198 lines
9.6 KiB
TeX
198 lines
9.6 KiB
TeX
\chapter{Introduction}
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\label{ch:Introduction}
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\acresetall
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% Intro to quantum computing
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In 1982, Richard Feynman, motivated by the difficulty of simulating
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quantum-mechanical systems on classical hardware, put forward the
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idea of building computers that are themselves quantum mechanical
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\cite{feynman_simulating_1982}.
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The use of such quantum computers has since been shown to offer promising
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prospects not only with regard to simulating quantum systems but also
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for solving certain kinds of problems that are classically intractable.
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The most prominent example is Shor's algorithm for integer
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factorization \cite{shor_algorithms_1994}.
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Similar to the way classical computers are built from bits and gates,
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quantum computers are built from \emph{qubits} and \emph{quantum gates}.
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Because of quantum entanglement, it does not suffice to consider the
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qubits individually, we also have to consider correlations between them.
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For a system of $n$ qubits, this makes the state space grow with
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$2^n$ instead of linearly with $n$, as would be the case for a classical system
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\cite[Sec.~1]{gottesman_stabilizer_1997}.
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This is both the reason quantum systems are difficult to simulate and
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what provides them with their power \cite[Sec.~2.1]{roffe_decoding_2020}.
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% The need for QEC
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Realizing algorithms that leverage these quantum-mechanical effects
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requires hardware that can execute long quantum computations reliably.
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This poses a problem, because the qubits making up current devices
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consistently interact with their environment \cite[Sec.~1]{roffe_quantum_2019}.
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This interaction acts as a continuous small-scale measurement, an
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effect we call \emph{decoherence} of the stored quantum state, which
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results in errors on the qubits.
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Decoherence is the reason large systems do not exhibit visible quantum
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properties at human scales \cite[Sec.~1]{gottesman_stabilizer_1997}.
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% Intro to QEC
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\Ac{qec} has emerged as a leading candidate in solving this problem.
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It addresses the issue by encoding the information of $k$
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\emph{logical qubits} into a larger number $n>k$ of \emph{physical
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qubits}, in close analogy to classical channel coding
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\cite[Sec.~1]{roffe_quantum_2019}.
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The redundancy introduced this way can then be used to detect and
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correct a corrupted the quantum state.
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The quantum setting imposes some important constraints that do not exist in the
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classical case, however \cite[Sec.~2.4]{roffe_quantum_2019}:
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\begin{itemize}
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\item The no-cloning theorem prohibits the duplication of quantum states.
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\item In addition to the bit-flip errors we know from the
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classical setting, qubits are subject to \emph{phase-flips}.
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\item We are not allowed to directly measure the encoded qubits,
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as that would collapse their quantum states.
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\end{itemize}
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We can deal with the first constraint by not duplicating information, instead
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spreading the quantum state across the physical qubits
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\cite[Sec.~I]{calderbank_good_1996}.
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To deal with phase-flip errors, we must take special care when
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constructing \ac{qec} codes.
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Using \ac{css} codes, for example, we can use two separate classical
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binary linear codes to protect against the two kinds of errors
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\cite[Sec. 10.5.6]{nielsen_quantum_2010}.
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Finally, we can get around the last issue by using \emph{stabilizer
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measurements}.
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These are parity measurements that give us information about
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potential errors without revealing the underlying qubit states
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\cite[Sec.~II.C.]{babar_fifteen_2015}.
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This way, we perform a \emph{syndrome extraction} and base the
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subsequent decoding process on the measured syndrome.
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Another difference between \ac{qec} and classical channel coding is
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the resource constraints.
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For \ac{qec}, achieving low latency matters more than having a low
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overall computational complexity, due to the backlog problem
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\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Certain gates turn
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single-qubit errors into multi-qubit ones, so errors must be
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corrected beforehand.
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A \ac{qec} system that is too slow accumulates a backlog at these points,
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causing exponential slowdown.
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Several code constructions have been proposed for \ac{qec} codes over the years.
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Topological codes, such as surface codes, have been the industry
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standard for experimental applications for a long time
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\cite[Sec.~I]{koutsioumpas_colour_2025}, due to their
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reliance on only local connections between qubits
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\cite[Sec.~5]{roffe_decoding_2020}.
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Recently, \ac{qldpc} codes have been getting increasing
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attention as they have been shown to offer comparable thresholds with
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substantially improved encoding rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
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\ac{qldpc} codes are generally decoded using a syndrome-based variant
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of the \ac{bp} algorithm \cite[Sec.~1]{roffe_decoding_2020}.
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We focus on \ac{qldpc} codes in our work and specifically \ac{bb} codes,
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as they are promising candidates for practical QEC due to their high
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encoding rates, large minimum distances, and short-depth syndrome
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extraction circuits \cite[Sec.~1]{bravyi_high-threshold_2024}.
