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Add project structure and write part of introduction

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# This script undoes the effect of .setup_local_env.sh
rm ./src/**/lib
rm ./src/**/.latexmkrc
rm ./src/**/src

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.latexmkrc Normal file
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$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S";
$out_dir = 'build';
$pdf_mode = 1;

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# This script sets up the environment to be able to compile the latex source
# files from within the 'src' directory with latexmk.
#
# Context: The project is set up in such a way that running make from the root
# directory compiles all documents. When using vimtex, however, when compiling
# a document it is always assumed that the root directory is that of the
# document currently being edited. Running this script enables the documents in
# 'src' to be edited and compiled with vimtex as well.
SRC_DIRS=$(ls src)
for SRC_DIR in $SRC_DIRS; do
# Create the latexmkrc file. Note the -cd option which changes the working
# directory
echo '$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S -cd ./../..";' >./src/$SRC_DIR/.latexmkrc
echo '$out_dir = "build";' >>./src/$SRC_DIR/.latexmkrc
echo '$pdf_mode = 1;' >>./src/$SRC_DIR/.latexmkrc
# Changing the working driectory using the -cd option seems to only affect
# source files.
#ln -s `pwd`/src ./src
ln -s $(pwd)/lib ./src/$SRC_DIR/lib
ln -s $(pwd)/src ./src/$SRC_DIR/src
done

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Dockerfile Normal file
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FROM ubuntu:24.04
ARG DEBIAN_FRONTEND=noninteractive
RUN apt update -y && apt upgrade -y
RUN apt install make texlive latexmk texlive-pictures -y
RUN apt install texlive-publishers texlive-science texlive-fonts-extra texlive-latex-extra -y
RUN apt install biber texlive-bibtex-extra -y
RUN apt install texlive-lang-german -y
RUN apt install python3 python3-pygments -y
# The 'bbm' package insists on generating stuff in the home directory. In
# order to guarantee access to the home directory no matter the user the
# docker container is run with, create a temporary one anyone can write to
RUN mkdir /tmp/home
RUN chmod -R 777 /tmp/home
ENV HOME=/tmp/home

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Makefile Normal file
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DOCUMENTS := $(patsubst src/%/main.tex,build/%.pdf,$(wildcard src/*/main.tex))
.PHONY: all
all: $(DOCUMENTS)
build/%.pdf: src/%/main.tex build/prepared
latexmk $<
mv build/main.pdf $@
build/prepared:
mkdir -p build
touch build/prepared
.PHONY: clean
clean:
rm -rf build

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README.md Normal file
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src/intro/.latexmkrc Normal file
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$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S -cd ./../..";
$out_dir = "build";
$pdf_mode = 1;

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src/intro/MA.bib Normal file
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@misc{derks_designing_2025,
title = {Designing fault-tolerant circuits using detector error models},
url = {http://arxiv.org/abs/2407.13826},
doi = {10.48550/arXiv.2407.13826},
abstract = {Quantum error-correcting codes, such as subspace, subsystem, and Floquet codes, are typically constructed within the stabilizer formalism, which does not fully capture the idea of fault-tolerance needed for practical quantum computing applications. In this work, we explore the remarkably powerful formalism of detector error models, which fully captures fault-tolerance at the circuit level. We introduce the detector error model formalism in a pedagogical manner and provide several examples. Additionally, we apply the formalism to three different levels of abstraction in the engineering cycle of fault-tolerant circuit designs: finding robust syndrome extraction circuits, identifying efficient measurement schedules, and constructing fault-tolerant procedures. We enhance the surface code's resistance to measurement errors, devise short measurement schedules for color codes, and implement a more efficient fault-tolerant method for measuring logical operators.},
