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Final readthrough corrections of quantum fundamentals

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@@ -91,13 +91,13 @@ We can arrange the coefficients of these equations in a
equivalently define the code as \cite[Sec.~3.1.1]{ryan_channel_2009} equivalently define the code as \cite[Sec.~3.1.1]{ryan_channel_2009}
\begin{align*} \begin{align*}
\mathcal{C} := \text{kern}(\bm{H}) = \left\{ \bm{x} \in \mathbb{F}_2^n : \mathcal{C} := \text{kern}(\bm{H}) = \left\{ \bm{x} \in \mathbb{F}_2^n :
\bm{H}\bm{x}^\text{T} = \bm{0} \right\} \bm{H}\bm{x}^\mathsf{T} = \bm{0} \right\}
.% .%
\end{align*} \end{align*}
In general, we have linearly dependent parity checks, In general, we have linearly dependent parity checks,
prompting us to define the \ac{pcm} as $\bm{H} \in prompting us to define the \ac{pcm} as $\bm{H} \in
\mathbb{F}_2^{m\times n}$ with $\hspace{2mm} m \ge n-k$ instead. \mathbb{F}_2^{m\times n}$ with $\hspace{2mm} m \ge n-k$ instead.
The \textit{syndrome} $\bm{s} = \bm{H} \bm{v}^\text{T}$ describes The \textit{syndrome} $\bm{s} = \bm{H} \bm{v}^\mathsf{T}$ describes
which parity checks a vector $\bm{v} \in \mathbb{F}_2^n$ violates. which parity checks a vector $\bm{v} \in \mathbb{F}_2^n$ violates.
The representation using the \ac{pcm} has the benefit of providing a The representation using the \ac{pcm} has the benefit of providing a
description of the code, the memory complexity of which does not grow description of the code, the memory complexity of which does not grow
@@ -601,7 +601,7 @@ decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}.
\label{sec:Quantum Mechanics and Quantum Information Science} \label{sec:Quantum Mechanics and Quantum Information Science}
Designing codes and decoders for \ac{qec} is generally performed on a Designing codes and decoders for \ac{qec} is generally performed on a
layer of abstraction far removed from the quantum mechanical layer of mathematical abstraction far removed from the quantum mechanical
processes underlying the actual physics. processes underlying the actual physics.
Nevertheless, having a fundamental understanding of the related Nevertheless, having a fundamental understanding of the related
quantum mechanical concepts is useful to grasp the unique constraints quantum mechanical concepts is useful to grasp the unique constraints
@@ -621,39 +621,41 @@ function and the observable world:
$\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of finding a particle at $\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of finding a particle at
position $x$ and time $t$ \cite[Sec.~1.2]{griffiths_introduction_1995}. position $x$ and time $t$ \cite[Sec.~1.2]{griffiths_introduction_1995}.
Note that this presupposes a normalization of $\psi$ such that Note that this presupposes a normalization of $\psi$ such that
$\int_{-\infty}^{\infty} \lvert \psi(x,t) \rvert^2 dx = 1$. \begin{align*}
\int_{-\infty}^{\infty} \lvert \psi(x,t) \rvert^2 dx = 1
.%
\end{align*}
% Dirac notation % Dirac notation
Much of the related mathematics can be very elegantly expressed The language of linear algebra allows one to express the related
using the language of linear algebra. mathematics particularly elegantly.
The so-called Bra-ket or Dirac notation is especially appropriate, The so-called Bra-ket or Dirac notation, introducced
having been proposed by Paul Dirac in 1939 for the express purpose by Paul Dirac in 1939 for the express purpose of simplifying quantum
of simplifying quantum mechanical notation \cite{dirac_new_1939}. mechanical notation \cite{dirac_new_1939}, is especially appropriate.
Two new symbols are defined, \emph{bra}s $\bra{\cdot}$ and Two new symbols are defined, \emph{bra} $\bra{\cdot}$ and
\emph{ket}s $\ket{\cdot}$. \emph{ket} $\ket{\cdot}$.
Kets denote column vectors, while bras denote their Hermitian conjugates. Kets denote column vectors, while bras denote their Hermitian conjugates.
For example, two vectors specified by the labels $a$ and $b$ For example, two vectors specified by the labels $a$ and $b$,
respectively are written as $\ket{a}$ and $\ket{b}$. respectively, are written as $\ket{a}$ and $\ket{b}$.
Their inner product is $\braket{a\vert b}$. Their inner product is $\braket{a\vert b}$.
% Expressing wave functions using linear algebra % Expressing wave functions using linear algebra
The connection we will make between quantum mechanics and linear The connection we will make between quantum mechanics and linear
algebra is that we will model the state space of a system as a algebra is that we will model the state space of a system as a
\emph{function space}, the Hilbert space $L_2$. \emph{function space}, namely the Hilbert space $L_2$.
We will represent the state of a particle with wave function The state of a particle with wave function $\psi(x,t)$ is represented
$\psi(x,t)$ using the vector $\ket{\psi}$ by the vector $\ket{\psi}$ \cite[Sec.~3.3]{griffiths_introduction_1995}.
\cite[Sec.~3.3]{griffiths_introduction_1995}.
% Operators % Operators
Another important notion is that of an \emph{operator}, a transformation Another important notion is that of an \emph{operator}, a transformation
that takes a function as an input and returns another function as an that maps a function onto another function
output \cite[Sec.~3.2.2]{griffiths_introduction_1995}. \cite[Sec.~3.2.2]{griffiths_introduction_1995}.
A prominent example of this is the differential operator $\partial x$.
Operators are useful to describe the relations between different Operators are useful to describe the relations between different
quantities relating to a particle. quantities relating to a particle.
An example of this is the differential operator $\partial x$.