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% DEMs and fault tolerance
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The syndrome extraction itself is implemented on quantum hardware and
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is therefore subject to the same noise as the data qubits.
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As a consequence, the \ac{qec} procedure, meant to protect the quantum
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state, itself introduces new \emph{internal errors}.
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A procedure is called \emph{fault-tolerant} if it remains effective
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even in the presence of these internal errors
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\cite[Sec.~4]{gottesman_introduction_2009}.
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To deal with internal errors that flip syndrome bits, multiple rounds
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of syndrome measurements are performed.
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One approach of implementing fault tolerance is using \acp{dem}.
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A \ac{dem} abstracts away the underlying circuit,
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focusing only on the relationship between possible errors
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and their effects on the syndrome \cite[Sec.~1.4.3]{higgott_practical_2024}.
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A \emph{detector error matrix} is generated from the circuit, which is
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used for decoding instead of the original check matrix.
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The detector error matrix is much larger than the
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check matrix of the underlying code, since it needs to represent many
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more error locations.
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For example, in our experiments using the $\llbracket 144,12,12
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\rrbracket$ \ac{bb} code with $12$ syndrome measurement rounds, the
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number of \acp{vn} grew from $144$ to $9504$ and the number of
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\acp{cn} grew from $72$ to $1008$.
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Therefore, decoding under a \ac{dem} poses a challenge with respect to the
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latency constraint.
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To keep the latency of \ac{dem} decoding manageable, one approach is
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\emph{sliding-window decoding}.
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Instead of decoding on the entire detector error matrix at once,
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it is partitioned into several overlapping windows.
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Once decoding of one window is complete, error estimates on the initial part
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that is no longer needed are committed, and the next window is processed.
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This way, decoding can start as soon as the syndrome bits required
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for the first window have been extracted.
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The idea originates with the \emph{overlapping recovery} scheme
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proposed for the surface code in
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\cite[Sec.~IV.B]{dennis_topological_2002} and has since been studied
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for surface and toric codes \cite{kuo_fault-tolerant_2024} as well as
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for \ac{qldpc} codes under both phenomenological and circuit-level
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noise \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}.
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% Reseach gap + our work
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We observe a structural similarity between sliding-window decoding for
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\acp{dem} and window decoding for \ac{sc}-\acs{ldpc} codes.
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In contrast to the latter, however, where \ac{bp} messages are
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carried between windows \cite[Sec.~III.~C.]{hassan_fully_2016},
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the existing realizations of sliding-window decoding for \ac{qec}
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discard the soft information produced inside one window before moving
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to the next.
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We propose \emph{warm-start sliding-window decoding}, in which the
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\ac{bp} messages from the overlap region of the previous window are
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reused to initialize \ac{bp} in the current window in place of the
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standard cold-start initialization.
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We formulate the warm start for standard \ac{bp} and for
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\ac{bpgd}, a variant of \ac{bp} with better convergence properties
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for \ac{qec} codes.
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The decoders are evaluated by Monte Carlo simulation on the
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$\llbracket 144,12,12 \rrbracket$ \ac{bb} code under standard
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circuit-based depolarizing noise over $12$ syndrome extraction rounds.
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The main finding is that warm-starting yields a consistent
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improvement at low iteration budgets, which is the regime relevant for
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low-latency operation.
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% Outline of the Thesis
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This thesis is structured as follows:
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\Cref{ch:Fundamentals} reviews the fundamentals of classical and
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quantum error correction.
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On the classical side, it covers binary linear block codes,
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\ac{ldpc} and \ac{sc}-\ac{ldpc} codes, and the \ac{bp} decoding
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algorithm.
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On the quantum side, it introduces the relevant quantum mechanical
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notation, stabilizer measurements, stabilizer codes, \acf{css} codes,
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\ac{qldpc} codes, and the \ac{bpgd} algorithm.
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\Cref{ch:Fault tolerance} introduces fault-tolerant \ac{qec}.
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It formalizes the notion of fault tolerance, presents the noise
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models considered in this work, and develops the \ac{dem} formalism
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through the measurement syndrome matrix, the detector matrix, and the
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detector error matrix.
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The chapter closes with a discussion of practical considerations
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including the choice of noise model, the per-round \acf{ler}, and the
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Stim toolchain.
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\Cref{ch:Decoding} considers practical aspects of decoding under \acp{dem}.
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It reviews the existing literature on sliding-window decoding for
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\ac{qec}, develops the formal windowing construction we build upon,
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introduces the proposed warm-start sliding-window decoder for
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plain \ac{bp} and for \ac{bpgd}, and reports numerical results on the
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$\llbracket 144,12,12 \rrbracket$ \ac{bb} code.
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% TODO: Possibly extend to mention specific proposed research directions
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\Cref{ch:Conclusion} concludes the thesis and outlines directions for
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further research.
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