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codelists/descendants/classical/list_self_dual.yml

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list_id: self_dual
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title: Self-dual objects
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title: Classical self-dual objects
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intro: |
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Here is a list of self-dual objects from classical coding theory --- codes, lattices, and polytopes.
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codelists/descendants/classical/q-ary_digits/list_q-ary_linear.yml

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list_id: 'q-ary_linear'
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title: |
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\(q\)-ary linear codes
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Non-binary linear codes
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Here is a list of \hyperref[code:q-ary_linear]{\(q\)-ary linear codes} over the Galois field \(\mathbb{F}_q\) for \(q > 2\). For \(q=2\), see \hyperref[codelist:binary_linear]{Binary linear codes}.

codes/classical/bits/cyclic/quad_residue/extended_golay.yml

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- code_id: binary_quad_residue
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detail: 'The extended Golay code is an extended binary quadratic-residue code \cite[Ch. 16]{preset:MacSlo}.'
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- code_id: icosahedron
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detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{post} by J. Baez for more details. To construct the code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.'
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detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \cite{manual:{J. Baez, "Golay Code", Visual Insight (2015) \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{URL}}} for more details. To construct the code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.'
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- code_id: dodecahedron
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detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{post} by J. Baez for more details. To construct the code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.'
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detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \cite{manual:{J. Baez, "Golay Code", Visual Insight (2015) \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{URL}}} for more details. To construct the code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.'
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- code_id: golay
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detail: 'The extended Golay code is an extension of the Golay code by a parity-check bit.'
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- code_id: group

codes/classical/bits/easy/cordaro_wagner.yml

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A linear binary code with \(k=2\) that is optimal among binary linear \([n,2,d]\) codes.
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- 'Implementation in Komm Python software library for communication systems \cite{manual:{R. W. Nobrega, Komm, 2025 \href{https://github.com/rwnobrega/komm}{URL}}}.'
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- 'Implementation in Komm Python software library for communication systems \cite{manual:{R. W. Nobrega, Komm Software Package, 2025 \href{https://github.com/rwnobrega/komm}{URL}}}.'
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codes/quantum/groups/group_quantum.yml

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- code_id: homogeneous_space_quantum
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detail: 'Homogeneous spaces \(G/H\) for trivial \(H\) reduce to group spaces. A group-\(G\) space can also be thought of as a multiplicity-free homogeneous space \((G\times G) / G\) \cite[pg. 60]{preset:Diaconis88}.'
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- code_id: category_quantum
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detail: 'Finite-group-based quantum codes, whose basis states are parameterized by a finite group, correspond to category-based codes for the fusion category \(Vec G\). Extensions of such categories to Lie groups can also be done \cite{manual:{K. Fredenhagen, “Superselection sectors with infinite statistical dimension”, Subfactors (Kyuzeso, 1993) (1994): 242-258},arxiv:2106.12577,manual:{K. Walker, “Lie group symmetries and non-unital higher categories. Subfactors and Fusion (2-)Categories”, Banff International Research Station, 2023 \href{https://www.birs.ca/events/2023/5-day-workshops/23w5091/videos/watch/202312061131-Walker.html}{URL}},arxiv:2503.14596} (see also \cite{arxiv:gr-qc/0303060}).'
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detail: 'Finite-group-based quantum codes, whose basis states are parameterized by a finite group, correspond to category-based codes for the fusion category \(Vec G\). Extensions of such categories to Lie groups can also be done \cite{manual:{K. Fredenhagen, “Superselection sectors with infinite statistical dimension”, Subfactors (Kyuzeso, 1993) (1994): 242-258},arxiv:2106.12577,manual:{K. Walker, “Lie group symmetries and non-unital higher categories”, Banff International Research Station, 2023 \href{https://www.birs.ca/events/2023/5-day-workshops/23w5091/videos/watch/202312061131-Walker.html}{URL}},arxiv:2503.14596} (see also \cite{arxiv:gr-qc/0303060}).'
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# Begin Entry Meta Information

