Expansion of High-Dimensional Cubical Complexes: with Application to Quantum Locally Testable Codes

Irit Dinur, Ting Chun Lin, Thomas Vidick

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We introduce a high-dimensional cubical complex, for any dimension t in N, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes H_{1}, ·,ht as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t=4 our construction gives a new family of 'almost-good' quantum LTCs - with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

Original languageEnglish
Title of host publicationProceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024
PublisherIEEE Computer Society
Pages379-385
Number of pages7
ISBN (Electronic)9798331516741
DOIs
StatePublished - 2024
Event65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024 - Chicago, United States
Duration: 27 Oct 202430 Oct 2024

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024
Country/TerritoryUnited States
CityChicago
Period27/10/2430/10/24

All Science Journal Classification (ASJC) codes

  • General Computer Science

Cite this