number = {{arXiv}:2407.13826},
publisher = {{arXiv}},
author = {Derks, Peter-Jan H. S. and Townsend-Teague, Alex and Burchards, Ansgar G. and Eisert, Jens},
urldate = {2025-10-28},
date = {2025-10-25},
eprinttype = {arxiv},
eprint = {2407.13826 [quant-ph]},
keywords = {Quantum Physics, /s1, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/NLEMWTH8/Derks et al. - 2025 - Designing fault-tolerant circuits using detector error models.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/QEBN4FJT/2407.html:text/html},
}
@online{nielsen_quantum_2010,
title = {Quantum Computation and Quantum Information: 10th Anniversary Edition},
url = {https://www.cambridge.org/highereducation/books/quantum-computation-and-quantum-information/01E10196D0A682A6AEFFEA52D53BE9AE},
shorttitle = {Quantum Computation and Quantum Information},
abstract = {One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.},
titleaddon = {Cambridge Aspire website},
author = {Nielsen, Michael A. and Chuang, Isaac L.},
urldate = {2025-10-28},
date = {2010-12-09},
langid = {english},
doi = {10.1017/CBO9780511976667},
note = {{ISBN}: 9780511976667
Publisher: Cambridge University Press},
keywords = {\#{FND}, \#{QM}, \#{QEC}},
file = {PDF:/home/andreas/Zotero/storage/2FGWZ5CC/Nielsen and Chuang - 2010 - Quantum Computation and Quantum Information 10th Anniversary Edition.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/RFPYY4AS/01E10196D0A682A6AEFFEA52D53BE9AE.html:text/html},
}
@thesis{klinke_neural_2025,
location = {Karlsruhe},
title = {Neural Belief Propagation Ensemble Decoding of Quantum {LDPC} Codes},
institution = {{KIT}},
type = {Bachelor's Thesis},
author = {Klinke, Jeremi},
date = {2025-09-26},
keywords = {/s1, \#{QEC}},
file = {PDF:/home/andreas/Zotero/storage/ENJG2F8D/Klinke - Neural Belief Propagation Ensemble Decoding of Quantum LDPC Codes.pdf:application/pdf},
}
@article{miao_quaternary_2025,
title = {Quaternary Neural Belief Propagation Decoding of Quantum {LDPC} Codes with Overcomplete Check Matrices},
volume = {13},
issn = {2169-3536},
url = {http://arxiv.org/abs/2308.08208},
doi = {10.1109/ACCESS.2025.3539475},
abstract = {Quantum low-density parity-check ({QLDPC}) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing {QLDPC} codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode {QLDPC} codes with a belief propagation ({BP}) decoder operating on overcomplete check matrices. Then, we extend the neural {BP} ({NBP}) decoder, which was originally studied for suboptimal binary {BP} decoding of {QLPDC} codes, to quaternary {BP} decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length {QLDPC} codes.},
pages = {25637--25649},
journaltitle = {{IEEE} Access},
shortjournal = {{IEEE} Access},
author = {Miao, Sisi and Schnerring, Alexander and Li, Haizheng and Schmalen, Laurent},
urldate = {2025-10-28},
date = {2025-02-05},
eprinttype = {arxiv},
eprint = {2308.08208 [quant-ph]},
note = {{TLDR}: This work proposes to decode {QLDPC} codes with a belief propagation ({BP}) decoder operating on overcomplete check matrices and extends the neural {BP} decoder, which was originally studied for suboptimal binary {BP} decoding of {QLPDC} codes, to quaternary {BP} decoders.},
keywords = {Quantum Physics, Computer Science - Information Theory, Mathematics - Information Theory, /unread, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/SJXAPQ9Z/Miao et al. - 2025 - Quaternary Neural Belief Propagation Decoding of Quantum LDPC Codes with Overcomplete Check Matrices.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/ZVHBIEHR/2308.html:text/html},
}
@article{roffe_quantum_2019,
title = {Quantum error correction: an introductory guide},
volume = {60},
issn = {0010-7514},
url = {https://doi.org/10.1080/00107514.2019.1667078},
doi = {10.1080/00107514.2019.1667078},
shorttitle = {Quantum error correction},
