# Minimum-Weight Parity Factor Decoder for Quantum Error Correction

arXiv: https://arxiv.org/abs/2508.04969 · HTML: https://mlsanity.com/paper/2508.04969/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2025.

## Built on by (most cited first, 1 of 1)

- [A scalable and real-time neural decoder for topological quantum codes](https://mlsanity.com/paper/2512.07737/) (2025)

## Abstract (arXiv)

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.