[Paper Review] Fast erasure decoder for hypergraph product codes

Nicholas Connolly, Vivien Londe+2 more|arXiv (Cornell University)|Aug 1, 2022

Quantum Computing Algorithms and Architecture11 references4 citations

TL;DR

This paper presents a fast erasure decoder for hypergraph product (HGP) codes, a leading class of quantum LDPC codes, using a hybrid approach combining pruned peeling and a novel VH decoder. The method achieves near-maximum-likelihood performance with $O(N^2)$ bit operations, and $O(N^{1.5})$ using probabilistic Gaussian elimination, enabling efficient correction of qubit erasures in quantum error correction systems.

ABSTRACT

We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in O(N^2) bit operations where N is the length of the quantum code. A probabilistic version of this decoder can be implemented in O(N^1.5) bit operations.

Motivation & Objective

• To design a fast, practical decoder for correcting qubit erasures in hypergraph product (HGP) codes, which are promising for reducing qubit overhead in fault-tolerant quantum computing.
• To overcome the high computational cost of maximum-likelihood (ML) decoding by developing a low-complexity alternative that approximates ML performance.
• To address the challenge of stopping sets in classical and quantum LDPC decoding by introducing a cluster-based correction strategy using isolated, dangling, and free clusters.
• To enable efficient implementation in quantum architectures where erasure is a dominant error source, such as photonic and neutral-atom systems.
• To reduce decoding complexity from $O(N^3)$ to $O(N^2)$ or $O(N^{1.5})$ while maintaining high logical error rate performance.

Proposed method

• The decoder uses a pruned peeling algorithm that iteratively corrects isolated and dangling clusters in the Tanner graph of the HGP code, delaying correction of clusters with free checks until later.
• A novel VH (Vertex-Heuristic) decoder is introduced that corrects clusters in two phases: first correcting internal checks and then resolving free checks using a stack-based deferred correction mechanism.
• The algorithm maintains a syndrome vector $s$ and updates it incrementally as errors are corrected, ensuring consistency with the original syndrome.
• Clusters are processed based on their connectivity: isolated or dangling clusters are corrected immediately, while free clusters are deferred using a stack to preserve syndrome consistency.
• The Gaussian elimination step for correcting each cluster is optimized using probabilistic techniques, reducing the per-cluster cost from $O(n^3)$ to $O(n^2)$, leading to overall $O(N^{1.5})$ complexity.
• The decoder fails only when the erasure pattern contains cycles in the VH-graph, which are shown to be rare at low erasure rates.

Experimental results

Research questions

• RQ1 Can a fast decoder be designed for HGP codes that achieves near-maximum-likelihood performance for erasure correction?
• RQ2 How can the complexity of decoding be reduced from $O(N^3)$ to $O(N^2)$ or better while maintaining high accuracy?
• RQ3 What role do stopping sets—especially those formed by clusters with free checks—play in decoder failure, and how can they be mitigated?
• RQ4 Can a hybrid approach combining peeling and cluster-based correction improve performance and scalability in quantum LDPC decoding?
• RQ5 To what extent do cycles in the VH-graph structure limit decoder performance, and can pruning strategies reduce their impact?

Key findings

• The proposed decoder achieves a logical error rate very close to that of the maximum-likelihood decoder, especially at low erasure rates.
• The decoder runs in $O(N^2)$ bit operations for the deterministic version, and $O(N^{1.5})$ using probabilistic Gaussian elimination, significantly improving over standard $O(N^3)$ decoding.
• The combination of pruned peeling and VH decoding performs nearly as well as the ML decoder, with only a small performance gap at low erasure rates.
• Decoder failure occurs only when the erasure pattern contains cycles in the VH-graph, which are infrequent in low-error regimes due to the sparsity of small-weight error clusters.
• The use of deferred correction for free checks via a stack allows the decoder to maintain syndrome consistency and correct complex cluster patterns without increasing complexity unduly.
• The implementation is available publicly and can be used as a subroutine in other decoders, such as Union-Find, to accelerate decoding in quantum LDPC codes.