abstract = {Quantum error correction protocols will play a central role in the realisation of quantum computing; the choice of error correction code will influence the full quantum computing stack, from the layout of qubits at the physical level to gate compilation strategies at the software level. As such, familiarity with quantum coding is an essential prerequisite for the understanding of current and future quantum computing architectures. In this review, we provide an introductory guide to the theory and implementation of quantum error correction codes. Where possible, fundamental concepts are described using the simplest examples of detection and correction codes, the working of which can be verified by hand. We outline the construction and operation of the surface code, the most widely pursued error correction protocol for experiment. Finally, we discuss issues that arise in the practical implementation of the surface code and other quantum error correction codes.},
pages = {226--245},
number = {3},
journaltitle = {Contemporary Physics},
author = {Roffe, Joschka},
urldate = {2025-11-04},
date = {2019-07-03},
keywords = {/s1, \#{FND}, \#{QEC}},
file = {Full Text PDF:/home/andreas/Zotero/storage/DW4EYDQ8/Roffe - 2019 - Quantum error correction an introductory guide.pdf:application/pdf},
}
@misc{calderbank_quantum_1997,
title = {Quantum Error Correction via Codes over {GF}(4)},
url = {http://arxiv.org/abs/quant-ph/9608006},
doi = {10.48550/arXiv.quant-ph/9608006},
abstract = {The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field {GF}(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.},
number = {{arXiv}:quant-ph/9608006},
publisher = {{arXiv}},
author = {Calderbank, A. R. and Rains, E. M. and Shor, P. W. and Sloane, N. J. A.},
urldate = {2025-11-05},
date = {1997-09-10},
eprinttype = {arxiv},
eprint = {quant-ph/9608006},
keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/5IM4A6FA/Calderbank et al. - 1997 - Quantum Error Correction via Codes over GF(4).pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/LLQUFH76/9608006.html:text/html},
}
@article{shor_scheme_1995,
title = {Scheme for reducing decoherence in quantum computer memory},
volume = {52},
rights = {http://link.aps.org/licenses/aps-default-license},
issn = {1050-2947, 1094-1622},
url = {https://link.aps.org/doi/10.1103/PhysRevA.52.R2493},
doi = {10.1103/PhysRevA.52.R2493},
pages = {R2493--R2496},
number = {4},
journaltitle = {Physical Review A},
shortjournal = {Phys. Rev. A},
author = {Shor, Peter W.},
urldate = {2025-11-05},
date = {1995-10-01},
langid = {english},
note = {{TLDR}: In the mid-1990s, theorists devised methods to preserve the integrity of quantum bits{\textbackslash}char22\{\}techniques that may become the key to practical quantum computing on a large scale.},
keywords = {/s2, \#{FND}, \#{QEC}},
file = {PDF:/home/andreas/Zotero/storage/DG6QT7UX/Shor - 1995 - Scheme for reducing decoherence in quantum computer memory.pdf:application/pdf},
}
@article{divincenzo_fault-tolerant_1996,
title = {Fault-Tolerant Error Correction with Efficient Quantum Codes},
volume = {77},
issn = {0031-9007, 1079-7114},
url = {http://arxiv.org/abs/quant-ph/9605031},
doi = {10.1103/PhysRevLett.77.3260},
abstract = {We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced group-theoretic framework for unifying all known quantum codes.},
pages = {3260--3263},
number = {15},
journaltitle = {Physical Review Letters},
shortjournal = {Phys. Rev. Lett.},
author = {{DiVincenzo}, David P. and Shor, Peter W.},
urldate = {2025-11-05},
date = {1996-10-07},
eprinttype = {arxiv},
eprint = {quant-ph/9605031},
note = {{TLDR}: This work exhibits a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes, derived using a recently introduced group-theoretic framework for unifying all known quantum codes.},
keywords = {Quantum Physics, /unread, \#{FND}, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/KNGHIXB3/DiVincenzo and Shor - 1996 - Fault-Tolerant Error Correction with Efficient Quantum Codes.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/6JURUG3K/9605031.html:text/html},
}
@misc{shor_fault-tolerant_1997,
