2026-07-23T18:55:30Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8170 2024-03-06T10:39:18Z ddc:004 open_access

Quantum Codes from High-Dimensional Manifolds Hastings, Matthew B. quantum codes random lattices Rankin invariants We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on N qubits with logarithmic weight stabilizers and distance N^{1-\epsilon} for any \epsilon>0. The conjecture is that there is a constant C>0 such that for any n-dimensional torus {\mathbb T}^n={\mathbb R}^n/\Lambda, where \Lambda is a lattice, the least volume unoriented n/2-dimensional cycle (using the Euclidean metric) representing nontrivial homology has volume at least C^n times the volume of the least volume n/2-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for \Lambda an integral lattice with the cycle restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes. We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Matthew B. Hastings 2017 Is Part Of LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2017.25 urn:nbn:de:0030-drops-81708 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.25 eng https://creativecommons.org/licenses/by/3.0/legalcode