We define the \emph{commutator} of two operators $P_1$ and $P_2$ as We define the \emph{commutator} of two operators $P_1$ and $P_2$ as
\begin{align*} \begin{align*}
[P_1,P_2] = P_1P_2 - P_2P_1 [P_1,P_2] = P_1P_2 - P_2P_1
@@ -672,22 +674,21 @@ We say the two operators \emph{commute} iff $[P_1,P_2] = 0$, and they
% Observable quantities % Observable quantities
An \emph{observable quantity} $Q(x,p,t)$ is a quantity of a quantum An \emph{observable} $Q(x,p,t)$ is a quantity of a quantum
mechanical system that we can measure, such as the position $x$ or mechanical system that we can measure, e.g., the position $x$ or
momentum $p$ of a particle. momentum $p$ of a particle.
In general, such measurements are not deterministic, i.e., In general, such measurements are not deterministic, i.e.,
measurements on identically prepared states can yield different results. measurements on identically prepared states can yield different results.
There are some states, however, that are \emph{determinate} for a However, some states are \emph{determinate} for a
specific observable: measuring those will always yield identical specific observable: Measuring those will always yield identical
observations \cite[Sec.~3.3]{griffiths_introduction_1995}. outcomes \cite[Sec.~3.3]{griffiths_introduction_1995}.
% General expression for expected value of observable quantity % General expression for expected value of observable quantity
If we know the wave function of a particle, we should be able to If the wave function of a particle is known, the expected value
compute the expected value $\braket{Q}$ of any observable quantity we wish. $\braket{Q}$ of any observable quantity can be computed.
It can be shown that for any $Q$, we can find a Indeed, for any $Q$, there exists a corresponding Hermitian operator
corresponding Hermitian operator $\hat{Q}$ such that $\hat{Q}$ such that \cite[Sec.~3.3]{griffiths_introduction_1995}
\cite[Sec.~3.3]{griffiths_introduction_1995}
\begin{align} \begin{align}
\label{eq:gen_expr_Q_exp} \label{eq:gen_expr_Q_exp}
\braket{Q} = \int_{-\infty}^{\infty} \psi^*(x,t) \hat{Q} \psi(x,t) dx \braket{Q} = \int_{-\infty}^{\infty} \psi^*(x,t) \hat{Q} \psi(x,t) dx
@@ -703,9 +704,9 @@ operator to $\hat{Q} = x$, we can write
= \int_{-\infty}^{\infty} x \lvert \psi(x,t) \rvert ^2 dx = \int_{-\infty}^{\infty} x \lvert \psi(x,t) \rvert ^2 dx
.% .%
\end{align*} \end{align*}
Note that $\lvert \psi(x,t) \rvert^2 $ represents the \ac{pdf} of Note that $\lvert \psi(x,t) \rvert^2 $ is the \ac{pdf} of
finding a particle in a specific state. We immediately see that the finding a particle at position $x$. Hence, we immediately see that
formula simplifies to the direct calculation of the expected value. the formula simplifies to the direct calculation of the expected value.
% Determinate states and eigenvalues % Determinate states and eigenvalues
@@ -719,40 +720,40 @@ We begin by translating \Cref{eq:gen_expr_Q_exp} into linear algebra as
.% .%
\end{align} \end{align}
\Cref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic \Cref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic
relationship. relationship, whereas the determinate states are inherently deterministic.
The determinate states are inherently deterministic.
To relate the two, we note that since determinate states should To relate the two, we note that since determinate states should
always yield the same measurement results, the variance of the always yield the same measurement results, the variance of the
observable should be zero. observable must be zero.
We thus compute \cite[Eq.~3.116]{griffiths_introduction_1995} We thus compute \cite[Eq.~3.116]{griffiths_introduction_1995}
\begin{align} \begin{align}
0 &\overset{!}{=} \braket{(Q - \braket{Q})^2} 0 &\overset{!}{=} \braket{(Q - \braket{Q})^2}
= \braket{e_n \vert (\hat{Q} - \braket{Q})^2 e_n} \nonumber\\ = \braket{e_n \vert (\hat{Q} - \braket{Q})^2 e_n} \nonumber\\
&= \braket{(Q - \braket{Q})e_n \vert (\hat{Q} - \braket{Q}) &= \braket{(\hat{Q} - \braket{Q})e_n \vert (\hat{Q} - \braket{Q})
e_n} \nonumber\\ e_n} \nonumber\\
&= \lVert (Q - \braket{Q}) e_n \rVert^2 \nonumber\\[3mm] &= \lVert (\hat{Q} - \braket{Q}) e_n \rVert^2 \nonumber\\[3mm]
&\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{e_n} = &\hspace{-14mm}\iff (\hat{Q} - \braket{Q}) \ket{e_n}
0 \nonumber\\ = 0 \nonumber\\
\label{eq:observable_eigenrelation} \label{eq:observable_eigenrelation}
&\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{e_n} &\hspace{-14mm}\iff \hat{Q}\ket{e_n}
= \underbrace{\braket{Q}}_{\lambda_n} \ket{e_n} = \underbrace{\braket{Q}}_{\lambda_n} \ket{e_n}
.% .%
\end{align}% \end{align}%
% %
Because we have assumed the variance to be zero, the expected value By setting the variance to zero, the expected value
$\braket{Q}$ is now the deterministic measurement result $\braket{Q}$ becomes a deterministic measurement result
corresponding to the determinate state corresponding to the determinate state
$\ket{e_n},~n\in \mathbb{N}$. $\ket{e_n},~n\in \mathbb{N}$.
We can see that the determinate states are the \emph{eigenstates} of The determinate states are precisely the \emph{eigenstates} of
the observable operator $\hat{Q}$ and that the measurement values are the observable operator $\hat{Q}$, and the associated measurement
the corresponding \emph{eigenvalues} $\lambda_n$ values are the corresponding \emph{eigenvalues} $\lambda_n$
\cite[Sec.~3.3]{griffiths_introduction_1995}. \cite[Sec.~3.3]{griffiths_introduction_1995}.
% Determinate states as a basis % Determinate states as a basis
As we are modelling the wave function $\psi(x,t)$ as a vector As we model the wave function $\psi(x,t)$ as a vector
$\ket{\psi}$, we can find a set of basis vectors to decompose it into. $\ket{\psi}$, 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% In particular, we can use the determinate states for this purpose,
expressing the state as%
\footnote{ \footnote{
We only consider 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. spectrum} here, i.e., having a discrete set of eigenvalues and vectors.
