Noisy-Syndrome Decoding of Hypergraph Product Codes

TL;DR

This work addresses noisy-syndrome decoding for quantum hypergraph product (HGP) codes by reducing the quantum problem to classical noisy-syndrome decoding for suitable base codes. It presents two concrete frameworks: (i) a noisier-start version requiring base codes with noisy-syndrome decodability, and (ii) a dual-friendly version requiring C and C^⊥ to be syndrome-decodable, each yielding HGP codes with distance Θ(N^{1/2}) and near-linear decoding times; expander, Reed-Solomon, and folded RS codes are instantiated. Concrete instantiations achieve decodability from Θ(N^{1/2}) errors with near-linear or polynomial-time decoders, leveraging expander-based and polynomial-code structures to maintain asymptotically optimal distance for HGP codes. A key technical thread is the ideal-theoretic framing of folded RS codes that enables efficient syndrome decoding, enabling practical decoders for quantum LDPC HGP codes with robust error-correction capabilities.

Abstract

Hypergraph product codes are a prototypical family of quantum codes with state-of-the-art decodability properties. Recently, Golowich and Guruswami (FOCS 2024) showed a reduction from quantum decoding to syndrome decoding for a general class of codes, which includes hypergraph product codes. In this work we consider the "noisy" syndrome decoding problem for hypergraph product codes, and show a similar reduction in the noisy setting, addressing a question posed by Golowich and Guruswami. Our results hold for a general family of codes wherein the code and the dual code are "simultaneously nice"; in particular, for codes admitting good syndrome decodability and whose duals look "similar". These include expander codes, Reed-Solomon codes, and variants.

Theorems & Definitions (16)

• Theorem 1.1: Informal, noisy version
• Theorem 1.2: Informal, non-noisy version
• Theorem 3.1: Formal, noisy version
• proof
• Claim 3.2
• proof
• Claim 3.3
• proof
• Corollary 3.4
• Claim 4.1