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