From 72acea0321648c3eb09da74a99697a6dd21adaba Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 4 May 2026 16:31:31 +0200 Subject: [PATCH] Incorporate Jonathan's corrections to the introduction --- src/thesis/chapters/1_introduction.tex | 34 +++++++++++++------------- src/thesis/main.tex | 5 ++++ 2 files changed, 22 insertions(+), 17 deletions(-) diff --git a/src/thesis/chapters/1_introduction.tex b/src/thesis/chapters/1_introduction.tex index 5faee6d..74380f0 100644 --- a/src/thesis/chapters/1_introduction.tex +++ b/src/thesis/chapters/1_introduction.tex @@ -17,7 +17,7 @@ factorization \cite{shor_algorithms_1994}. Similar to the way classical computers are built from bits and gates, quantum computers are built from \emph{qubits} and \emph{quantum gates}. -Because of quantum entanglement, it is not enough to consider the +Because of quantum entanglement, it does not suffice to consider the qubits individually, we also have to consider correlations between them. For a system of $n$ qubits, this makes the state space grow with $2^n$ instead of linearly with $n$, as would be the case for a classical system @@ -30,12 +30,11 @@ what provides them with their power \cite[Sec.~2.1]{roffe_decoding_2020}. Realizing algorithms that leverage these quantum-mechanical effects requires hardware that can execute long quantum computations reliably. This poses a problem, because the qubits making up current devices -are difficult to sufficiently isolate from their environment -\cite[Sec.~1]{roffe_quantum_2019}. -Their interaction with the environment acts as a continuous small-scale -measurement, an effect we call \emph{decoherence} of the stored quantum -state. -Decoherence is the reason large systems don't exhibit visible quantum +consistently interact with their environment \cite[Sec.~1]{roffe_quantum_2019}. +This interaction acts as a continuous small-scale measurement, an +effect we call \emph{decoherence} of the stored quantum state, which +results in errors on the qubits. +Decoherence is the reason large systems do not exhibit visible quantum properties at human scales \cite[Sec.~1]{gottesman_stabilizer_1997}. % Intro to QEC @@ -45,8 +44,8 @@ It addresses the issue by encoding the information of $k$ \emph{logical qubits} into a larger number $n>k$ of \emph{physical qubits}, in close analogy to classical channel coding \cite[Sec.~1]{roffe_quantum_2019}. -The redundancy introduced this way can then be used to restore -the quantum state, should it be disturbed. +The redundancy introduced this way can then be used to detect and +correct a corrupted the quantum state. The quantum setting imposes some important constraints that do not exist in the classical case, however \cite[Sec.~2.4]{roffe_quantum_2019}: \begin{itemize} @@ -54,7 +53,7 @@ classical case, however \cite[Sec.~2.4]{roffe_quantum_2019}: \item In addition to the bit-flip errors we know from the classical setting, qubits are subject to \emph{phase-flips}. \item We are not allowed to directly measure the encoded qubits, - as that would disturb their quantum states. + as that would collapse their quantum states. \end{itemize} We can deal with the first constraint by not duplicating information, instead spreading the quantum state across the physical qubits @@ -74,8 +73,8 @@ subsequent decoding process on the measured syndrome. Another difference between \ac{qec} and classical channel coding is the resource constraints. -For \ac{qec}, low latency matters more than low overall computational -complexity, due to the backlog problem +For \ac{qec}, achieving low latency matters more than having a low +overall computational complexity, due to the backlog problem \cite[Sec.~II.G.3.]{terhal_quantum_2015}: Certain gates turn single-qubit errors into multi-qubit ones, so errors must be corrected beforehand. @@ -83,7 +82,7 @@ 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. -Topological codes such as surface codes have been the industry +Topological codes, such as surface codes, have been the industry standard for experimental applications for a long time \cite[Sec.~I]{koutsioumpas_colour_2025}, due to their reliance on only local connections between qubits @@ -116,15 +115,15 @@ 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 +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 $1008$. +Therefore, decoding under a \ac{dem} poses a challenge with respect to the +latency constraint. To keep the latency of \ac{dem} decoding manageable, one approach is \emph{sliding-window decoding}. @@ -154,7 +153,7 @@ 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 +We formulate the warm start for standard \ac{bp} and for \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 @@ -166,6 +165,7 @@ low-latency operation. % Outline of the Thesis +This thesis is structured as follows: \Cref{ch:Fundamentals} reviews the fundamentals of classical and quantum error correction. On the classical side, it covers binary linear block codes, diff --git a/src/thesis/main.tex b/src/thesis/main.tex index b83143f..4a2d15b 100644 --- a/src/thesis/main.tex +++ b/src/thesis/main.tex @@ -123,6 +123,11 @@ % \listoftables % \include{abbreviations} +% \cleardoublepage +% \phantomsection +% \addcontentsline{toc}{chapter}{List of Abbreviations} +% \printacronyms + \bibliography{lib/cel-thesis/IEEEabrv,src/thesis/bibliography} \end{document}