diff --git a/src/thesis/chapters/1_introduction.tex b/src/thesis/chapters/1_introduction.tex index c7eaf33..605bd3c 100644 --- a/src/thesis/chapters/1_introduction.tex +++ b/src/thesis/chapters/1_introduction.tex @@ -3,14 +3,13 @@ % Intro to quantum computing -% TODO: Rephrase In 1982, Richard Feynman, motivated by the difficulty of simulating quantum-mechanical systems on classical hardware, put forward the idea of building computers from quantum hardware themselves \cite{feynman_simulating_1982}. The use of such quantum computers has since been shown to offer promising prospects not only with regard to simulating quantum systems but also -for solving certain kinds of problems that are classicaly intractable. +for solving certain kinds of problems that are classically intractable. The most prominent example is Shor's algorithm for integer factorization \cite{shor_algorithms_1994}. @@ -73,12 +72,12 @@ subsequent decoding process on the measured syndrome. Another difference between \ac{qec} and classical channel coding is the resource constraints. -For QEC, low latency matters more than low overall computational +For \ac{qec}, low latency matters more than low overall computational complexity, due to the backlog problem -\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Some gates may turn +\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Certain gates turn single-qubit errors into multi-qubit ones, so errors must be corrected beforehand. -A QEC system that is too slow accumulates a backlog at these points, +A \ac{qec} system that is too slow accumulates a backlog at these points, causing exponential slowdown. Several code constructions have been proposed for \ac{qec} codes over the years. @@ -87,43 +86,75 @@ standard for experimental applications for a long time \cite[Sec.~I]{koutsioumpas_colour_2025}, due to their reliance on only local connections between qubits \cite[Sec.~5]{roffe_decoding_2020}. -Recently, \ac{qldpc} codes have been getting increasingly more +Recently, \ac{qldpc} codes have been getting increasing attention as they have been shown to offer comparable thresholds with substantially improved encoding rates \cite[Sec.~1]{bravyi_high-threshold_2024}. \ac{qldpc} codes are generally decoded using a syndrome-based variant of the \ac{bp} algorithm \cite[Sec.~1]{roffe_decoding_2020}. +We focus on \ac{qldpc} codes in our work and specifically \ac{bb} codes, +as they are promising candidates for practical QEC due to their high +encoding rates, large minimum distances, and short-depth syndrome +extraction circuits \cite[Sec.~1]{bravyi_high-threshold_2024}. % DEMs and fault tolerance -\content{Syndrome extraction can also be faulty -> Need for fault tolerance} -\content{Have to repeat syndrome measurements} -\content{DEMs one way of implementing fault tolerance: Model more -error locations -> Larger resulting codes} -\content{Literature deals with latency problem for fault tolerance by -sliding-window decoding} +The syndrome extraction itself is implemented on quantum hardware and +is therefore subject to the same noise as the data qubits. +As a consequence, the \ac{qec} procedure, meant to protect the quantum +state, itself introduces new \emph{internal errors}. +A procedure is called \emph{fault-tolerant} if it remains effective +even in the presence of these internal errors +\cite[Sec.~4]{gottesman_introduction_2009}. +To deal with internal errors that flip syndrome bits, multiple rounds +of syndrome measurements are performed. + +One approach of implementing fault tolerance is using \acp{dem}. +A \ac{dem} abstracts away the underlying circuit, +focusing only on the relationship between possible errors +and their effects on the syndrome \cite[Sec.~1.4.3]{higgott_practical_2024}. +A \emph{detector error matrix} is generated from the circuit, which is +used for decoding instead of the original check matrix. +Decoding under a \ac{dem} poses a challenge with respect to the +latency constraint. +This is because the detector error matrix is much larger than the +check matrix of the underlying code, since it needs to represent many +more error locations. +For example, in our experiments using the $\llbracket 144,12,12 +\rrbracket$ \ac{bb} code with $12$ syndrome measurement rounds, the +number of \acp{vn} grew from $144$ to $9504$ and the number of +\acp{cn} grew from $72$ to $1008$. + +To keep the latency of \ac{dem} decoding manageable, one approach is +\emph{sliding-window decoding}. +Instead of decoding on the entire detector error matrix at once, +it is partitioned into several overlapping windows. +Once decoding of one window is complete, error estimates on the initial part +that is no longer needed are committed, and the next window is processed. +This way, decoding can start as soon as the syndrome bits required +for the first window have been extracted. +The idea originates with the \emph{overlapping recovery} scheme +proposed for the surface code in +\cite[Sec.