Minimum-Weight Parity Factor Decoder for Quantum Error Correction

TL;DR

HyperBlossom is a novel decoding framework for quantum error correction that formulates maximum-likelihood decoding as a minimum-weight parity factor problem, unifying existing decoders and achieving lower error rates with efficient runtime.

Contribution

The paper introduces HyperBlossom, a unified framework generalizing the blossom algorithm to hypergraphs for quantum error correction decoding, with implementation and improved performance.

Findings

Achieves 4.8x lower logical error rate than MWPM on surface code.

Achieves 1.6x lower logical error rate than BPOSD on bicycle code.

Provides almost-linear runtime scaling up to large code distances.

Abstract

Fast and accurate quantum error correction (QEC) decoding is crucial for scalable fault-tolerant quantum computation. Most-Likely-Error (MLE) decoding, while being near-optimal, is intractable on general quantum Low-Density Parity-Check (qLDPC) codes and typically relies on approximation and heuristics. We propose HyperBlossom, a unified framework that formulates MLE decoding as a Minimum-Weight Parity Factor (MWPF) problem and generalizes the blossom algorithm to hypergraphs via a similar primal-dual linear programming model with certifiable proximity bounds. HyperBlossom unifies all the existing graph-based decoders like (Hypergraph) Union-Find decoders and Minimum-Weight Perfect Matching (MWPM) decoder, thus bridging the gap between heuristic and certifying decoders. We implement HyperBlossom in software, namely Hyperion. Hyperion achieves a 4.8x lower logical error rate compared to the MWPM decoder on the distance-11 surface code and 1.6x lower logical error rate compared to a fine-tuned BPOSD decoder on the $[[90, 8, 10]]$ bivariate bicycle code under code-capacity noise. It also achieves an almost-linear average runtime scaling on both the surface code and the color code, with numerical results up to sufficiently large code distances of 99 and 31 for code-capacity noise and circuit-level noise, respectively.