codes/quantum/groups/stabilizer/qldpc/general_qldpc.yml

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features:
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decoders:
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- 'Non-binary decoding algorithm for CSS-type QLDPC codes \cite{doi:10.1109/ACCESS.2015.2503267}.'
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- 'GD-CSS Decoder for Galois-qudit CSS QLDPC codes \cite{arxiv:2507.11534,manual:{K. Kasai, “GD-CSS Decoder (Quantum Error Correction using Non-binary LDPC over GF(q))”, 2025 \href{https://github.com/kasaikenta/gd-css-decoder}{URL}}}'
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- 'GD-CSS Decoder for Galois-qudit CSS QLDPC codes \cite{arxiv:2507.11534,manual:{K. Kasai, “GD-CSS Decoder (Quantum Error Correction using Non-binary LDPC over GF(q)) Software Package”, 2025 \href{https://github.com/kasaikenta/gd-css-decoder}{URL}}}'
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- 'Infleqtion QLDPC software library for estimating distance and creating various qubit and Galois-qudit QLDPC CSS codes \cite{manual:{M. A. Perlin, qLDPC, 2023 \href{https://github.com/Infleqtion/qLDPC}{URL}}}'
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- 'Infleqtion QLDPC software library for estimating distance and creating various qubit and Galois-qudit QLDPC CSS codes \cite{manual:{M. A. Perlin, qLDPC Software Package, 2023 \href{https://github.com/Infleqtion/qLDPC}{URL}}}'
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- 'LDPC Python software library for decoding LDPC and QLDPC codes \cite{arxiv:2005.07016,preset:Roffe22}.'
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- 'Reviews of QLDPC codes provided in Refs. \cite{doi:10.1109/ACCESS.2015.2503267,arxiv:2103.06309,arxiv:2510.14090}.'
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codes/quantum/oscillators/qsc/cat/two-legged-cat.yml

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# - 'Approximate ("kitten") optical realization of the odd cat state \cite{doi:10.1126/science.1122858}.'
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- 'Pedagogical introduction to cat codes in the context of microwave cavities can be found in Refs. \cite{arxiv:2203.03222,manual:{S. Puri, “QEC when the noise is biased”, 2019 \href{https://physinfo.fr/houches2019/files/Puri_v1.pdf}{URL}}}, and in the context of optical systems in books \cite{doi:10.1093/acprof:oso/9780198509141.001.0001,doi:10.1142/9781860948169_0009,doi:10.1002/9783527695805}.'
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- 'Pedagogical introduction to cat codes in the context of microwave cavities can be found in Refs. \cite{arxiv:2203.03222,manual:{S. Puri, “QEC when the noise is biased”, Les Houches Summer School (2019) \href{https://physinfo.fr/houches2019/files/Puri_v1.pdf}{URL}}}, and in the context of optical systems in books \cite{doi:10.1093/acprof:oso/9780198509141.001.0001,doi:10.1142/9781860948169_0009,doi:10.1002/9783527695805}.'
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- 'Ground states of the fluxonium superconducting qubit resemble two-component cat codewords \cite{arxiv:2501.16425}.'
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codes/quantum/properties/hamiltonian/self_correct.yml