@@ -790,7 +791,7 @@ $Q(x,t,p)$ using a corresponding operator $\hat{Q}$, which allows us
to compute the expected value as $\braket{Q} = \braket{\psi to compute the expected value as $\braket{Q} = \braket{\psi
\vert \hat{Q} \psi}$. \vert \hat{Q} \psi}$.
The eigenvectors of $\hat{Q}$ are the determinate states The eigenvectors of $\hat{Q}$ are the determinate states
$\ket{e_n},~n\in \mathbb{N}$ and the eigenvalues are the respective $\ket{e_n},~n\in \mathbb{N}$, and the eigenvalues are the respective
measurement outcomes. measurement outcomes.
We can decompose an arbitrary state as $\ket{\psi} = \sum_{n=1}^{\infty} c_n We can decompose an arbitrary state as $\ket{\psi} = \sum_{n=1}^{\infty} c_n
\ket{e_n}$, where $\lvert c_n \rvert ^2$ represents the probability \ket{e_n}$, where $\lvert c_n \rvert ^2$ represents the probability
@@ -808,16 +809,16 @@ The measurements we considered in the previous section, for which
\Cref{eq:gen_expr_Q_exp_lin} holds, belong to the category of \Cref{eq:gen_expr_Q_exp_lin} holds, belong to the category of
\emph{projective measurements}. \emph{projective measurements}.
For these, certain restrictions such as repeatability apply: the act For these, certain restrictions such as repeatability apply: the act
of measuring a quantum state should \emph{collapse} it onto one of of measuring a quantum state \emph{collapses} it onto one of
the determinate states. the determinate states.
Further measurements should then yield the same value. Further measurements then yield the same value.
More general methods of modelling measurements exist, e.g., describing More general methods of modelling measurements exist, e.g.,
destructive measurements \cite[Box~2.5]{nielsen_quantum_2010}, but destructive measurements \cite[Box~2.5]{nielsen_quantum_2010}, but
they are not relevant to this work. they are not relevant to this work.
% Projection operators % Projection operators
We can model the collapse of the original state onto one of the We model the collapse of the original state onto one of the
superimposed basis states as a \emph{projection}. superimposed basis states as a \emph{projection}.
To see this, we use To see this, we use
\Cref{eq:determinate_basis,eq:observable_eigenrelation} to compute \Cref{eq:determinate_basis,eq:observable_eigenrelation} to compute
@@ -836,9 +837,9 @@ the separate components as
using \emph{projection operators} \cite[Eq.~3.160]{griffiths_introduction_1995} using \emph{projection operators} \cite[Eq.~3.160]{griffiths_introduction_1995}
\begin{align*} \begin{align*}
\hat{P}_n := \ket{e_n}\bra{e_n}, \hspace{3mm} n\in \mathbb{N} \hat{P}_n := \ket{e_n}\bra{e_n}, \hspace{3mm} n\in \mathbb{N}
. ,
\end{align*}% \end{align*}%
These project a vector onto the subspace spanned by $\ket{e_n}$. which project a vector onto the subspace spanned by $\ket{e_n}$.
% % Using projection operators to measure if a state has a component % % Using projection operators to measure if a state has a component
% % along a basis vector % % along a basis vector
@@ -864,10 +865,9 @@ These project a vector onto the subspace spanned by $\ket{e_n}$.
% Intro % Intro
% TODO: Make sure `quantum gate` is proper terminology
A central concept for quantum computing is that of the \emph{qubit}. A central concept for quantum computing is that of the \emph{qubit}.
We employ it analogously to the classical \emph{bit}. It takes the place of the classical \emph{bit}.
For classical computers, we alter bits' states using \emph{gates}. For classical computers, we alter the state of a bit using \emph{gates}.
We can chain multiple of these gates together to build up more complex logic, We can chain multiple of these gates together to build up more complex logic,
such as half-adders or eventually a full processor. such as half-adders or eventually a full processor.
In principle, quantum computers work in a similar fashion, only that In principle, quantum computers work in a similar fashion, only that
@@ -898,10 +898,10 @@ A qubit is defined to be a system with quantum state
\alpha \\ \alpha \\
\beta \beta
\end{pmatrix} \end{pmatrix}
= \alpha \ket{0} + \beta \ket{1} = \alpha \ket{0} + \beta \ket{1}, \hspace{5mm} \alpha,\beta \in \mathbb{C}
.% .%
\end{align} \end{align}
The overall state of a composite quantum system is described using The overall state of a multi-qubit quantum system is described using
the \emph{tensor product}, denoted as $\otimes$ the \emph{tensor product}, denoted as $\otimes$
\cite[Sec.~2.2.8]{nielsen_quantum_2010}. \cite[Sec.~2.2.8]{nielsen_quantum_2010}.
Take for example the two qubits Take for example the two qubits
@@ -930,9 +930,9 @@ i.e.,
.% .%
\end{split} \end{split}
\end{align} \end{align}
We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the We call $\ket{x_0, \ldots, x_n},~x_i \in \{0,1\}$ the
\emph{computational basis states} \cite[Sec.~4.6]{nielsen_quantum_2010}. \emph{computational basis states} \cite[Sec.~4.6]{nielsen_quantum_2010}.
To additionally simplify set notation, we define To simplify set notation, we define
\begin{align*} \begin{align*}
\mathcal{M}^{\otimes n} := \underbrace{\mathcal{M}\otimes \ldots \mathcal{M}^{\otimes n} := \underbrace{\mathcal{M}\otimes \ldots
\otimes \mathcal{M}}_{n \text{ times}} \otimes \mathcal{M}}_{n \text{ times}}
@@ -941,7 +941,7 @@ To additionally simplify set notation, we define
% Entanglement % Entanglement
States that are not able to be decomposed into such products States that are not able to be decomposed into products of single-qubit states
are called \emph{entangled} \cite[Sec.~2.2.8]{nielsen_quantum_2010}. are called \emph{entangled} \cite[Sec.~2.2.8]{nielsen_quantum_2010}.
