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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
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}
\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}
.%
\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.
Assuming the qubits are independent, this is a \emph{product state}
$\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$.
@@ -1374,7 +1370,6 @@ We can describe it using the check matrix
\right]
.%
\end{align}
% TODO: Check X vs. Z
The first $n$ columns correspond to $X$ operators acting on the
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
\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{$\ket{0}_{\text{A}_2}$} & \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.
As $Z$ errors anti-commute with $X$ operators in the stabilizers and
vice versa, this property translates into being able to split the
stabilizers into some being made up of only $X$
operators and some only of $Z$ operators.
stabilizers into a subset being made up of only $X$
operators and the rest only of $Z$ operators.
We call such codes \ac{css} codes.
We can see this property in \autoref{eq:steane} in the check matrix
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\}
,%
\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*}
\left[
\begin{array}{c|c}
\bm{H}(\mathcal{C}_2^\perp) & \bm{0} \\
\bm{0} & \bm{H}(\mathcal{C}_1)
\bm{H}_X & \bm{0} \\
\bm{0} & \bm{H}_Z
\end{array}
\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
\begin{align}
\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}
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
% TODO: Introduce H_X and H_Z above
A recent addition to the class of \ac{qldpc} codes is that of \ac{bb}
codes \cite[Sec.~3]{bravyi_high-threshold_2024}.
These are a special type of \ac{css} code, where $\bm{H}_X$ and