+<div class="video" title="Changing correctable errors" data-videoid="7RyusTzOw6Y"></div>
+
 In Section \@ref(three-qubit-codes), when Alice used a random choice of four isometries to produce a three-qubit output, notice how we can write
 $$
   \begin{aligned}
@@ -744,6 +754,8 @@ which, when we error correct, split into two cases: with probability $\cos^2\the
 
 ## The classical repetition code {#the-classical-repetition-code}
 
+<div class="video" title="Classical repetition codes" data-videoid="11VOJrHAlYI"></div>
+
 To give a sense of how quantum error correction actually works, we first need a brief detour to introduce some concepts and methods from the classical theory of error correction.^[Even though quantum problems often require innovative solutions, it is always a good idea to look at the classical world to see if there is anything analogous there, and how it is dealt with.]
 In this particular case, once we have digitised quantum errors, we can see that quantum decoherence is a bit like classical noise (i.e. bit-flips), except that we have *two* types of errors: bit-flips and phase-flips; the former essentially classical, the latter purely quantum.
 But we also know that phase-flips can be turned into bit-flips if sandwiched between Hadamard transforms: $HZH=X$.
@@ -902,6 +914,8 @@ For example, in our $3$-bit repetition code, we have the two codewords $000$ and
 
 ## Correcting bit-flips {#correcting-bit-flips}
 
+<div class="video" title="Three-qubit repetition code for bit-flip errors" data-videoid="9mr9c35xJ2g"></div>
+
 In order to protect a qubit against bit-flips (thought of as incoherent $X$ rotations), we rely on the same classical repetition code as in Section \@ref(the-classical-repetition-code), but both encoding and error correction are now implemented by quantum operations.^[All the codes we will study have encoding circuits that can be constructed out of controlled-$\texttt{NOT}$ and Hadamard gates: we are dealing with Clifford circuits (recall Section \@ref(clifford-walks-on-stabiliser-states)).]
 Let's return to the example we introduced in Section \@ref(three-qubit-codes).
 We take a qubit in some unknown pure state $\alpha\ket{0}+\beta\ket{1}$ and encode it into three qubits, introducing two auxiliary qubits:
@@ -1089,6 +1103,8 @@ $$
 
 ## Correcting phase-flips {#correcting-phase-flips}
 
+<div class="video" title="Three-qubit repetition code for phase-flip errors" data-videoid="az_JPhNpWFo"></div>
+
 We have seen how the classical $[3,1,3]$-code can be adapted to detect and correct for a single quantum bit flip, but in Section \@ref(quantum-errors) we said that there are three possible errors that we need to worry about: bit-flips, phase-flips, and bit-and-phase flips.
 Having dealt with the first, we now deal with the second; finding a way to combine these two solutions to deal with the third is the subject of Section \@ref(correcting-bit-phase-flips).
 
@@ -1328,6 +1344,8 @@ In general, the number of errors that can be corrected by a distance $d$ code is
 
 ## Correcting any single error: Shor [[9,1,3]] {#correcting-bit-phase-flips}
 
+<div class="video" title="Nine-qubit code" data-videoid="fX8J_n2pwS0"></div>
+
 In Section \@ref(composing-correctable-channels) we derived the encoding circuit for the Shor $[[9,1,3]]$-code, so now let's go from the top and put all the pieces together to understand how this gives an error correction procedure for all possible single-qubit errors.^[Although nine qubits is actually more than necessary (we can achieve the same result with a different scheme that only uses five), this code, proposed by Shor in 1995, allows us to more easily see what's really going on.]
 
 To start, we encode our qubit with the phase-flip code

source/14-fault-tolerant-quantum-computation.Rmd

@@ -6,30 +6,8 @@
 <!-- TO-DO -->
 :::
 
-<div class="video" title="Overview of quantum error correction" data-videoid="WJr7jqpjge0"></div>
-
-<div class="video" title="Classical repetition codes" data-videoid="11VOJrHAlYI"></div>
-
-<div class="video" title="Three-qubit repetition code for bit-flip errors" data-videoid="9mr9c35xJ2g"></div>
-
-<div class="video" title="Three-qubit repetition code for phase-flip errors" data-videoid="az_JPhNpWFo"></div>
-
-<div class="video" title="Nine-qubit code" data-videoid="fX8J_n2pwS0"></div>
-
-<div class="video" title="Inverting quantum channels" data-videoid="dBqLrfFNJko"></div>
-
-<div class="video" title="Inverting quantum channels (revisited)" data-videoid="c4ruTbmVl_Q"></div>
-
-<div class="video" title="Correctable errors" data-videoid="gJYfrnpLNHw"></div>
-
-<div class="video" title="Changing correctable errors" data-videoid="7RyusTzOw6Y"></div>
-
 <div class="video" title="Pauli operators in error correction" data-videoid="kd3ahY55DYA"></div>
 
-<div class="video" title="Stabilizers" data-videoid="HsizVGW2sk0"></div>
-
-<div class="video" title="Normalizers (and centralizers)" data-videoid="_8QnQG0jS1Q"></div>
-
 <div class="video" title="Stabilizer codes" data-videoid="jI8S-2oB7fg"></div>
 
 <div class="video" title="Seven-qubit code and transversal constructions" data-videoid="vZ5p_fJbTMc"></div>