Optimization Using Locally-Quantum Decoders

TL;DR

This paper introduces a new quantum decoding technique for LDPC codes that outperforms classical belief propagation on average-case instances of D-regular max-k-XORSAT, but does not yet achieve quantum advantage.

Contribution

It develops an intrinsically quantum decoding method for LDPC codes that surpasses classical algorithms in certain average-case optimization problems.

Findings

Quantum decoder outperforms belief propagation for many (k,D) values.

Approximate optima surpass Prange's algorithm and simulated annealing.

Quantum advantage remains unachieved due to a tie with Prange's algorithm.

Abstract

It was pointed out in [JSW+25] that widely-studied optimization problems such as D-regular max-k-XORSAT can be reduced to decoding of LDPC codes, using quantum algorithms related to Regev's reduction. LDPC codes have very good decoders, such as Belief Propagation (BP), and this therefore makes D-regular max-k-XORSAT an enticing target for this class of quantum algorithms. However, BP was found insufficient to achieve quantum advantage. Here, we develop an intrinsically quantum decoding technique, which decodes classical LDPC codes subject to coherent superpositions of bit flip errors. For average-case instances of D-regular max-k-XORSAT drawn from Gallager's ensemble, this quantum decoder strongly outperforms classical belief propagation at many values of k and D. For some (k,D) the approximate optima achievable using this decoder surpass both Prange's algorithm and simulated annealing. However, we stop short of achieving quantum advantage because we identify an enhancement to Prange's algorithm that recovers a precise tie, much as a precise tie was observed between the standard version of Prange's algorithm and a more limited version of locally-quantum decoding in [CT24].