An example of such states are the \emph{Bell states} An example of such states are the \emph{Bell states}
\begin{align*} \begin{align*}
@@ -979,7 +979,7 @@ we now shift our focus to describing the evolution of their states.
We model state changes as operators. We model state changes as operators.
Unlike classical systems, where there are only two possible states and Unlike classical systems, where there are only two possible states and
thus the only possible state change is a bit-flip, a general qubit thus the only possible state change is a bit-flip, a general qubit
state as shown in \Cref{eq:gen_qubit_state} lives on a continuum of values. state as shown in \Cref{eq:gen_qubit_state} lies on a continuum of values.
We thus technically also have an infinite number of possible state changes. We thus technically also have an infinite number of possible state changes.
Fortunately, we can express any operator as a linear combination of the Fortunately, we can express any operator as a linear combination of the
\emph{Pauli operators} \cite[Sec.~2.2]{gottesman_stabilizer_1997} \emph{Pauli operators} \cite[Sec.~2.2]{gottesman_stabilizer_1997}
@@ -1016,13 +1016,15 @@ Fortunately, we can express any operator as a linear combination of the
In fact, if we allow for complex coefficients, the $X$ and $Z$ In fact, if we allow for complex coefficients, the $X$ and $Z$
operators are sufficient to express any other operator as a linear operators are sufficient to express any other operator as a linear
combination \cite[Sec.~2.2]{roffe_quantum_2019}. combination \cite[Sec.~2.2]{roffe_quantum_2019}.
$I$ is the identity operator and $X$ and $Z$ are referred to as Hereby, $I$ is the identity operator and $X$ and $Z$ are referred to as
\emph{bit-flips} and \emph{phase-flips} respectively. \emph{bit-flips} and \emph{phase-flips} respectively.
We call the set $\mathcal{G}_n = \left\{ \pm I,\pm \mathrm{i}I, \pm We call the set
X,\pm \mathrm{i}X, \begin{align}
\pm Y,\pm \mathrm{i}Y, \pm Z, \pm \mathrm{i}Z \right\}^{\otimes n}$ \mathcal{G}_n = \left\{ \pm I,\pm \mathrm{i}I, \pm
the \emph{Pauli X,\pm \mathrm{i}X,
group} over $n$ qubits. \pm Y,\pm \mathrm{i}Y, \pm Z, \pm \mathrm{i}Z \right\}^{\otimes n}
\end{align}
the \emph{Pauli group} over $n$ qubits.
In the context of modifying qubit states, we also call operators \emph{gates}. 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 When working with multi-qubit systems, we can also apply Pauli gates
@@ -1052,7 +1054,7 @@ Other important operators include the \emph{Hadamard} and
\centering \centering
\begin{align*} \begin{align*}
\begin{array}{c} \begin{array}{c}
CNOT\text{ Operator} \\ \text{CNOT Operator} \\
\hline\\ \hline\\
\ket{00} \mapsto \ket{00} \\ \ket{00} \mapsto \ket{00} \\
\ket{01} \mapsto \ket{01} \\ \ket{01} \mapsto \ket{01} \\
@@ -1063,7 +1065,9 @@ Other important operators include the \emph{Hadamard} and
\end{minipage}% \end{minipage}%
\end{figure} \end{figure}
\vspace{-4mm} \vspace{-4mm}
\noindent Many more operators relevant to quantum computing exist, but they are \noindent The CNOT operator is a 2-qubit gate that applies a bit-flip to the
second qubit conditioned on the state of the first one.
Many more operators relevant to quantum computing exist, but they are
not covered here as they are not central to this work. not covered here as they are not central to this work.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
@@ -1096,9 +1100,8 @@ The control connection is represented by a vertical line connecting
the gate to the corresponding qubit, where a filled dot is placed. the gate to the corresponding qubit, where a filled dot is placed.
A controlled gate applies the respective operation only if the A controlled gate applies the respective operation only if the
control qubit is in state $\ket{1}$. control qubit is in state $\ket{1}$.
An example of this is the CNOT gate introduced in \Cref{fig:cnot_circuit} depicts an example of this: The CNOT gate
\Cref{subsec:Qubits and Multi-Qubit States}, which is depicted in introduced in \Cref{subsec:Qubits and Multi-Qubit States}.
\Cref{fig:cnot_circuit}.
\begin{figure}[t] \begin{figure}[t]
\centering \centering
@@ -1120,7 +1123,7 @@ An example of this is the CNOT gate introduced in
% General motivation behind QEC % General motivation behind QEC
One of the major barriers on the road to building a functioning One of the major barriers on the road to building a functioning and scalable
quantum computer is the inevitability of errors during quantum quantum computer is the inevitability of errors during quantum
computation. These arise due to the difficulty in sufficiently isolating the computation. These arise due to the difficulty in sufficiently isolating the
qubits from external noise \cite[Sec.~1]{roffe_quantum_2019}. qubits from external noise \cite[Sec.~1]{roffe_quantum_2019}.
@@ -1129,7 +1132,7 @@ with the environment act as small measurements, an effect called
\emph{decoherence} of the quantum state \emph{decoherence} of the quantum state
\cite[Sec.~1]{gottesman_stabilizer_1997}. \cite[Sec.~1]{gottesman_stabilizer_1997}.
\ac{qec} is one approach of dealing with this problem, by protecting \ac{qec} is one approach of dealing with this problem, by protecting
the quantum state in a similar fashion to information in classical error a quantum state in a similar fashion to information in classical error
correction. correction.
% The unique challenges of QEC % The unique challenges of QEC
@@ -1149,9 +1152,10 @@ Three main restrictions apply \cite[Sec.~2.4]{roffe_quantum_2019}:
% General idea (logical vs. physical gates) + notation % General idea (logical vs. physical gates) + notation
Much like in classical error correction, in \ac{qec} information Much like in classical error correction, \ac{qec} protects information by
is protected by mapping it onto codewords in a higher-dimensional space, introducing redundancy.
thereby introducing redundancy. The information, represented by a state in a low-dimensional space,
is mapped onto an encoded state in a higher-dimensional space.
