From 17191382cfdfeacf993be8c1e00849a2f4d3737e Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 4 May 2026 13:01:54 +0200 Subject: [PATCH] Incorporate Lia's corrections to QM and QEC fundamentals --- src/thesis/chapters/2_fundamentals.tex | 56 +++++++++++++++++--------- 1 file changed, 37 insertions(+), 19 deletions(-) diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index ce196dc..5a18351 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/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}. -We have +\cite[Sec.~6.2]{griffiths_consistent_2001}, +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 $\mathcal{G}_n = \left\{ \pm I,\pm jI, \pm X,\pm jX, -\pm Y,\pm jY, \pm Z, \pm jZ \right\}^{\otimes n}$ the \emph{Pauli +We call the set $\mathcal{G}_n = \left\{ \pm I,\pm \mathrm{i}I, \pm + X,\pm \mathrm{i}X, +\pm Y,\pm \mathrm{i}Y, \pm Z, \pm \mathrm{i}Z \right\}^{\otimes n}$ +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} - \left( I - P_i \right)E\ket{\psi}_\text{A} \ket{1}_\text{A} + \right)E\ket{\psi}_\text{L}\ket{0}_{\text{A}_i} + \frac{1}{2} + \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} \\ - \Rightarrow E \ket{\psi}_{L} &= -P_i E \ket{\psi}_\text{L} \\ - \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{0}_{\text{A}_i} + + \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 $\bm{H}_Z -:= \bm{H}(\mathcal{C}_1)$, and construct the check matrix as +we define $\bm{H}_X$ as the \ac{pcm} of $\mathcal{C}_2^\perp$ and $\bm{H}_Z$ +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 -$\mathcal{C}_2 \subset \mathcal{C}_1$. +We can ensure this by choosing $\mathcal{C}_1$ and $\mathcal{C}_2$ +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