Minimum-Weight Parity Factor Decoder for Quantum Error Correction

2025-08-06

AI Summary

In quantum error correction (QEC), maximum-likelihood estimation (MLE) decoders for general qLDPC codes face a trade-off between computational efficiency and theoretical guarantees. This work introduces HyperBlossom: the first decoder that unifies MLE decoding as the minimum-weight perfect parity factor (MWPF) problem on hypergraphs and generalizes Edmonds’ blossom algorithm to hypergraph structures. Leveraging a primal-dual linear programming framework, HyperBlossom provides provable approximation ratio bounds—bridging the gap between heuristic and certifiable decoders. Implemented in the open-source software Hyperion, it achieves lower logical error rates than MWPM and BPOSD on surface codes and doubly-even bicycle codes. Notably, it maintains near-linear average-case time complexity even for codes of distance up to 99 and 31, respectively.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Fast and accurate quantum error correction decoding
• Intractability of Most-Likely-Error decoding on qLDPC codes
• Unifying heuristic and certifying decoders for quantum codes

Innovation

• Unified MLE decoding via Minimum-Weight Parity Factor
• Generalizes blossom algorithm to hypergraphs
• Achieves lower error rates and linear runtime scaling