title = {Fault-tolerant quantum computation},
url = {http://arxiv.org/abs/quant-ph/9605011},
doi = {10.48550/arXiv.quant-ph/9605011},
abstract = {Recently, it was realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties of realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, thus making long computations impossible. A futher difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering the output of long computations unreliable. It was previously known that a quantum circuit with t gates could tolerate O(1/t) amounts of inaccuracy and decoherence per gate. We show, for any quantum computation with t gates, how to build a polynomial size quantum circuit that can tolerate O(1/(log t){\textasciicircum}c) amounts of inaccuracy and decoherence per gate, for some constant c. We do this by showing how to compute using quantum error correcting codes. These codes were previously known to provide resistance to errors while storing and transmitting quantum data.},
number = {{arXiv}:quant-ph/9605011},
publisher = {{arXiv}},
author = {Shor, Peter W.},
urldate = {2025-11-05},
date = {1997-03-05},
eprinttype = {arxiv},
eprint = {quant-ph/9605011},
keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/CSLTPZU5/Shor - 1997 - Fault-tolerant quantum computation.pdf:application/pdf},
}
@misc{gottesman_stabilizer_1997,
title = {Stabilizer Codes and Quantum Error Correction},
url = {http://arxiv.org/abs/quant-ph/9705052},
doi = {10.48550/arXiv.quant-ph/9705052},
abstract = {Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.},
number = {{arXiv}:quant-ph/9705052},
publisher = {{arXiv}},
author = {Gottesman, Daniel},
urldate = {2025-11-06},
date = {1997-05-28},
eprinttype = {arxiv},
eprint = {quant-ph/9705052},
keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/JT582GBB/Gottesman - 1997 - Stabilizer Codes and Quantum Error Correction.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/5GCZHHTH/9705052.html:text/html},
}
@article{gottesman_theory_1998,
title = {Theory of fault-tolerant quantum computation},
volume = {57},
url = {https://link.aps.org/doi/10.1103/PhysRevA.57.127},
doi = {10.1103/PhysRevA.57.127},
abstract = {In order to use quantum error-correcting codes to improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a theory of fault-tolerant operations on stabilizer codes based on symmetries of the code stabilizer. This allows a straightforward determination of which operations can be performed fault-tolerantly on a given code. I demonstrate that fault-tolerant universal computation is possible for any stabilizer code. I discuss a number of examples in more detail, including the five-quantum-bit code.},
pages = {127--137},
number = {1},
journaltitle = {Physical Review A},
shortjournal = {Phys. Rev. A},
author = {Gottesman, Daniel},
urldate = {2025-11-06},
date = {1998-01-01},
note = {Publisher: American Physical Society
{TLDR}: It is demonstrated that fault-tolerant universal computation is possible for any stabilizer code, including the five-quantum-bit code.},
keywords = {/s1, \#{FND}, \#{QEC}},
file = {APS Snapshot:/home/andreas/Zotero/storage/BP7CHBIU/PhysRevA.57.html:text/html;Full Text PDF:/home/andreas/Zotero/storage/7E5TUIMN/Gottesman - 1998 - Theory of fault-tolerant quantum computation.pdf:application/pdf},
}
@misc{gottesman_introduction_2009,
title = {An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation},
url = {http://arxiv.org/abs/0904.2557},
doi = {10.48550/arXiv.0904.2557},
abstract = {Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over {GF}(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.},
number = {{arXiv}:0904.2557},
publisher = {{arXiv}},
author = {Gottesman, Daniel},
urldate = {2025-11-06},
date = {2009-04-16},
eprinttype = {arxiv},
eprint = {0904.2557 [quant-ph]},
keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/AGETMT4C/Gottesman - 2009 - An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/UHAPPP5S/0904.html:text/html},
}
@article{calderbank_good_1996,
title = {Good quantum error-correcting codes exist},
volume = {54},
url = {https://link.aps.org/doi/10.1103/PhysRevA.54.1098},
doi = {10.1103/PhysRevA.54.1098},