~IV.B]{dennis_topological_2002} and has since been studied +for surface and toric codes \cite{kuo_fault-tolerant_2024} as well as +for \ac{qldpc} codes under both phenomenological and circuit-level +noise \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}. % Reseach gap + our work -\content{Use BP for decoding, but has convergence issues -> Modify BP} - -\content{We note a striking similarity between sliding-window -decoding for DEMs and the way SC-LDPC codes are decoded} -\content{Extend QEC sliding-window decoding by warm start, inspired -by SC-LDPC decoders} -The existing realizations of sliding-window decoding for \ac{qec} +We observe a structural similarity between sliding-window decoding for +\acp{dem} and window decoding for \ac{sc}-\acs{ldpc} codes. +In contrast to the latter, however, where \ac{bp} messages are +carried between windows \cite[Sec.~III.~C.]{hassan_fully_2016}, +the existing realizations of sliding-window decoding for \ac{qec} discard the soft information produced inside one window before moving -on to the next, in contrast to the analogous \ac{sc}-\ac{ldpc} -decoders, which carry messages between windows -\cite[Sec.~III.~C.]{hassan_fully_2016}. -This thesis investigates whether the same idea can be carried over to -the \ac{qec} setting. +to the next. We propose \emph{warm-start sliding-window decoding}, in which the \ac{bp} messages from the overlap region of the previous window are reused to initialize \ac{bp} in the current window in place of the standard cold-start initialization. We formulate the warm start first for plain \ac{bp} and then for -\ac{bpgd}, where some care is needed in deciding which information to -carry over. +\ac{bpgd}, a variant of \ac{bp} with better convergence properties +for \ac{qec} codes. The decoders are evaluated by Monte Carlo simulation on the $\llbracket 144,12,12 \rrbracket$ \ac{bb} code under standard circuit-based depolarizing noise over $12$ syndrome extraction rounds. @@ -131,140 +162,6 @@ The main finding is that warm-starting yields a consistent improvement at low iteration budgets, which is the regime relevant for low-latency operation. -% The need for fault tolerance - -% A naive picture of \ac{qec} treats the syndrome extraction circuit as -% ideal and only considers errors on the data qubits. -% In reality, every gate, every ancilla, and every measurement involved -% in extracting the syndrome can itself fail, introducing new faults -% into the procedure that is supposed to correct them -% \cite[Sec.~III]{shor_scheme_1995}. -% A \ac{qec} procedure is called \emph{fault-tolerant} if it remains -% effective in the presence of these internal faults -% \cite[Sec.~4]{gottesman_introduction_2009}. - -% Fault tolerance - -% The standard formal definition requires the number of output errors -% to remain bounded as long as the combined number of input and -% internal errors does not exceed the correction capability of the code -% \cite[Def.~4.2]{derks_designing_2025}. -% To deal with internal errors that flip syndrome bits, multiple rounds -% of syndrome measurements are performed, and the resulting space-time -% history of detector outcomes is decoded jointly. -% The probabilities of errors at each location in the circuit are -% collected in a \emph{noise model}. -% The most general such model, in which an arbitrary Pauli error is -% allowed after each gate, is referred to as \emph{circuit-level noise} -% \cite[Def.~2.5]{derks_designing_2025} and is the noise model that -% should be used for fault-tolerance simulations -% \cite[Sec.~4.2]{derks_designing_2025}. - -% DEMs - -% The combination of circuit-level noise and multiple syndrome -% measurement rounds yields a complicated, code- and circuit-specific -% decoding problem. -% A recent line of work argues that this problem is most cleanly -% expressed through a \acf{dem} \cite[Sec.~6]{derks_designing_2025}. -% A \ac{dem} abstracts away the underlying circuit and lists the -% independent error mechanisms together with the detectors they flip -% and the logical observables they affect. -% From the decoder's perspective, decoding under a \ac{dem} is again a -% classical decoding problem on a parity-check matrix, with the -% detectors playing the role of \acfp{cn} and the error mechanisms -% playing the role of \acfp{vn}. -% The standard tool for generating \acp{dem} from arbitrary stabilizer -% circuits is Stim \cite{gidney_stim_2021}, in which the \ac{dem} -% formalism was originally introduced. - -% The issues with deocoding under DEMs - -% For \ac{qec}, the binding constraint on the decoder is latency, not -% raw computational complexity. -% This is the \emph{backlog problem}: certain gates can transform -% existing single-qubit errors into multi-qubit errors, and any -% correction must be applied before such gates are reached. -% A decoder that fails to keep up with the rate at which the hardware -% produces syndromes leads to an exponential slowdown of the computation -% \cite[Sec.