Asymptotically Good Quantum and Locally Testable Classical LDPC Codes

TL;DR

This paper demonstrates that certain classical and quantum LDPC codes constructed via the lifted product over non-abelian groups are asymptotically good, with quantum codes confirming the qLDPC conjecture and classical codes being locally testable.

Contribution

It proves the asymptotic goodness of quantum LDPC codes and the local testability of classical LDPC codes, resolving two longstanding conjectures.

Findings

Quantum LDPC codes are asymptotically good.

Classical LDPC codes are locally testable with constant query and soundness.

The results confirm the qLDPC and local testability conjectures.

Abstract

We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes.