Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes (2406.18655v1)

Abstract: Quantum low-density parity-check codes are a promising candidate for fault-tolerant quantum computing with considerably reduced overhead compared to the surface code. However, the lack of a practical decoding algorithm remains a barrier to their implementation. In this work, we introduce localized statistics decoding, a reliability-guided inversion decoder that is highly parallelizable and applicable to arbitrary quantum low-density parity-check codes. Our approach employs a parallel matrix factorization strategy, which we call on-the-fly elimination, to identify, validate, and solve local decoding regions on the decoding graph. Through numerical simulations, we show that localized statistics decoding matches the performance of state-of-the-art decoders while reducing the runtime complexity for operation in the sub-threshold regime. Importantly, our decoder is more amenable to implementation on specialized hardware, positioning it as a promising candidate for decoding real-time syndromes from experiments.

• The paper presents a novel LSD algorithm that partitions the decoding graph into clusters, enabling parallel on-the-fly matrix elimination.
• The method significantly reduces runtime complexity compared to BP+OSD while achieving competitive logical error rates for various QLDPC codes.
• The algorithm's parallel implementation paves the way for real-time error correction on specialized hardware, advancing fault-tolerant quantum computing.

Localized Statistics Decoding: A Parallel Decoding Algorithm for Quantum Low-Density Parity-Check Codes

The paper "Localized Statistics Decoding: A Parallel Decoding Algorithm for Quantum Low-Density Parity-Check Codes" presents a novel approach for decoding quantum low-density parity-check (QLDPC) codes. These codes are a promising candidate for achieving fault-tolerant quantum computing with lower overhead compared to the widely considered surface code. However, a significant challenge hindering their implementation is the lack of a practical decoding algorithm suitable for real-time error correction. This paper introduces the Localized Statistics Decoding (LSD) algorithm, which is a highly parallelizable decoder designed to address this challenge.

Context and Motivation

Quantum Low-Density Parity-Check codes (QLDPC) have garnered attention due to their potential for fault tolerance with relatively lower overhead. Unlike surface codes, which encode a single logical qubit per block, QLDPC codes encode multiple logical qubits, offering a more efficient path to fault tolerance. Despite this promise, the current gold standard for decoding these codes—Belief Propagation plus Ordered Statistics Decoder (BP+OSD)—suffers from high runtime complexity primarily due to the Gaussian elimination step involved in the OSD algorithm. This inefficiency is particularly problematic for large circuit-level decoding graphs, which can contain tens of thousands of nodes, rendering BP+OSD impractical for real-time decoding.

Localized Statistics Decoding (LSD) Algorithm

The LSD algorithm proposed in this paper is a reliability-guided decoder that targets the sub-threshold operational regime of QLDPC codes, where errors are sparse. The key idea is to partition the decoding problem into smaller, independent sub-problems called clusters. Each cluster is a localized region of the decoding graph containing a subset of the error syndromes. The LSD algorithm applies a parallel matrix factorization strategy referred to as on-the-fly elimination, which allows the independent and concurrent inversion of matrices associated with these clusters.

1. Cluster Formation: The algorithm begins by forming clusters around each flipped detector node. Each cluster grows by adding the most probable fault node from its neighborhood, as determined by a given reliability vector. This growth continues until the local syndrome within the cluster becomes linearly dependent on the cluster's decoding matrix, rendering the cluster valid.
2. On-the-Fly Elimination: This novel technique enables the algorithm to maintain the PLU factorization of cluster matrices dynamically. It allows the transformation of Gaussian elimination into a parallel process, managing the merging of clusters efficiently without recomputing the entire factorization from scratch.
3. Parallel Implementation: The clusters grow and merge in parallel, utilizing a parallel union-find data structure to handle collisions (detecting whether clusters overlap) and merges efficiently. Once all clusters are valid, the solution for each cluster is computed in parallel, significantly reducing the overall runtime complexity.

Numerical Results

The paper benchmarks the performance of the LSD algorithm against BP+OSD and other decoders across multiple scenarios, including surface codes, hypergraph product codes, and bivariate bicycle codes. The findings are as follows:

• Surface Codes: BP+LSD achieves logical error rates and thresholds nearly identical to BP+OSD while being significantly more efficient in runtime, indicating its suitability for practical implementation.
• Hypergraph Product Codes: BP+LSD exhibits impressive performance gains over the BP plus Small Set Flip (BP+SSF) decoder, achieving almost two orders of magnitude improvement in logical error suppression at low physical error rates.
• Bivariate Bicycle Codes: The performance of BP+LSD is comparable to BP+OSD-CS-7 (a combination sweep strategy for OSD up to order 7), demonstrating that LSD can still offer competitive performance without the computational overhead of higher-order OSD.

Theoretical and Practical Implications

The introduction of LSD provides a promising avenue for real-time decoding of QLDPC codes, particularly relevant for high-throughput quantum computers. The reduction in runtime complexity through parallelized matrix factorization opens up the possibility of implementing real-time decoders on specialized hardware such as FPGAs or ASICs. This can significantly mitigate the backlog problem, where syndrome data accumulates due to insufficient decoding speed.

Theoretically, LSD contributes to the body of work on reliability-guided decoders, offering a more granular and parallelizable approach to matrix factorization. The on-the-fly elimination routine, in particular, has potential applications beyond quantum error correction, such as in efficiently solving sparse linear systems in various computational fields.

Future Directions

Future work could explore the implementation of the LSD algorithm on specialized hardware to further validate its real-time performance capabilities. Additionally, adapting LSD to other noise models, such as erasure-biased systems, could widen its applicability. Investigating the integration of maximum-likelihood decoding at the cluster level, as suggested in recent works, could further enhance the efficiency and accuracy of the LSD algorithm.

Conclusion

The Localized Statistics Decoding algorithm represents a significant advancement in the parallel decoding of QLDPC codes. By addressing the runtime inefficiencies of current decoding approaches, it paves the way for more practical, real-time quantum error correction solutions. The thorough numerical testing and potential for practical implementation highlight its importance in the ongoing development of fault-tolerant quantum computing technologies.