~II.G.3.]{terhal_quantum_2015}. - -% Decoding under a \ac{dem} aggravates this constraint, because the -% matrix that results from unrolling several rounds of syndrome -% extraction is much larger than the parity-check matrix of the -% underlying code. -% Each error mechanism in the circuit becomes a separate \ac{vn} and -% each detector becomes a separate \ac{cn}. -% For the $\llbracket 144,12,12 \rrbracket$ \acf{bb} code -% \cite[Sec.~3]{bravyi_high-threshold_2024} with $12$ syndrome -% measurement rounds, the number of \acp{vn} grows from $144$ to $9504$ -% and the number of \acp{cn} grows from $72$ to $1008$. - -% Exiting solutions to these issues (sliding-window decoding + BP modifications) - -% The dominant strategy for keeping the latency of \ac{dem} decoding -% manageable is \emph{sliding-window decoding}. -% Instead of decoding the entire space-time history at once, the -% decoder operates on a window that spans only a few syndrome -% measurement rounds. -% After each round, the window slides forward, and the corrections in -% the part of the previous window that is no longer needed are committed. -% The idea originates with the \emph{overlapping recovery} scheme -% proposed for the surface code in \cite[Sec.~IV.B]{dennis_topological_2002} -% and has since been studied for surface and toric codes -% \cite{kuo_fault-tolerant_2024} as well as for \ac{qldpc} codes under -% both phenomenological and circuit-level noise -% \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}. -% The structure of the decoding problem inside each window is -% reminiscent of \acf{sc}-\acf{ldpc} decoding from classical -% communications \cite[Intro.]{costello_spatially_2014}, where similar -% windowing techniques are used and where soft information is passed -% between consecutive windows -% \cite[Sec.~III.~C.]{hassan_fully_2016}. - -% We focus on QLDPC codes - -% In this work we focus on \acf{qldpc} codes, of which the \ac{bb} code -% mentioned above is one example. -% \ac{qldpc} codes have emerged as leading candidates for practical -% \ac{qec} due to their high encoding rates and large minimum distances -% at short syndrome-extraction-circuit depths -% \cite[Sec.~1]{bravyi_high-threshold_2024}. -% The natural decoder for them is \acf{bp}, which is well suited to -% sparse parity-check matrices and admits an efficient and parallel -% implementation, but is known to converge poorly on quantum codes due -% to quantum degeneracy and the unavoidable short cycles in the Tanner -% graph \cite[Sec.~II.C.]{babar_fifteen_2015}\cite[Sec.~V]{roffe_decoding_2020}. -% Several modifications of \ac{bp} have been proposed to address this: -% combining \ac{bp} with \acf{osd} \cite{roffe_decoding_2020}, decoding -% multiple variations of the code in parallel as in \acf{aed} -% \cite{koutsioumpas_automorphism_2025}, or extending \ac{bp} with -% guided decimation as in \acf{bpgd} \cite{yao_belief_2024}. - -% Contributions of this Thesis - -% The existing realizations of sliding-window decoding for \ac{qec} -% discard the soft information produced inside one window before moving -% on to the next, in contrast to the analogous \ac{sc}-\ac{ldpc} -% decoders, which carry messages between windows -% \cite[Sec.~III.~C.]{hassan_fully_2016}. -% This thesis investigates whether the same idea can be carried over to -% the \ac{qec} setting. -% -% We propose \emph{warm-start sliding-window decoding}, in which the -% \ac{bp} messages from the overlap region of the previous window are -% reused to initialize \ac{bp} in the current window in place of the -% standard cold-start initialization. -% We formulate the warm start first for plain \ac{bp} and then for -% \ac{bpgd}, where some care is needed in deciding which information to -% carry over. -% The decoders are evaluated by Monte Carlo simulation on the -% $\llbracket 144,12,12 \rrbracket$ \ac{bb} code under standard -% circuit-based depolarizing noise over $12$ syndrome extraction rounds. -% The main finding is that warm-starting yields a consistent -% improvement at low iteration budgets, which is the regime relevant for -% fault-tolerant operation. - % Outline of the Thesis \Cref{ch:Fundamentals} reviews the fundamentals of classical and @@ -292,6 +189,7 @@ introduces the proposed warm-start sliding-window decoder for plain \ac{bp} and for \ac{bpgd}, and reports numerical results on the $\llbracket 144,12,12 \rrbracket$ \ac{bb} code. +% TODO: Possibly extend to mention specific proposed research directions \Cref{ch:Conclusion} concludes the thesis and outlines directions for further research.