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Remove TODOs, formatting, minor changes

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Andreas Tsouchlos
2026-04-24 17:58:30 +02:00
parent 494a639329
commit 5875066581
1 changed files with 8 additions and 13 deletions
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21
src/thesis/chapters/2_fundamentals.tex
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@@ -6,8 +6,6 @@ communications engineering and quantum information science.
This chapter provides the relevant theoretical background on both of This chapter provides the relevant theoretical background on both of
these topics and subsequently introduces the fundamentals of \ac{qec}. these topics and subsequently introduces the fundamentals of \ac{qec}.
% TODO: Is an explanation of BP with guided decimation needed in this chapter?
% TODO: Is an explanation of OSD needed chapter?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Classical Error Correction} \section{Classical Error Correction}
\label{sec:Classical Error Correction} \label{sec:Classical Error Correction}
@@ -872,8 +870,6 @@ Take for example the two qubits
\ket{\psi_2} = \alpha_2 \ket{0} + \beta_2 \ket{1} \ket{\psi_2} = \alpha_2 \ket{0} + \beta_2 \ket{1}
.% .%
\end{align*} \end{align*}
% TODO: Fix the fact that \psi is used above for the single-qubit
% case and below for the multi-qubit case
We examine the state $\ket{\psi}$ of the composite system. We examine the state $\ket{\psi}$ of the composite system.
Assuming the qubits are independent, this is a \emph{product state} Assuming the qubits are independent, this is a \emph{product state}
$\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$. $\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$.
@@ -1374,7 +1370,6 @@ We can describe it using the check matrix
\right] \right]
.% .%
\end{align} \end{align}
% TODO: Check X vs. Z
The first $n$ columns correspond to $X$ operators acting on the The first $n$ columns correspond to $X$ operators acting on the
corresponding physical qubit, the rest to the $Z$ operators. corresponding physical qubit, the rest to the $Z$ operators.
@@ -1383,7 +1378,7 @@ corresponding physical qubit, the rest to the $Z$ operators.
% tex-fmt: off % tex-fmt: off
\begin{quantikz} \begin{quantikz}
\lstick[2]{$E\ket{\psi}_\text{L}$} & & \gate[2]{P_1} & \gate[2]{P_2} & \gate[style={draw=none},2]{\ldots} & \gate[2]{P_{n-k}} & & & \\ \lstick[2]{$E\ket{\psi}_\text{L}$} & & \gate[2]{P_1} & \gate[2]{P_2} & \gate[style={draw=none},2]{\ldots} & \gate[2]{P_{n-k}} & & & \\
& & & & & & & & \\ & & & & & & & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \gate{H} & \ctrl{-1} & & & & \gate{H} & \meter{} & \setwiretype{c} \\ \lstick{$\ket{0}_{\text{A}_1}$} & \gate{H} & \ctrl{-1} & & & & \gate{H} & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & \gate{H} & & \ctrl{-2} & & & \gate{H} & \meter{} & \setwiretype{c} \\ \lstick{$\ket{0}_{\text{A}_2}$} & \gate{H} & & \ctrl{-2} & & & \gate{H} & \meter{} & \setwiretype{c} \\
@@ -1409,8 +1404,8 @@ Stabilizer codes are especially practical to work with when they can
handle $X$ and $Z$ type errors independently. handle $X$ and $Z$ type errors independently.
As $Z$ errors anti-commute with $X$ operators in the stabilizers and As $Z$ errors anti-commute with $X$ operators in the stabilizers and
vice versa, this property translates into being able to split the vice versa, this property translates into being able to split the
stabilizers into some being made up of only $X$ stabilizers into a subset being made up of only $X$
operators and some only of $Z$ operators. operators and the rest only of $Z$ operators.
We call such codes \ac{css} codes. We call such codes \ac{css} codes.
We can see this property in \autoref{eq:steane} in the check matrix We can see this property in \autoref{eq:steane} in the check matrix
of the Steane code. of the Steane code.
@@ -1428,12 +1423,13 @@ 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\} \bm{x}' \bm{x}^\text{T} = 0 ~\forall \bm{x} \in \mathcal{C}_2 \right\}
,% ,%
\end{align*} \end{align*}
we can construct the check matrix as 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
\begin{align*} \begin{align*}
\left[ \left[
\begin{array}{c|c} \begin{array}{c|c}
\bm{H}(\mathcal{C}_2^\perp) & \bm{0} \\ \bm{H}_X & \bm{0} \\
\bm{0} & \bm{H}(\mathcal{C}_1) \bm{0} & \bm{H}_Z
\end{array} \end{array}
\right] \right]
.% .%
@@ -1442,7 +1438,7 @@ In order to yield a valid stabilizer code, $\mathcal{C}_1$ and
$\mathcal{C}_2$ must satisfy the commutativity condition $\mathcal{C}_2$ must satisfy the commutativity condition
\begin{align} \begin{align}
\label{eq:css_condition} \label{eq:css_condition}
\bm{H}(\mathcal{C}_2^\perp) \bm{H}(\mathcal{C}_1)^\text{T} = \bm{0} \bm{H}_X \bm{H}_Z^\text{T} = \bm{0}
.% .%
\end{align} \end{align}
We can ensure this is the case by choosing them such that We can ensure this is the case by choosing them such that
@@ -1470,7 +1466,6 @@ code, scaling up of which would be prohibitively expensive
% Bivariate Bicycle codes % Bivariate Bicycle codes
% TODO: Introduce H_X and H_Z above
A recent addition to the class of \ac{qldpc} codes is that of \ac{bb} A recent addition to the class of \ac{qldpc} codes is that of \ac{bb}
codes \cite[Sec.~3]{bravyi_high-threshold_2024}. codes \cite[Sec.~3]{bravyi_high-threshold_2024}.
These are a special type of \ac{css} code, where $\bm{H}_X$ and These are a special type of \ac{css} code, where $\bm{H}_X$ and