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Move 3-qubit repetition code check matrix; Rewrite DEM intro

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Andreas Tsouchlos
2026-04-28 18:58:16 +02:00
parent 42a689d811
commit 87e48b5ac6
1 changed files with 47 additions and 36 deletions
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83
src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -160,9 +160,22 @@ different error locations in the circuit.
We will illustrate the most widely used types of error models on the
example of the three-qubit repetition code for $X$ errors.
This code has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
\autoref{fig:pure_syndrome_extraction} shows the respective
check matrix and syndrome extraction circuit.
This is a code with check matrix
\begin{align*}
\bm{H} =
\left[
\begin{array}{ccc|ccc}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1
\end{array}
\right]
.
\end{align*}
We can see that it has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
\autoref{fig:pure_syndrome_extraction} shows the corresponding
syndrome extraction circuit.
We refer to the qubits carrying the logical state
$\ket{\psi}_\text{L}$ as \emph{data qubits}.
Note that this is a concrete implementation using CNOT gates, as
@@ -247,30 +260,15 @@ error locations.
\begin{figure}[t]
\centering
\begin{minipage}{0.5\textwidth}
\begin{align*}
\bm{H} =
\left[
\begin{array}{ccc|ccc}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1
\end{array}
\right]
\end{align*}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
% tex-fmt: off
\begin{quantikz}%[row sep=4mm, column sep=4mm]
\lstick[3]{$\ket{\psi}_\text{L}$} & \ctrl{3} & & & & & \\
& & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
\end{quantikz}
% tex-fmt: on
\end{minipage}%
% tex-fmt: off
\begin{quantikz}%[row sep=4mm, column sep=4mm]
\lstick[3]{$\ket{\psi}_\text{L}$} & \ctrl{3} & & & & & \\
& & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
\end{quantikz}
% tex-fmt: on
\caption{
Syndrome extraction circuit for the three-qubit repetition
@@ -400,16 +398,29 @@ error locations.
\section{Detector Error Models}
\label{sec:Detector Error Models}
\emph{Detector error models} constitue a standardized framework for
passing information about the circuit used for \ac{qec} to a decoder.
They are also useful in the design of fault-tolerant \ldots such as
fault-tolerant quantum computing schemes \cite[Sec.~1]{derks_designing_2025}.
% While alternate ways of considering fault tolerance exist, detector
% error models
% benefit from the fact that
\content{Benefits of this approach \cite[Sec.~4.2]{derks_designing_2025}}
\emph{Detector error models} (\acsp{dem}) constitue a standardized framework for
passing information about a circuit used for \ac{qec} to a decoder.
They are also useful as a theoretical tool to aid in the design of
fault-tolerant \ac{qec} schemes.
E.g., they can be used to easily determine whether a measurement
schedule is fault-tolerant \cite[Example~12]{derks_designing_2025}.
\content{Where they were introduced originally}
Other approaches of implementing fault tolerance exist, such as
flag error correction, which uses additional ancilla qubits to detect
potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}.
However, \acp{dem} offer some unique advantages
\cite[Sec.~4.2]{derks_designing_2025}:
\begin{itemize}
\item They distinguish between errors based on their effect on
the measurements, not based on their location in the circuit.
This allows for merging equivalent errors, which decreases
decoding complexity.
\item Errors on the data qubits and on the measurements are
treated in a unified manner. This leads to a more powerful
description of the overall circuit.
\end{itemize}
In this work, we only consider the process of decoding under the
\ac{dem} framework.
% Core idea