To this end, $k \in \mathbb{N}$ \emph{logical qubits} are mapped onto To this end, $k \in \mathbb{N}$ \emph{logical qubits} are mapped onto
$n \in \mathbb{N}$ \emph{physical qubits}, $n>k$. $n \in \mathbb{N}$ \emph{physical qubits}, $n>k$.
We circumvent the no-cloning restriction by not copying the state of any of We circumvent the no-cloning restriction by not copying the state of any of
@@ -1172,8 +1176,9 @@ This is due to the \emph{backlog problem}
\cite[Sec.~II.G.3.]{terhal_quantum_2015}: There are certain gates \cite[Sec.~II.G.3.]{terhal_quantum_2015}: There are certain gates
at which the effect of existing errors on single qubits may be at which the effect of existing errors on single qubits may be
exacerbated by transforming them to multi-qubit errors. exacerbated by transforming them to multi-qubit errors.
We wish to correct the errors before passing qubits through such gates. If we ensure decoding with sufficiently low latency, we can correct
If the \ac{qec} system is not fast enough, there will be an increasing the errors before passing qubits through such gates.
However, if the \ac{qec} system is not fast enough, there will be an increasing
backlog of information at this point in the circuit, leading to an backlog of information at this point in the circuit, leading to an
exponential slowdown in computation. exponential slowdown in computation.
@@ -1203,8 +1208,8 @@ Note that this code is only able to detect single $X$-type errors.
% Measuring stabilizers % Measuring stabilizers
To determine if an error occurred, we want to measure To determine if an error occurred, we aim at to measuring whether a
whether a state belongs state belongs
% TODO: Remove footnote? % TODO: Remove footnote?
% \footnote{ % \footnote{
% It is possible for a state to not completely lie in either subspace. % It is possible for a state to not completely lie in either subspace.
@@ -1213,11 +1218,12 @@ whether a state belongs
% } % }
to $\mathcal{C}$ or $\mathcal{F}$. to $\mathcal{C}$ or $\mathcal{F}$.
As explained in \Cref{subsec:Observables}, physical measurements As explained in \Cref{subsec:Observables}, physical measurements
can be mathematically described using operators whose eigenvalues can be mathematically described using operators, whose eigenvalues
are the possible measurement results. are the possible measurement results.
Here, we need an operator with two eigenvalues and the corresponding Here, we need an operator with two eigenvalues and the corresponding
eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively. eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively.
For the two-qubit code, $Z_1Z_2$ is such an operator: For the two-qubit repetition code, $Z_1Z_2 \in \mathcal{G}_2$ is such
an operator:
\begin{align} \begin{align}
Z_1Z_2 E \ket{\psi}_\text{L} &= (+1) E \ket{\psi}_\text{L} Z_1Z_2 E \ket{\psi}_\text{L} &= (+1) E \ket{\psi}_\text{L}
\hspace*{3mm} \forall \hspace*{3mm} \forall
@@ -1228,13 +1234,14 @@ For the two-qubit code, $Z_1Z_2$ is such an operator:
.% .%
\end{align} \end{align}
$E \in \left\{ X,I \right\}$ is an operator describing a possible $E \in \left\{ X,I \right\}$ is an operator describing a possible
error and $E \ket{\psi}_\text{L}$ is the resulting state after that error. single-qubit error and $E \ket{\psi}_\text{L}$ is the resulting state
after that error.
By measuring the corresponding eigenvalue, we can determine if By measuring the corresponding eigenvalue, we can determine if
$E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$. $E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$.
% TODO: If necessary, cite \cite[Sec.~3]{roffe_quantum_2019} for the % TODO: If necessary, cite \cite[Sec.~3]{roffe_quantum_2019} for the
% non-compromising meausrement of the information % non-compromising meausrement of the information
To do this without directly observing (and thus potentially To do this without directly observing and, thus potentially
collapsing) the logical state $\ket{\psi}_\text{L}$, we prepare an collapsing, the logical state $\ket{\psi}_\text{L}$, we prepare an
ancilla qubit with state $\ket{0}_\text{A}$ and entangle it with ancilla qubit with state $\ket{0}_\text{A}$ and entangle it with
$\ket{\psi}_\text{L}$ in such a way that the eigenvalue is indicated $\ket{\psi}_\text{L}$ in such a way that the eigenvalue is indicated
by measuring the ancilla qubit instead. by measuring the ancilla qubit instead.
@@ -1299,11 +1306,11 @@ This effect is referred to as error \emph{digitization}
% The stabilizer group % The stabilizer group
Operators such as $Z_1Z_2$ above are called \emph{stabilizers}. 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 More generally, an operator $P_i \in \mathcal{G}_n$ is a stabilizer of an
$\llbracket n, k, d_\text{min} \rrbracket$ code $\mathcal{C}$, if $\llbracket n, k, d_\text{min} \rrbracket$ code $\mathcal{C}$, if
\begin{itemize} \begin{itemize}
\item It stabilizes all logical states, i.e., \item It stabilizes all logical states, i.e.,
$P_i\ket{\psi}_\text{L} = (+1)\ket{\psi}_\text{L} ~\forall~ $P_i\ket{\psi}_\text{L} = (+1)\ket{\psi}_\text{L}, ~\forall~
\ket{\psi}_\text{L} \in \mathcal{C}$. \ket{\psi}_\text{L} \in \mathcal{C}$.
\item It commutes with all other stabilizers $P_j$ of the code, \item It commutes with all other stabilizers $P_j$ of the code,
i.e., $[P_i, P_j] = 0$. i.e., $[P_i, P_j] = 0$.
@@ -1319,8 +1326,8 @@ Formally, we define the \emph{stabilizer group} $\mathcal{S}$ as
[P_i,P_j] = 0 \forall i,j\right\} [P_i,P_j] = 0 \forall i,j\right\}
.% .%
\end{align*} \end{align*}
We care in particular about the commuting properties of stabilizers We care about the commuting properties of stabilizers with respect to
with respect to possible errors. possible errors, in particular.
