2013/10/21 by Larry Guth, Alexander Lubotzky · 2 citations
Computer Science · Mathematics · #Block code #Coding theory #Coding theory and cryptography #Cryptography #Cryptography and Data Security #Error detection and correction #Parity (physics) #Quantum #Quantum Computing Algorithms and Architecture #Quantum computer #Quantum error correction #math.DG #msc:53C23
paper · pdf · doi:10.1063/1.4891487
21 pages
arxiv created 2013/10/21 · openalex publication_date 2014/08/01 · arxiv updated 2015/06/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance nε. Their rate is evaluated via Euler characteristic arguments and their distance using \documentclass[12pt]minimal\begindocument\mathbb Z2\enddocumentZ2-systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), Lecture Notes in Computer Science Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.