From a6092a51831e6c4668a186728d155ad98ebc7d8d Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 20 Apr 2026 22:49:30 +0200 Subject: [PATCH] Finish first draft of stabilizer measurement section --- src/thesis/chapters/2_fundamentals.tex | 61 +++++++++++++++++++------- 1 file changed, 45 insertions(+), 16 deletions(-) diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index c0bb9ff..b6dd501 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/src/thesis/chapters/2_fundamentals.tex @@ -871,7 +871,7 @@ Take for example the two qubits \end{align*} % TODO: Fix the fact that \psi is used above for the single-qubit % case and below for the multi-qubit case -We denote the state of the composite system as $\ket{\psi}$. +We examine the state $\ket{\psi}$ of the composite system as. 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 @@ -896,7 +896,7 @@ We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the % Entanglement -States that are not able to be decomposed into such a product +States that are not able to be decomposed into such products are called \emph{entangled} \cite[Sec.~2.2.8]{nielsen_quantum_2010}. An example of such states are the \emph{Bell states} \begin{align*} @@ -1068,7 +1068,7 @@ thereby introducing redundancy. To this end, $k \in \mathbb{N}$ \emph{logical qubits} are mapped onto $n \in \mathbb{N},~n>k$ \emph{physical qubits}. We circumvent the no-cloning restriction by not copying the state of -the $k$ logical qubits, but rather spreading it out over all $n$ +the $k$ logical qubits, instead spreading it out over all $n$ physical ones \cite[Intro.]{calderbank_good_1996} To differentiate a quantum codes from classical ones, we denote a code with parameters $k,n$ and minimum distance $d_\text{min}$ using @@ -1237,24 +1237,51 @@ E.g., $P_\mathcal{C}$ will eliminate all components of $E \ket{\psi}_\text{L}$ that lie in $\mathcal{F}$. This process, together with the fact that any coherent error can be decomposed into a linear combination of $X$ and $Z$ errors, means -that it is enough for a \ac{qec} scheme only needs to be able to -correct $X$ and $Z$ errors. +that it is enough for a \ac{qec} to be able to correct only $X$ and $Z$ errors. This effect is referred to as error \emph{digitization} \cite[Sec.~2.2]{roffe_quantum_2019}. % The stabilizer group -\indent\red{[Conditions for the stabilizer group -\cite[Sec.~4.1]{roffe_quantum_2019}]} \\ -\indent\red{[Why we care about (anti-)commutativity of the - stabilizers with errors + Z-type operators for X type errors and vice -versa ]} \\ -\indent\red{[(?) Stabilizer generators]} - -% A general stabilizer measurement circuit - -\indent\red{[General stabilizer measurement circuit -\cite[Figure~4]{roffe_quantum_2019}]} +Operators such as $Z_1Z_2$ above are called \emph{stabilizers}. +An operator $P_i \in \mathcal{G}_n$ is called a stabilizer of an +$[[n, k, d_\text{min}]]$ code $\mathcal{C}$, if +\begin{itemize} + \item It stabilizes all logical states, i.e., + $P_i\ket{\psi}_\text{L} = (+1)\ket{\psi}_\text{L} ~\forall~ + \ket{\psi}_\text{L} \in \mathcal{C}$. + \item It commutes with all other stabilizers of the code. This + property is important to be able to measure the eigenvalue of + a stabilizer without disturbing the eigenvectors of the + others \cite[Sec.~1.2]{gottesman_stabilizer_1997}. +\end{itemize} +Formally, we define the \emph{stabilizer group} $\mathcal{S}$ as +\cite[Sec.~4.1]{roffe_quantum_2019} +\begin{align*} + \mathcal{S} = \left\{P_i \in \mathcal{G}_n ~:~ P_i \ket{\psi}_\text{L} = + (+1)\ket{\psi}_\text{L} \forall \ket{\psi}_\text{L} ~\cap~ + [P_i,P_j] = 0 \forall i,j\right\} + .% +\end{align*} +We care in particular about the commuting properties of stabilizers +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} + \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} + .% +\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 +\end{align*} +and the stabilizer measurement returns 1. %%%%%%%%%%%%%%%% \subsection{Stabilizer Codes} @@ -1290,6 +1317,7 @@ A full \emph{syndrome extraction circuit} is depicted in \autoref{fig:sec}. the encoded computations \cite[Sec.~2.6]{derks_designing_2025}]} \\ \indent\red{[X and Z measurements can be performed with only CNOT and Hadamard gates \cite[Sec.~10.5.8]{nielsen_quantum_2010}]} \\ +\indent\red{[(?) Stabilizer generators]} \\ \indent\red{[Parity-check matrix \cite[Sec.~10.5.1]{nielsen_quantum_2010}]} \begin{figure}[t] @@ -1329,6 +1357,7 @@ handle $X$- and $Z$-type errors independently. We can then separate the stabilizer generators into some with only $Z$ operators and some with only $X$ operators. +\indent\red{[Z-type operators for X type errors and vice versa ]} \\ \indent\red{[Construction from two binary linear codes \cite[p.~452,469]{nielsen_quantum_2010}]}