The measurement circuit for an arbitrary stabilizer $P_i$ modifies The measurement circuit for an arbitrary stabilizer $P_i$ modifies
the state as \cite[Eq.~29]{roffe_quantum_2019} the state as \cite[Eq.~29]{roffe_quantum_2019}
\begin{align*} \begin{align*}
@@ -1353,6 +1360,7 @@ If a given error $E$ anticommutes with $P_i$, we have
\end{align*} \end{align*}
and measuring the ancilla $\text{A}_i$ corresponding to stabilizer and measuring the ancilla $\text{A}_i$ corresponding to stabilizer
$P_i$ returns 1. $P_i$ returns 1.
Similarly, if it commutes, the ancilla measurement returns 0.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
\subsection{Stabilizer Codes} \subsection{Stabilizer Codes}
@@ -1360,9 +1368,10 @@ $P_i$ returns 1.
% Structure of a stabilizer code % Structure of a stabilizer code
For classical binary linear block codes, we use $n-k$ parity-checks Stabilizer codes are the quantum analogue of classical binary linear
block codes, for which we use $n-k$ parity checks
to reduce the degrees of freedom introduced by the encoding operation. to reduce the degrees of freedom introduced by the encoding operation.
Effectively, each parity-check defines a local code splitting the Effectively, each parity check defines a local code splitting the
vector space in half, with only one part containing valid codewords. vector space in half, with only one part containing valid codewords.
The global code is the intersection of all local codes. The global code is the intersection of all local codes.
We can do the same in the quantum case. We can do the same in the quantum case.
@@ -1380,19 +1389,23 @@ operators $P_i$, each using a circuit as explained in
\Cref{subsec:Stabilizer Measurements}. \Cref{subsec:Stabilizer Measurements}.
Note that this is an abstract representation of the syndrome extraction. Note that this is an abstract representation of the syndrome extraction.
For the actual implementation in hardware, we can transform this into For the actual implementation in hardware, we can transform this into
a circuit that requires only CNOT and H-gates a circuit that requires only CNOT and $H$-gates
\cite[Sec.~10.5.8]{nielsen_quantum_2010}. \cite[Sec.~10.5.8]{nielsen_quantum_2010}.
% Logical operators % Logical operators
In order to modify the logical state encoded using the physical In order to modify the logical state encoded using the physical
qubits, we can use \emph{logical operators} \cite[Sec.~4.2]{roffe_quantum_2019}. qubits, we can use \emph{logical operators} \cite[Sec.~4.2]{roffe_quantum_2019}.
For each qubit, there are two logical operators, $X_i$ and $Z_j$. For a $\llbracket n,k \rrbracket$ stabilizer code, there exist
These are operators that logical operators generated by $2k$ representatives $X_i,
Z_j,~i,j\in[1:k]$ such that
\begin{itemize} \begin{itemize}
\item Commute with all the stabilizers in $\mathcal{S}$. \item They commute with all stabilizers in $\mathcal{S}$.
\item Anti-commute with one another, i.e., $[ \overline{X}_i, \item For $i=j$, they anti-commute with one another, i.e., $[
\overline{Z}_i ]_{+} = \overline{X}_i \overline{Z}_i + \overline{X}_i, \overline{Z}_i ]_{+} = \overline{X}_i
\overline{Z}_i + \overline{Z}_i \overline{X}_i = 0$.
\item For $i\neq j$, they commute with one another, i.e., $[ \overline{X}_i,
\overline{Z}_i ] = \overline{X}_i \overline{Z}_i -
\overline{Z}_i \overline{X}_i = 0$. \overline{Z}_i \overline{X}_i = 0$.
\end{itemize} \end{itemize}
We can also measure these operators to find out the logical state a We can also measure these operators to find out the logical state a
@@ -1402,22 +1415,22 @@ physical state corresponds to \cite[Sec.~2.6]{derks_designing_2025}.
% TODO: Do I have to introduce before that stabilizers only need X % TODO: Do I have to introduce before that stabilizers only need X
% and Z operators? % and Z operators?
We can represent stabilizer codes using a \emph{check matrix} We can represent stabilizer codes using a binary \emph{check matrix}
\cite[Sec.~10.5.1]{nielsen_quantum_2010} $\bm{H} \in \mathbb{F}_2^{(n-k)\times(2n)}$
\cite[Sec.~10.5.1]{nielsen_quantum_2010} with
\begin{align*} \begin{align*}
\bm{H} = \left[ \bm{H} = \left[
\begin{array}{c|c} \begin{array}{c|c}
\bm{H}_X & \bm{H}_Z \bm{H}_X & \bm{H}_Z
\end{array} \end{array}
\right] \right]
,% .%
\end{align*} \end{align*}
with $\bm{H} \in \mathbb{F}_2^{(n-k)\times(2n)}$.
This is similar to a classical \ac{pcm} in that it contains $n-k$ This is similar to a classical \ac{pcm} in that it contains $n-k$
rows, each describing one constraint. Each constraint restricts an additional rows, each describing one constraint. Each constraint restricts an additional
degree of freedom of the higher-dimensional space we use to introduce degree of freedom of the higher-dimensional space we use to introduce
redundancy. redundancy.
In contrast to the classical case, this matrix now has $2n$ columns, In contrast to the classical case, this matrix has $2n$ columns,
as we have to consider both the $X$ and $Z$ type operators that make up as we have to consider both the $X$ and $Z$ type operators that make up
the stabilizers. the stabilizers.
Take for example the Steane code \cite[Eq.~10.83]{nielsen_quantum_2010}. Take for example the Steane code \cite[Eq.~10.83]{nielsen_quantum_2010}.
@@ -1436,8 +1449,8 @@ We can describe it using the check matrix
\right] \right]
.% .%
\end{align} \end{align}
The first $n$ columns correspond to $X$ operators acting on the The first $n$ columns correspond to $X$ stabilizers acting on the
corresponding physical qubit, the rest to the $Z$ operators. corresponding physical qubit, the rest to the $Z$ stabilizers.
\begin{figure}[t] \begin{figure}[t]
\centering \centering
@@ -1466,27 +1479,27 @@ corresponding physical qubit, the rest to the $Z$ operators.
