List Decodable Quantum LDPC Codes

TL;DR

This paper introduces a novel construction of quantum LDPC codes that achieve a near-optimal rate and distance tradeoff, with efficient list decoding capabilities up to the Johnson bound, using advanced algorithmic techniques.

Contribution

It presents a new method for constructing list decodable quantum LDPC codes that do not require classical side channels and leverage the quantum distance amplification scheme.

Findings

Achieves polynomial-time list decoding up to the Johnson bound.

Constructs quantum LDPC codes with near-optimal rate and distance.

Uses convex relaxations and the Sum-of-Squares hierarchy for decoding.

Abstract

We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time. Previous constructions of list decodable good distance quantum codes either required access to a classical side channel or were based on algebraic constructions that preclude the LDPC property. Our construction relies on new algorithmic results for codes obtained via the quantum analog of the distance amplification scheme of Alon, Edmonds, and Luby [FOCS 1995]. These results are based on convex relaxations obtained using the Sum-of-Squares hierarchy, which reduce the problem of list decoding the distance amplified codes to unique decoding the starting base codes. Choosing these base codes to be the recent breakthrough constructions of good QLDPC codes with efficient unique decoders, we get efficiently list decodable QLDPC codes.