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\chapter{Introduction}
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\label{ch:Introduction}
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% Intro to quantum computing
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% TODO: Rephrase
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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 from quantum hardware themselves
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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 classicaly 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 is not enough 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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are difficult to sufficiently isolate from their environment
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\cite[Sec.~1]{roffe_quantum_2019}.
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Their interaction with the environment acts as a continuous small-scale
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measurement, an effect we call \emph{decoherence} of the stored quantum
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state.
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Decoherence is the reason large systems don't 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 restore
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the quantum state, should it be disturbed.
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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 disturb 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 QEC, low latency matters more than low overall computational
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complexity, due to the backlog problem
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\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Some gates may 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 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 increasingly more
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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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% DEMs and fault tolerance
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\content{Syndrome extraction can also be faulty -> Need for fault tolerance}
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\content{Have to repeat syndrome measurements}
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\content{DEMs one way of implementing fault tolerance: Model more
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error locations -> Larger resulting codes}
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\content{Literature deals with latency problem for fault tolerance by
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sliding-window decoding}
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% Reseach gap + our work
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\content{Use BP for decoding, but has convergence issues -> Modify BP}
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\content{We note a striking similarity between sliding-window
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decoding for DEMs and the way SC-LDPC codes are decoded}
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\content{Extend QEC sliding-window decoding by warm start, inspired
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by SC-LDPC decoders}
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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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on to the next, in contrast to the analogous \ac{sc}-\ac{ldpc}
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decoders, which carry messages between windows
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\cite[Sec.~III.~C.]{hassan_fully_2016}.
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This thesis investigates whether the same idea can be carried over to
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the \ac{qec} setting.
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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 first for plain \ac{bp} and then for
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\ac{bpgd}, where some care is needed in deciding which information to
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carry over.
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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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% The need for fault tolerance
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% A naive picture of \ac{qec} treats the syndrome extraction circuit as
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% ideal and only considers errors on the data qubits.
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% In reality, every gate, every ancilla, and every measurement involved
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% in extracting the syndrome can itself fail, introducing new faults
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% into the procedure that is supposed to correct them
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% \cite[Sec.~III]{shor_scheme_1995}.
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% A \ac{qec} procedure is called \emph{fault-tolerant} if it remains
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% effective in the presence of these internal faults
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% \cite[Sec.~4]{gottesman_introduction_2009}.
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% Fault tolerance
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% The standard formal definition requires the number of output errors
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% to remain bounded as long as the combined number of input and
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% internal errors does not exceed the correction capability of the code
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% \cite[Def.~4.2]{derks_designing_2025}.
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% To deal with internal errors that flip syndrome bits, multiple rounds
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% of syndrome measurements are performed, and the resulting space-time
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% history of detector outcomes is decoded jointly.
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% The probabilities of errors at each location in the circuit are
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% collected in a \emph{noise model}.
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% The most general such model, in which an arbitrary Pauli error is
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% allowed after each gate, is referred to as \emph{circuit-level noise}
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% \cite[Def.~2.5]{derks_designing_2025} and is the noise model that
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% should be used for fault-tolerance simulations
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% \cite[Sec.~4.2]{derks_designing_2025}.
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% DEMs
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% The combination of circuit-level noise and multiple syndrome
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% measurement rounds yields a complicated, code- and circuit-specific
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% decoding problem.
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% A recent line of work argues that this problem is most cleanly
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% expressed through a \acf{dem} \cite[Sec.~6]{derks_designing_2025}.
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% A \ac{dem} abstracts away the underlying circuit and lists the
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% independent error mechanisms together with the detectors they flip
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% and the logical observables they affect.
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% From the decoder's perspective, decoding under a \ac{dem} is again a
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% classical decoding problem on a parity-check matrix, with the
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% detectors playing the role of \acfp{cn} and the error mechanisms
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% playing the role of \acfp{vn}.
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% The standard tool for generating \acp{dem} from arbitrary stabilizer
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% circuits is Stim \cite{gidney_stim_2021}, in which the \ac{dem}
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% formalism was originally introduced.