% Intro % Intro
Stabilizer codes are especially practical to work with when they can Stabilizer codes are especially practical to work with when the
handle $X$ and $Z$ type errors independently. stabilizers can be split into one subset consisting only of
$Z$ stabilizers and one consisting only of $X$ stabilizers.
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 correct $X$
stabilizers into a subset being made up of only $X$ or $Z$ errors independently.
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 \Cref{eq:steane} in the check matrix We can see this property in \Cref{eq:steane} in the check matrix
of the Steane code. of the Steane code.
% Construction % Construction
We can exploit this separate consideration of $X$ and $Z$ errors in We can exploit this separate consideration of $X$ and $Z$ stabilizers in
the construction of \ac{css} codes. the construction of \ac{css} codes.
We combine two binary linear codes $\mathcal{C}_1$ and We combine two binary linear codes $\mathcal{C}_1$ and
$\mathcal{C}_2$, each responsible for correcting one type of error $\mathcal{C}_2$, each responsible for correcting either $Z$ or $X$ errors
\cite[Sec.~10.5.6]{nielsen_quantum_2010}. \cite[Sec.~10.5.6]{nielsen_quantum_2010}.
Using the dual code of $\mathcal{C}_2$ \cite[Eq.~3.4]{ryan_channel_2009} Using the dual code of $\mathcal{C}_2$ \cite[Eq.~3.4]{ryan_channel_2009}
\begin{align*} \begin{align*}
\mathcal{C}_2^\perp := \left\{ \bm{x}' \in \mathbb{F}^2 : \mathcal{C}_2^\perp := \left\{ \bm{x}' \in \mathbb{F}^2 :
\bm{x}' \bm{x}^\text{T} = 0 ~\forall \bm{x} \in \mathcal{C}_2 \right\} \bm{x}' \bm{x}^\mathsf{T} = 0 ~\forall \bm{x} \in \mathcal{C}_2 \right\}
,% ,%
\end{align*} \end{align*}
we define $\bm{H}_X$ as the \ac{pcm} of $\mathcal{C}_2^\perp$ and $\bm{H}_Z$ we define $\bm{H}_X$ as the \ac{pcm} of $\mathcal{C}_2^\perp$ and $\bm{H}_Z$
@@ -1504,7 +1517,7 @@ 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}_X \bm{H}_Z^\text{T} = \bm{0} \bm{H}_X \bm{H}_Z^\mathsf{T} = \bm{0}
.% .%
\end{align} \end{align}
We can ensure this by choosing $\mathcal{C}_1$ and $\mathcal{C}_2$ We can ensure this by choosing $\mathcal{C}_1$ and $\mathcal{C}_2$
@@ -1519,15 +1532,15 @@ such that $\mathcal{C}_2 \subset \mathcal{C}_1$.
Various methods of constructing \ac{qec} codes exist Various methods of constructing \ac{qec} codes exist
\cite{swierkowska_eccentric_2025}. \cite{swierkowska_eccentric_2025}.
Topological codes, for example, encode information in the features of Topological codes, for example, encode information in the features of
a lattice and are intrinsically robust against local errors. a lattice in a way that allows for local interactions between qubits.
Among these, the \emph{surface code} is the most widely studied. Among these, the \emph{surface code} is the most widely studied.
Another example are concatenated codes, which nest one code within Another example are concatenated codes, which nest one code within
another, allowing for especially simple and flexible constructions another, allowing for especially simple and flexible constructions
\cite[Sec.~3.2]{swierkowska_eccentric_2025}. \cite[Sec.~3.2]{swierkowska_eccentric_2025}.
An area of research that has recently seen more attention is that of An area of research that has recently seen more attention is that of
quantum \ac{ldpc} (\acs{qldpc}) codes. quantum \ac{ldpc} (\acs{qldpc}) codes.
They have much better encoding efficiency than, e.g., the surface They have much higher rate than, e.g., surface codes, scaling up of
code, scaling up of which would be prohibitively expensive which would be prohibitively expensive
\cite[Sec.~I]{bravyi_high-threshold_2024}. \cite[Sec.~I]{bravyi_high-threshold_2024}.
% Bivariate Bicycle codes % Bivariate Bicycle codes
@@ -1539,7 +1552,7 @@ $\bm{H}_Z$ are constructed from two matrices $\bm{A}$ and $\bm{B}$ as
\begin{align*} \begin{align*}
\bm{H}_X = [\bm{A} \vert \bm{B}] \bm{H}_X = [\bm{A} \vert \bm{B}]
\hspace*{5mm} \text{and} \hspace*{5mm} \hspace*{5mm} \text{and} \hspace*{5mm}
\bm{H}_Z = [\bm{B}^\text{T} \vert \bm{A}^\text{T}] \bm{H}_Z = [\bm{B}^\mathsf{T} \vert \bm{A}^\mathsf{T}]
.% .%
\end{align*} \end{align*}
This way, we can guarantee the satisfaction of the commutativity This way, we can guarantee the satisfaction of the commutativity
@@ -1579,16 +1592,17 @@ This necessitates a modification of the standard \ac{bp} algorithm
introduced in \Cref{subsec:Iterative Decoding} introduced in \Cref{subsec:Iterative Decoding}
\cite[Sec.~3.1]{yao_belief_2024}. \cite[Sec.~3.1]{yao_belief_2024}.