abstract = {A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n=1-2𝐻2(2t/n) where 𝐻2(p) is the binary entropy function -plog2p-(1-p)log2(1-p). Upper bounds on this asymptotic rate are given. © 1996 The American Physical Society.},
pages = {1098--1105},
number = {2},
journaltitle = {Physical Review A},
shortjournal = {Phys. Rev. A},
author = {Calderbank, A. R. and Shor, Peter W.},
urldate = {2025-11-06},
date = {1996-08-01},
note = {Publisher: American Physical Society
{TLDR}: The techniques investigated in this paper can be extended so as to reduce the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions.},
keywords = {/s1, \#{FND}, \#{QEC}},
file = {APS Snapshot:/home/andreas/Zotero/storage/IK4DH994/PhysRevA.54.html:text/html;Full Text PDF:/home/andreas/Zotero/storage/RLKB7SKX/Calderbank and Shor - 1996 - Good quantum error-correcting codes exist.pdf:application/pdf},
}
@book{griffiths_introduction_1995,
title = {Introduction to Quantum Mechanics},
isbn = {0-13-124405-1},
abstract = {Changes and additions to the new edition of this classic textbook include a new chapter on symmetries, new problems and examples, improved explanations, more numerical problems to be worked on a computer, new applications to solid state physics, and consolidated treatment of time-dependent potentials.},
publisher = {Prentice Hall},
author = {Griffiths, David J.},
date = {1995},
langid = {english},
keywords = {\#{FND}, \#{QM}, \#{MAT}},
file = {PDF:/home/andreas/Zotero/storage/ZLP4S5EB/Griffiths and Schroeter - 2018 - Introduction to Quantum Mechanics.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/V7B6J8YI/990799CA07A83FC5312402AF6860311E.html:text/html},
}
@online{bradley_tensor_2018,
title = {The Tensor Product, Demystified},
url = {https://www.math3ma.com/blog/the-tensor-product-demystified},
author = {Bradley, Tai-Danae},
urldate = {2025-11-11},
date = {2018-11-18},
keywords = {\#{FND}, \#{MAT}},
file = {Snapshot:/home/andreas/Zotero/storage/JWTQ4W7G/the-tensor-product-demystified.html:text/html},
}
@misc{camps-moreno_toward_2024,
title = {Toward Quantum {CSS}-T Codes from Sparse Matrices},
url = {http://arxiv.org/abs/2406.00425},
doi = {10.48550/arXiv.2406.00425},
abstract = {{CSS}-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A {CSS}-T code depends on a pair \$(C\_1, C\_2)\$ of binary linear codes \$C\_1\$ and \$C\_2\$ that satisfy certain conditions. We prove that \$C\_1\$ and \$C\_2\$ form a {CSS}-T pair if and only if \$C\_2 {\textbackslash}subset {\textbackslash}operatorname\{Hull\}(C\_1) {\textbackslash}cap {\textbackslash}operatorname\{Hull\}(C\_1{\textasciicircum}2)\$, where the hull of a code is the intersection of the code with its dual. We show that if \$(C\_1,C\_2)\$ is a {CSS}-T pair, and the code \$C\_2\$ is degenerated on \${\textbackslash}\{i{\textbackslash}\}\$, meaning that the \$i{\textasciicircum}\{th\}\$-entry is zero for all the elements in \$C\_2\$, then the pair of punctured codes \$(C\_1{\textbar}\_i,C\_2{\textbar}\_i)\$ is also a {CSS}-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum {LDPC} or {LDGM} {CSS}-T codes computationally.},
number = {{arXiv}:2406.00425},
publisher = {{arXiv}},
author = {Camps-Moreno, Eduardo and López, Hiram H. and Matthews, Gretchen L. and {McMillon}, Emily},
urldate = {2025-11-13},
date = {2024-06-04},
eprinttype = {arxiv},
eprint = {2406.00425 [cs]},
keywords = {Computer Science - Information Theory, /unread, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/C634YE7N/Camps-Moreno et al. - 2024 - Toward Quantum CSS-T Codes from Sparse Matrices.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/TD3KFLFZ/2406.html:text/html},
}
@misc{koutsioumpas_colour_2025,
title = {Colour Codes Reach Surface Code Performance using Vibe Decoding},
url = {http://arxiv.org/abs/2508.15743},
doi = {10.48550/arXiv.2508.15743},
