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Towards local testability for quantum coding

hal.structure.identifierCryptologie symétrique, cryptologie fondée sur les codes et information quantique [COSMIQ]
dc.contributor.authorLEVERRIER, Anthony
hal.structure.identifierCryptologie symétrique, cryptologie fondée sur les codes et information quantique [COSMIQ]
dc.contributor.authorLONDE, Vivien
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorZÉMOR, Gilles
dc.date.accessioned2024-04-04T02:57:49Z
dc.date.available2024-04-04T02:57:49Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/192598
dc.description.abstractEnWe introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the $p$-faces of the $n$-cube (for $n>p$) and stabilizer constraints with faces of dimension $(p\pm1)$. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into $N = 2^{n-p-1} \tbinom{n}{p}$ physical qubits and displays local testability with a soundness of $\Omega(\log^{-2} (N))$ beating the current state-of-the-art of $\log^{-3} (N)$ due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. We then extend this code family by considering the quotient of the $n$-cube by arbitrary linear classical codes of length $n$. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be $1/\log(N)$, similarly to the ordinary $n$-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.
dc.language.isoen
dc.title.enTowards local testability for quantum coding
dc.typeDocument de travail - Pré-publication
dc.subject.halPhysique [physics]/Physique Quantique [quant-ph]
dc.identifier.arxiv1911.03069
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-02432360
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-02432360v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=LEVERRIER,%20Anthony&LONDE,%20Vivien&Z%C3%89MOR,%20Gilles&rft.genre=preprint


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