
@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},
    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},
}

@article{dirac_new_1939,
    title = {A new notation for quantum mechanics},
    volume = {35},
    issn = {1469-8064, 0305-0041},
    url =
    {https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/new-notation-for-quantum-mechanics/4631DB9213D680D6332BA11799D76AFB},
    doi = {10.1017/S0305004100021162},
    abstract = {In mathematical theories the question of notation,
        while not of primary importance, is yet worthy of careful
        consideration, since a good notation can be of great value in
        helping the development of a theory, by making it easy to write
        down those quantities or combinations of quantities that are
        important, and difficult or impossible to write down those that
        are unimportant. The summation convention in tensor analysis is
    an example, illustrating how specially appropriate a notation can be.},
    language = {en},
    number = {3},
    journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
    author = {Dirac, P. a. M.},
    month = jul,
    year = {1939},
    keywords = {/unread},
    pages = {416--418},
}