abstract = {Two-dimensional quantum colour codes hold significant promise for quantum error correction, offering advantages such as planar connectivity and low overhead logical gates. Despite their theoretical appeal, the practical deployment of these codes faces challenges due to complex decoding requirements compared to surface codes. This paper introduces vibe decoding which, for the first time, brings colour code performance on par with the surface code under practical decoding. Our approach leverages an ensemble of belief propagation decoders - each executing a distinct serial message passing schedule - combined with localised statistics post-processing. We refer to this combined protocol as {VibeLSD}. The {VibeLSD} decoder is highly versatile: our numerical results show it outperforms all practical existing colour code decoders across various syndrome extraction schemes, noise models, and error rates. By estimating qubit footprints through quantum memory simulations, we show that colour codes can operate with overhead that is comparable to, and in some cases lower than, that of the surface code. This, combined with the fact that localised statistics decoding is a parallel algorithm, makes {VibeLSD} suitable for implementation on specialised hardware for real-time decoding. Our results establish the colour code as a practical architecture for near-term quantum hardware, providing improved compilation efficiency for both Clifford and non-Clifford gates without incurring additional qubit overhead relative to the surface code.},
number = {{arXiv}:2508.15743},
publisher = {{arXiv}},
author = {Koutsioumpas, Stergios and Noszko, Tamas and Sayginel, Hasan and Webster, Mark and Roffe, Joschka},
urldate = {2025-11-13},
date = {2025-08-22},
eprinttype = {arxiv},
eprint = {2508.15743 [quant-ph]},
note = {{TLDR}: The results establish the colour code as a practical architecture for near-term quantum hardware, providing improved compilation efficiency for both Clifford and non-Clifford gates without incurring additional qubit overhead relative to the surface code.},
keywords = {Quantum Physics, Computer Science - Information Theory, /s1, \#{QEC}},
file = {Preprint PDF:/home/andreas/Zotero/storage/C4K3XG2S/Koutsioumpas et al. - 2025 - Colour Codes Reach Surface Code Performance using Vibe Decoding.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/76EXKH3S/2508.html:text/html},
}
@article{koutsioumpas_automorphism_2025,
title = {Automorphism Ensemble Decoding of Quantum {LDPC} Codes},
author = {Koutsioumpas, Stergios and Sayginel, Hasan and Webster, Mark and Browne, Dan E},
date = {2025-03-04},
langid = {english},
keywords = {/s1, \#{QEC}},
file = {PDF:/home/andreas/Zotero/storage/SHUGDAU8/Koutsioumpas et al. - Automorphism Ensemble Decoding of Quantum LDPC Codes.pdf:application/pdf},
}
@article{geiselhart_automorphism_2021,
title = {Automorphism Ensemble Decoding of Reed–Muller Codes},
volume = {69},
issn = {1558-0857},
url = {https://ieeexplore.ieee.org/document/9492151},
doi = {10.1109/TCOMM.2021.3098798},
abstract = {Reed–Muller ({RM}) codes are known for their good maximum likelihood ({ML}) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for {RM} codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation ({MBBP}) and may use any polar or {RM} decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation ({SC})-, {SC}-list ({SCL})- and belief propagation ({BP})-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-{ML} performance for the {RM}(3,7)-code (e.g., 0.04 {dB} away from the {ML} bound at {BLER} of 10−3) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of {RM} codes and {SC} decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes.},
pages = {6424--6438},
number = {10},
journaltitle = {{IEEE} Transactions on Communications},
author = {Geiselhart, Marvin and Elkelesh, Ahmed and Ebada, Moustafa and Cammerer, Sebastian and Brink, Stephan ten},
urldate = {2025-11-13},
date = {2021-07-21},
note = {{TLDR}: A versatile decoding architecture for {RM} codes based on their rich automorphism group is presented and the theoretical limitations of this method with respect to polar codes are proved.},
keywords = {/unread, \#{FND}, Belief propagation, belief propagation decoding, code automorphisms, Complexity theory, Encoding, ensemble decoding, Generators, Iterative decoding, list decoding, Maximum likelihood decoding, polar codes, Polar codes, Reed-Muller Codes, successive cancellation decoding, \#{CEC}},
