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Full-Rank Constructions of Explicit Lossless Expanders and Good Small-Set qLTCs — arXiv:2611.02963

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.02963
License (as archived): CC0-1.0
Archived: 2026-11-03T18:51:28.543Z
SHA-256 of the archived text: a2547c79c95da21012f55f2daa254195576afd903a70db59bf36a96b241240b4
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How to cite

Full-Rank Constructions of Explicit Lossless Expanders and Good Small-Set qLTCs — arXiv:2611.02963 — arXiv (en), https://arxiv.org/abs/2611.02963 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_8e27fc030bb0 (Metraipolis archive dc_8e27fc030bb0, sha256 a2547c79c95da21012f55f2daa254195576afd903a70db59bf36a96b241240b4)

BibTeX

@misc{dc8e27fc030bb0,
  title = {Full-Rank Constructions of Explicit Lossless Expanders and Good Small-Set qLTCs — arXiv:2611.02963},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.02963},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_8e27fc030bb0; sha256 a2547c79c95da21012f55f2daa254195576afd903a70db59bf36a96b241240b4},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.02963. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We construct explicit two-sided lossless expanders whose binary adjacency matrices have full rank, and use them to obtain explicit asymptotically good quantum LDPC codes whose specified X- and Z-check matrices have full row rank and satisfy small-set soundness, or confinement, on a linear range. Our lossless expanders are a variant of those of Hsieh, Lubotzky, Mohanty, Reiner, and Zhang, with a free action of a finite 2-group. Under such an action, injectivity and surjectivity of the adjacency matrix are detected on a fixed-size matrix indexed by vertex orbits. Adding a sparse factor to the gadget at the middle vertices of a single type then enforces full rank, with a controlled loss in expansion. To obtain the 2-group action, we observe that reducing the generators of the Lubotzky-Phillips-Sarnak Ramanujan graphs modulo powers of 2, instead of modulo an odd prime, gives Ramanujan Cayley graphs on finite 2-groups. This construction may be of independent interest. Plugging two such expanders into the balanced product of Lin and Hsieh gives the quantum codes, since full rank of the input graphs is inherited by the check matrices. Our construction supplies the code family sought by Hong, Guo, and Lucas. Applying their analysis, within their framework and under their dynamical assumptions, gives quantum memories whose free-energy density is analytic at every positive temperature and whose guaranteed lifetime is $exp(\Omega(N))$ at sufficiently low fixed positive temperature, where N is the blocklength. This improves the previously guaranteed stretched-exponential timescale.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Quantum Physics (quant-ph) ; Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO); Group Theory (math.GR).
Pagina del lavoro: https://arxiv.org/abs/2611.02963 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T18:15:38.140Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).