Instead of attempting to find the most likely codeword directly, the Instead of attempting to find the most likely codeword directly, the
algorithm will now try to find an error pattern $\hat{\bm{e}} \in syndrome-based decoding algorithm tries to find an error pattern
\mathbb{F}_2^n$ that satisfies $\hat{\bm{e}} \in \mathbb{F}_2^n$ that satisfies
\begin{align*} \begin{align*}
\bm{H} \hat{\bm{e}}^\text{T} = \bm{s} \bm{H} \hat{\bm{e}}^\mathsf{T} = \bm{s}
.% .%
\end{align*} \end{align*}
To this end, we initialize the channel \acp{llr} as To this end, we initialize the channel \acp{llr} as
\begin{align*} \begin{align*}
\tilde{L}_i = \log{\frac{P(X_i = 0)}{P(X_i = 1)}} = \log{\frac{1 \tilde{L}_i = \log{\frac{P(X_i = 0)}{P(X_i = 1)}} = \log{
- p_i}{p_i}} \left( \frac{1 - p_i}{p_i} \right)
}
,% ,%
\end{align*} \end{align*}
where $p_i$ is the prior probability of error of \ac{vn} $i$. where $p_i$ is the prior probability of error of \ac{vn} $i$.
@@ -1645,7 +1659,7 @@ The resulting syndrome-based \ac{bp} algorithm is shown in
\right\}$ \right\}$
\EndFor \EndFor
\If{$\bm{H}\hat{\bm{e}}^\text{T} = \bm{s}$} \If{$\bm{H}\hat{\bm{e}}^\mathsf{T} = \bm{s}$}
\State \textbf{break} \State \textbf{break}
\EndIf \EndIf
@@ -1724,7 +1738,7 @@ This way, we obtain the \ac{ler}.
\mathbbm{1}\left\{ L^\text{total}_i \right\}$ \mathbbm{1}\left\{ L^\text{total}_i \right\}$
\EndFor \EndFor
\If{$\bm{H}\hat{\bm{e}}^\text{T} = \bm{s}$} \If{$\bm{H}\hat{\bm{e}}^\mathsf{T} = \bm{s}$}
\State \textbf{break} \State \textbf{break}
\Else \Else
\State $i_\text{max} \leftarrow \argmax_{i \in \mathcal{I}'} \lvert L^\text{total}_i \rvert $ \State $i_\text{max} \leftarrow \argmax_{i \in \mathcal{I}'} \lvert L^\text{total}_i \rvert $

8
src/thesis/chapters/3_fault_tolerant_qec.tex
View File

@@ -844,10 +844,10 @@ violate the same set of detectors, i.e.,
\begin{align*} \begin{align*}
\hspace{-15mm} \hspace{-15mm}
% tex-fmt: off % tex-fmt: off
&& \bm{H} \bm{e}_1^\text{T} & \neq \bm{H} \bm{e}_2^\text{T} \\ && \bm{H} \bm{e}_1^\mathsf{T} & \neq \bm{H} \bm{e}_2^\mathsf{T} \\
\iff \hspace{-33mm} && \bm{H} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\ \iff \hspace{-33mm} && \bm{H} \left( \bm{e}_1 - \bm{e}_2 \right)^\mathsf{T} & \neq 0 \\
\iff \hspace{-33mm} && \bm{D} \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\ \iff \hspace{-33mm} && \bm{D} \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\mathsf{T} & \neq 0 \\
\iff \hspace{-33mm} && \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \notin \text{kern} \{\bm{D}\} \iff \hspace{-33mm} && \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\mathsf{T} & \notin \text{kern} \{\bm{D}\}
% tex-fmt: on % tex-fmt: on
.% .%
\end{align*} \end{align*}

6
src/thesis/chapters/4_decoding_under_dems.tex
View File

@@ -701,7 +701,7 @@ estimates committed after decoding window $\ell$, we have to set
\begin{align*} \begin{align*}
\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}} = \left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}} =
\bm{H}_\text{overlap}^{(\ell)} \bm{H}_\text{overlap}^{(\ell)}
\left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\text{T} \left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\mathsf{T}
.% .%
\end{align*} \end{align*}
@@ -986,7 +986,7 @@ Note that the decoding procedure performed on the individual windows
\State $\displaystyle\left(\hat{\bm{e}}^\text{total}\right)_{\mathcal{I}^{(\ell)}_\text{commit}} \leftarrow \hat{\bm{e}}^{(\ell)}_\text{commit}$ \State $\displaystyle\left(\hat{\bm{e}}^\text{total}\right)_{\mathcal{I}^{(\ell)}_\text{commit}} \leftarrow \hat{\bm{e}}^{(\ell)}_\text{commit}$
\State $\displaystyle\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}} \State $\displaystyle\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}}
\leftarrow \bm{H}_\text{overlap}^{(\ell)} \leftarrow \bm{H}_\text{overlap}^{(\ell)}
\left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\text{T}$ \left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\mathsf{T}$
\If{$\ell < n_\text{win} - 1$} \If{$\ell < n_\text{win} - 1$}
\State $L^{(\ell+1)}_{i\leftarrow j} \leftarrow \State $L^{(\ell+1)}_{i\leftarrow j} \leftarrow
L^{(\ell)}_{i\leftarrow j} L^{(\ell)}_{i\leftarrow j}
@@ -1184,7 +1184,7 @@ decimation information after initializing the \ac{cn} to \ac{vn} messages.
% \State $\displaystyle\left(\hat{\bm{e}}^\text{total}\right)_{\mathcal{I}^{(\ell)}_\text{commit}} \leftarrow \hat{\bm{e}}^{(\ell)}_\text{commit}$ % \State $\displaystyle\left(\hat{\bm{e}}^\text{total}\right)_{\mathcal{I}^{(\ell)}_\text{commit}} \leftarrow \hat{\bm{e}}^{(\ell)}_\text{commit}$
% \State $\displaystyle\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}} % \State $\displaystyle\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}}
% \leftarrow \bm{H}_\text{overlap}^{(\ell)} % \leftarrow \bm{H}_\text{overlap}^{(\ell)}
% \left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\text{T}$ % \left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\mathsf{T}$
% \If{$\ell < n_\text{win} - 1$} % \If{$\ell < n_\text{win} - 1$}
% \State $L^{(\ell+1)}_{i\leftarrow j} \leftarrow % \State $L^{(\ell+1)}_{i\leftarrow j} \leftarrow
% L^{(\ell)}_{i\leftarrow j} % L^{(\ell)}_{i\leftarrow j}