file = {Full Text PDF:/home/andreas/Zotero/storage/KV3JR3MS/Geiselhart et al. - 2021 - Automorphism Ensemble Decoding of Reed–Muller Codes.pdf:application/pdf},
}

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\documentclass[dvipsnames]{article}
\usepackage[a4paper,left=3cm,right=2cm,top=2.5cm,bottom=2.5cm]{geometry}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mleftright}
\usepackage{bm}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepackage{acro}
\usepackage{braket}
\usepackage[
backend=biber,
style=ieee,
sorting=nty,
]{biblatex}
\usetikzlibrary{calc, positioning}
\title{A Brief Introduction to Quantum Error Correction}
\author{Andreas Tsouchlos}
\DeclareAcronym{qec}{
short=QEC,
long=quantum error correction
}
\addbibresource{src/intro/MA.bib}
\begin{document}
\maketitle
% TODO: Give context for why we need QC and QEC, introduce quantum
% gates, and the X,Z,Y operators
Much like in classical error correction, in \ac{qec} information
is protected by mapping it onto codewords in an expanded
Hilbert space, thereby introducing redundancy. Take for example, the
two-qubit code \cite{roffe_quantum_2019}, where we map%
%
\begin{align*}
\ket{\psi} = \underbrace{a \ket{0} + b \ket{1}}_{\in
\mathcal{H}_2} \hspace*{3mm}
\rightarrow
\hspace*{3mm} \ket{\psi}_\text{L} = \underbrace{a \ket{00} + b
\ket{11}}_{\in \mathcal{H}_4}
.%
\end{align*}%
%
We call $\ket{\psi}_L$ the logical state.
We define the codespace as $\mathcal{C} := \text{span}\mleft\{
\ket{00}, \ket{11} \mright\}$ and the error subspace as $\mathcal{F} :=
\text{span} \mleft\{\ket{01}, \ket{10} \mright\}$.
To determine if an error occurred, we want to know
whether a state belongs%
\footnote{
It is possible for a state to not completely lie in either subspace.
In this case, we can interpret the state as being in
$\mathcal{C}$ or $\mathcal{F}$ with a certain probability.
}
to $\mathcal{C}$ or $\mathcal{F}$.
In quantum mechanics, measurements of observable quantities are
described using operators \cite[Section
1.5]{griffiths_introduction_1995}. Because of the way these operators are
defined, their eigenvalues correspond to the possible outcomes of
measuring that observable, and the corresponding
eigenstates are the states that yield those values as measurements \cite[Section
3.3]{griffiths_introduction_1995}.
In our case, we need an operator with two eigenvalues, and the corresponding
eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively.
For the two-qubit code, $Z_1Z_2$ is such an operator:%
%
\begin{align*}
Z_1Z_2 E \ket{\psi}_\text{L} &= (+1) E \ket{\psi}_\text{L}
\hspace*{3mm} \forall
E \ket{\psi}_\text{L} \in \mathcal{C} \\
Z_1Z_2 E \ket{\psi}_\text{L} &= (-1) E \ket{\psi}_\text{L}
\hspace*{3mm} \forall
E \ket{\psi}_\text{L} \in \mathcal{F}
.%
\end{align*}%
%
$E$ is an operator describing a possible error and $E
\ket{\psi}_\text{L}$ is the resulting state after that error.
By measuring the corresponding eigenvalue, we can determine if
$E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$.
To do this without directly measuring (and thus disturbing) the
logical state $\ket{\psi}_\text{L}$, we prepare an ancilla
qubit with state $\ket{0}_\text{A}$ and we entangle it with
$\ket{\psi}_\text{L}$ in such a way that determining that instead
indicates the eigenvalue. More specifically, using a syndrome
extraction circuit as shown in fig. \ref{fig:syndrome extraction}, we
transform the state of the three-qubit system as%
%
\begin{align*}
E\ket{\psi}_\text{L} \ket{0}_\text{A} \hspace*{3mm} \rightarrow
\hspace*{3mm}
\underbrace{\frac{1}{2} \mleft( I_1I_2 + Z_1Z_2 \mright)}_{=:
P_\mathcal{C}} E\ket{\psi}_\text{L}
\ket{0}_\text{A}
+ \underbrace{\frac{1}{2} \mleft( I_1I_2 - Z_1Z_2 \mright)}_{=:
P_\mathcal{F}}
E\ket{\psi}_\text{L} \ket{1}_\text{A}
.%
\end{align*}%
%
If $E \ket{\psi}_\text{L} \in \mathcal{C}$, the second term will
cancel and we will deterministically measure $\ket{0}_\text{A}$ for
the ancilla qubit. Similarly, if $E \ket{\psi}_\text{L} \in
\mathcal{F}$ we will deterministically measure $\ket{1}_\text{A}$.