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detail: 'Self-correcting quantum codes do no require a symmetry for protection, so in that sense they are protected by a trivial symmetry.'
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cousins:
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- code_id: surface
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detail: 'The surface code is not thermally stable \cite{arxiv:quant-ph/0702102,arxiv:0709.2717,arxiv:0810.4584,arxiv:0911.3843,arxiv:2410.01206} because its string-like logical operators anti-commute with stabilizer generators supported only at their ends, and thus have a constant energy cost of creation. Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons in the code so as to prevent them from spreading \cite{arxiv:0812.4622,arxiv:0908.4264,arxiv:1101.6028,manual:{D. Poulin, “Decoding problem for topological quantum codes” [Conference talk], Simons Center for Geometry and Physics, Stony Brook University, 2011 \href{http://scgp.stonybrook.edu/archives/1087}{URL}},arxiv:1406.2338,arxiv:1511.05579,arxiv:1512.04528,arxiv:2510.08056}.'
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detail: 'The surface code is not thermally stable \cite{arxiv:quant-ph/0702102,arxiv:0709.2717,arxiv:0810.4584,arxiv:0911.3843,arxiv:2410.01206} because its string-like logical operators anti-commute with stabilizer generators supported only at their ends, and thus have a constant energy cost of creation. Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons in the code so as to prevent them from spreading \cite{arxiv:0812.4622,arxiv:0908.4264,arxiv:1101.6028,manual:{D. Poulin, “Decoding problem for topological quantum codes”, Simons Center for Geometry and Physics, Stony Brook University, 2011 \href{http://scgp.stonybrook.edu/archives/1087}{URL}},arxiv:1406.2338,arxiv:1511.05579,arxiv:1512.04528,arxiv:2510.08056}.'
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- code_id: 2d_stabilizer
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detail: '2D stabilizer codes \cite{arxiv:0810.1983} and encodings of frustration-free code Hamiltonians \cite{arxiv:1209.5750} admit only constant-energy excitations, and so do not have an energy barrier.'
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- code_id: quantum_double

codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml

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- 'See Refs. \cite{arxiv:quant-ph/0602096,doi:10.1002/9783527635283} for a review of cluster states and their applications.'
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- 'Cluster states are useful for entanglement purification \cite{arxiv:0705.4165}.'
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- 'The original one-way MBQC paper also noted that universal computation remains possible on irregular occupied 2D clusters above the percolation threshold, and that finite clusters can be reused by re-entangling successive computation segments \cite{doi:10.1103/PhysRevLett.86.5188}.'
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- 'Graph-State Compass (GSC) Python software library for manipulating local Clifford equivalence classes of cluster states \cite{arxiv:1910.03969,doi:10.5281/zenodo.2582616,manual:{S. Morley-Short, GSC, 2021 \href{https://github.com/sammorley-short/gsc/tree/master}{URL}}}.'
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- 'Graph-State Compass (GSC) Python software library for manipulating local Clifford equivalence classes of cluster states \cite{arxiv:1910.03969,doi:10.5281/zenodo.2582616,manual:{S. Morley-Short, GSC Software Package, 2021 \href{https://github.com/sammorley-short/gsc/tree/master}{URL}}}.'
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codes/quantum/qubits/stabilizer/qldpc/qldpc.yml

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- 'Links to code tables of notable QLDPC codes \cite{arxiv:2103.06309}.'
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- 'Collection of QLDPC qubit codes based on hyperbolic tilings in the QEC-Pages software library \cite{manual:{L. P. Pryadko and M. Woolls, Quantum_LDPC_Codes, 2023 \href{https://github.com/QEC-pages/Quantum_LDPC_Codes}{URL}}}.'
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- 'Collection of QLDPC qubit codes based on hyperbolic tilings in the QEC-Pages software library \cite{manual:{L. P. Pryadko and M. Woolls, Quantum_LDPC_Codes Software Package, 2023 \href{https://github.com/QEC-pages/Quantum_LDPC_Codes}{URL}}}.'
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- 'High-rate QLDPC codes can be used for Bell-pair distillation \cite{arxiv:2502.09542}.'
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- 'Qldpc code circUIT Simulator (QUITS) Python software library for simulating QLDPC code circuits \cite{arxiv:2504.02673,manual:{M. Kang, QUITS, 2025 \href{https://github.com/mkangquantum/quits}{URL}}}.'
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- 'Qldpc code circUIT Simulator (QUITS) Python software library for simulating QLDPC code circuits \cite{arxiv:2504.02673,manual:{M. Kang, QUITS Software Package, 2025 \href{https://github.com/mkangquantum/quits}{URL}}}.'
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