Incorporate Lia's corrections to QM and QEC fundamentals

src/thesis/chapters/2_fundamentals.tex

@@ -751,7 +751,7 @@ As we are modelling the wave function $\psi(x,t)$ as a vector
 $\ket{\psi}$, we can find a set of basis vectors to decompose it into.
 We can use the determinate states for this purpose, expressing the state as%
 \footnote{
-    We are only considering the case of having a \emph{discrete
+    We only consider the case of having a \emph{discrete
     spectrum} here, i.e., having a discrete set of eigenvalues and vectors.
     For continuous spectra, the procedure is analogous.
 }
@@ -912,8 +912,8 @@ Assuming the qubits are independent, this is a \emph{product state}
 $\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$.
 When not ambiguous, we may omit the tensor product symbol or even write
 the entire product state as a single ket
-\cite[Sec.~6.2]{griffiths_consistent_2001}.
+\cite[Sec.~6.2]{griffiths_consistent_2001},
-We have
+i.e.,
 \begin{align}
     \label{eq:product_state}
     \begin{split}
@@ -1015,14 +1015,17 @@ operators are sufficient to express any other operator as a linear
 combination \cite[Sec.~2.2]{roffe_quantum_2019}.
 $I$ is the identity operator and $X$ and $Z$ are referred to as
 \emph{bit-flips} and \emph{phase-flips} respectively.
-We call the set $\mathcal{G}_n = \left\{ \pm I,\pm jI, \pm X,\pm jX,
+We call the set $\mathcal{G}_n = \left\{ \pm I,\pm \mathrm{i}I, \pm
-\pm Y,\pm jY, \pm Z, \pm jZ \right\}^{\otimes n}$ the \emph{Pauli
+    X,\pm \mathrm{i}X,
+\pm Y,\pm \mathrm{i}Y, \pm Z, \pm \mathrm{i}Z \right\}^{\otimes n}$
+the \emph{Pauli
 group} over $n$ qubits.
 
 In the context of modifying qubit states, we also call operators \emph{gates}.
 When working with multi-qubit systems, we can also apply Pauli gates
 to individual qubits independently, which we write, e.g., as $I_1 X_2
 I_3 Z_4 Y_5$.
+Each operator is applied to the qubit denoted in the corresponding subscript.
 We often omit the identity operators, instead writing, e.g., $X_2 Z_4 Y_5$.
 Other important operators include the \emph{Hadamard} and
 \emph{controlled-NOT (CNOT)} gates \cite[Sec.~1.3]{nielsen_quantum_2010}
@@ -1189,7 +1192,7 @@ Consider the two-qubit repetition code
     \underbrace{\ket{11}}_{=:\ket{1}_\text{L}}
     .%
 \end{align*}
-We call $\ket{\psi}_L$ the logical state, and
+We call $\ket{\psi}_\text{L}$ the logical state, and
 we define the \emph{codespace} as $\mathcal{C} := \text{span}\mleft\{
 \ket{00}, \ket{11} \mright\}$ and the \emph{error subspace} as
 $\mathcal{F} := \text{span} \mleft\{\ket{01}, \ket{10} \mright\}$.
@@ -1294,7 +1297,7 @@ This effect is referred to as error \emph{digitization}
 