In general, the resulting state of the three-qubit system will be a
superposition of the two cases.
Note that the expressions $P_\mathcal{C}$ and $P_\mathcal{F}$ above
essentially constitute projection operators onto $\mathcal{C}$ and
$\mathcal{F}$. E.g., $P_\mathcal{C}$ will eliminate all
components of a superposition state $E \ket{\psi}_\text{L}$ that lie
in $\mathcal{F}$.
By measuring the ancilla qubit, we collapse the overall state
into one of two configurations. In each of those configurations, $E
\ket{\psi}_\text{L}$ is projected back onto either $\mathcal{C}$ or
$\mathcal{F}$. At the same time, because $Z_1Z_2 \ket{\psi}_\text{L}
= \ket{\psi}_\text{L}$, the projections leave the logical state
$\ket{\psi}_\text{L}$ untouched. We have thus managed to determine
whether an error occurred without disturbing the encoded
quantum information.
\begin{figure}[H]
\centering
\tikzset{
meter/.append style={
draw, rectangle,
font=\vphantom{A}, minimum width=8mm, minimum height=8mm,
path picture={
\draw[black]
([shift={(.1,.3)}]path picture bounding box.south west)
to[bend left=50]
([shift={(-.1,.3)}]path picture bounding box.south east);
\draw[black,-latex]
([shift={(0,.1)}]path picture bounding box.south)
-- ([shift={(.3,-.1)}]path picture bounding box.north);
}
}
}
\begin{tikzpicture}
\node[rectangle, minimum width=2cm, minimum height=3cm, draw]
(ZZ) {$Z_1Z_2$};
\coordinate (qi1) at (-3, 1);
\coordinate (qi2) at (-3, -1);
\coordinate (qi3) at (-3, -3);
\coordinate (qo1) at (4, 1);
\coordinate (qo2) at (4, -1);
\coordinate (qo3) at (4, -3);
\node[rectangle, minimum width=8mm, minimum height=8mm, draw]
(H1) at ($(qo3 -| ZZ) + (-2, 0)$) {H};
\node[rectangle, minimum width=8mm, minimum height=8mm, draw]
(H2) at ($(qo3 -| ZZ) + (2, 0)$) {H};
\node[circle, fill] (not) at (H1 -| ZZ) {};
\node[meter, right=5mm of H2] (mes) {};
\draw (qi1) -- (ZZ.west |- qi1);
\draw (qi2) -- (ZZ.west |- qi2);
\draw (qo1) -- (ZZ.east |- qo1);
\draw (qo2) -- (ZZ.east |- qo2);
\draw (qi3) -- (H1) -- (not) -- (H2) -- (mes);
\draw (not) -- (ZZ);
\coordinate (qo3u) at ($(qo3) + (0, .5mm)$);
\coordinate (qo3d) at ($(qo3) + (0, -.5mm)$);
\draw (mes.east |- qo3u) -- (qo3u);
\draw (mes.east |- qo3d) -- (qo3d);
\node[left] at (qi3) {$\ket{0}_\text{A}$};
\node[left] at ($(qi1)!.5!(qi2)$) {$E\ket{\psi}_\text{L}$};
\node[right] at ($(qo1)!.5!(qo2)$) {$E\ket{\psi}_\text{L}$};
\end{tikzpicture}
\caption{Syndrome extraction circuit for the two-qubit repetition code}
\label{fig:syndrome extraction}
\end{figure}
\printbibliography
\end{document}

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tabsize = 4