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% The issues with deocoding under DEMs
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% For \ac{qec}, the binding constraint on the decoder is latency, not
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% raw computational complexity.
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% This is the \emph{backlog problem}: certain gates can transform
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% existing single-qubit errors into multi-qubit errors, and any
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% correction must be applied before such gates are reached.
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% A decoder that fails to keep up with the rate at which the hardware
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% produces syndromes leads to an exponential slowdown of the computation
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% \cite[Sec.~II.G.3.]{terhal_quantum_2015}.
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% Decoding under a \ac{dem} aggravates this constraint, because the
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% matrix that results from unrolling several rounds of syndrome
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% extraction is much larger than the parity-check matrix of the
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% underlying code.
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% Each error mechanism in the circuit becomes a separate \ac{vn} and
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% each detector becomes a separate \ac{cn}.
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% For the $\llbracket 144,12,12 \rrbracket$ \acf{bb} code
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% \cite[Sec.~3]{bravyi_high-threshold_2024} with $12$ syndrome
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% measurement rounds, the number of \acp{vn} grows from $144$ to $9504$
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% and the number of \acp{cn} grows from $72$ to $1008$.
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% Exiting solutions to these issues (sliding-window decoding + BP modifications)
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% The dominant strategy for keeping the latency of \ac{dem} decoding
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% manageable is \emph{sliding-window decoding}.
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% Instead of decoding the entire space-time history at once, the
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% decoder operates on a window that spans only a few syndrome
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% measurement rounds.
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% After each round, the window slides forward, and the corrections in
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% the part of the previous window that is no longer needed are committed.
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% The idea originates with the \emph{overlapping recovery} scheme
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% proposed for the surface code in \cite[Sec.~IV.B]{dennis_topological_2002}
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% and has since been studied for surface and toric codes
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% \cite{kuo_fault-tolerant_2024} as well as for \ac{qldpc} codes under
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% both phenomenological and circuit-level noise
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% \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}.
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% The structure of the decoding problem inside each window is
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% reminiscent of \acf{sc}-\acf{ldpc} decoding from classical
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% communications \cite[Intro.]{costello_spatially_2014}, where similar
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% windowing techniques are used and where soft information is passed
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% between consecutive windows
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% \cite[Sec.~III.~C.]{hassan_fully_2016}.
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% We focus on QLDPC codes
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% In this work we focus on \acf{qldpc} codes, of which the \ac{bb} code
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% mentioned above is one example.
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% \ac{qldpc} codes have emerged as leading candidates for practical
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% \ac{qec} due to their high encoding rates and large minimum distances
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% at short syndrome-extraction-circuit depths
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% \cite[Sec.~1]{bravyi_high-threshold_2024}.
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% The natural decoder for them is \acf{bp}, which is well suited to
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% sparse parity-check matrices and admits an efficient and parallel
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% implementation, but is known to converge poorly on quantum codes due
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% to quantum degeneracy and the unavoidable short cycles in the Tanner
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% graph \cite[Sec.~II.C.]{babar_fifteen_2015}\cite[Sec.~V]{roffe_decoding_2020}.
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% Several modifications of \ac{bp} have been proposed to address this:
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% combining \ac{bp} with \acf{osd} \cite{roffe_decoding_2020}, decoding
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% multiple variations of the code in parallel as in \acf{aed}
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% \cite{koutsioumpas_automorphism_2025}, or extending \ac{bp} with
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% guided decimation as in \acf{bpgd} \cite{yao_belief_2024}.
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% Contributions of this Thesis
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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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% on to the next, in contrast to the analogous \ac{sc}-\ac{ldpc}
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% decoders, which carry messages between windows
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% \cite[Sec.~III.~C.]{hassan_fully_2016}.
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% This thesis investigates whether the same idea can be carried over to
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% the \ac{qec} setting.
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%
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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 first for plain \ac{bp} and then for
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% \ac{bpgd}, where some care is needed in deciding which information to
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% carry over.
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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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% fault-tolerant operation.
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% Outline of the Thesis
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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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\Cref{ch:Conclusion} concludes the thesis and outlines directions for
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further research.
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