 Operators such as $Z_1Z_2$ above are called \emph{stabilizers}.
 More generally, an operator $P_i \in \mathcal{G}_n$ is called a stabilizer of an
-$[[n, k, d_\text{min}]]$ code $\mathcal{C}$, if
+$\llbracket n, k, d_\text{min} \rrbracket$ code $\mathcal{C}$, if
 \begin{itemize}
     \item It stabilizes all logical states, i.e.,
         $P_i\ket{\psi}_\text{L} = (+1)\ket{\psi}_\text{L} ~\forall~
@@ -1318,20 +1321,35 @@ with respect to possible errors.
 The measurement circuit for an arbitrary stabilizer $P_i$ modifies
 the state as \cite[Eq.~29]{roffe_quantum_2019}
 \begin{align*}
-    E\ket{\psi}_\text{L}\ket{0}_\text{A}
+    E\ket{\psi}_\text{L}\ket{0}_{\text{A}_i}
     \hspace{3mm}\mapsto\hspace{3mm}
     \frac{1}{2} \left( I + P_i
-    \right)E\ket{\psi}_\text{L}\ket{0}_\text{A} + \frac{1}{2}
+    \right)E\ket{\psi}_\text{L}\ket{0}_{\text{A}_i} + \frac{1}{2}
-    \left( I - P_i \right)E\ket{\psi}_\text{A} \ket{1}_\text{A}
+    \left( I - P_i \right)E\ket{\psi}_\text{L} \ket{1}_{\text{A}_i}
     .%
 \end{align*}
 If a given error $E$ anticommutes with $P_i$, we have
 \begin{align*}
-    EP_i \ket{\psi}_{L} &= -P_i E \ket{\psi}_\text{L} \\
+    & \frac{1}{2} \left( I + P_i \right)
-    \Rightarrow E \ket{\psi}_{L} &= -P_i E \ket{\psi}_\text{L} \\
+    E\ket{\psi}_\text{L}\ket{0}_{\text{A}_i}
-    \Rightarrow \left( I + P_i \right)E\ket{\psi}_\text{L} &= 0
+    + \frac{1}{2} \left( I - P_i \right)
+    E\ket{\psi}_\text{L} \ket{1}_{\text{A}_i} \\
+    = & \frac{1}{2} \left(
+        E\ket{\psi}_\text{L} +  P_i E\ket{\psi}_\text{L}
+    \right) \ket{0}_{\text{A}_i}
+    + \frac{1}{2} \left(
+        E\ket{\psi}_\text{L} - P_i E\ket{\psi}_\text{L}
+    \right)\ket{1}_{\text{A}_i} \\
+    = & \frac{1}{2} \left(
+        E\ket{\psi}_\text{L} -  E\ket{\psi}_\text{L}
+    \right) \ket{0}_{\text{A}_i}
+    + \frac{1}{2} \left(
+        E\ket{\psi}_\text{L} + E\ket{\psi}_\text{L}
+    \right)\ket{1}_{\text{A}_i} \\
+    = & E\ket{\psi}_\text{L}\ket{1}_{\text{A}_i}
 \end{align*}
-and the stabilizer measurement returns 1.
+and measuring the ancilla $\text{A}_i$ corresponding to stabilizer
+$P_i$ returns 1.
 
 %%%%%%%%%%%%%%%%
 \subsection{Stabilizer Codes}
@@ -1468,8 +1486,8 @@ Using the dual code of $\mathcal{C}_2$ \cite[Eq.~3.4]{ryan_channel_2009}
     \bm{x}' \bm{x}^\text{T} = 0 ~\forall \bm{x} \in \mathcal{C}_2 \right\}
     ,%
 \end{align*}
-we define $\bm{H}_X := \bm{H}(\mathcal{C}_2^\perp)$ and $\bm{H}_Z
+we define $\bm{H}_X$ as the \ac{pcm} of $\mathcal{C}_2^\perp$ and $\bm{H}_Z$
-:= \bm{H}(\mathcal{C}_1)$, and construct the check matrix as
+as the \ac{pcm} of $\mathcal{C}_1$, and construct the check matrix as
 \begin{align*}
     \left[
         \begin{array}{c|c}
@@ -1486,8 +1504,8 @@ $\mathcal{C}_2$ must satisfy the commutativity condition
     \bm{H}_X \bm{H}_Z^\text{T} = \bm{0}
     .%
 \end{align}
-We can ensure this is the case by choosing them such that
+We can ensure this by choosing $\mathcal{C}_1$ and $\mathcal{C}_2$
-$\mathcal{C}_2 \subset \mathcal{C}_1$.
+such that $\mathcal{C}_2 \subset \mathcal{C}_1$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Quantum Low-Density Parity-Check Codes}
@@ -1522,7 +1540,7 @@ $\bm{H}_Z$ are constructed from two matrices $\bm{A}$ and $\bm{B}$ as
     .%
 \end{align*}
 This way, we can guarantee the satisfaction of the commutativity
-condition (\Cref{eq:css_condition}).
+condition \Cref{eq:css_condition}.
 To define $\bm{A}$ and $\bm{B}$ we first introduce some additional notation.
 We denote the identity matrix as $\bm{I_l} \in \mathbb{F}^{l\times l}$ and
 the \emph{cyclic shift matrix} as $\bm{S_l} \in \mathbb{F}^{l\times