FPGA-tailored algorithms for real-time decoding of quantum LDPC codes

TL;DR

This paper evaluates FPGA-optimized algorithms for real-time decoding of quantum LDPC codes, finding message passing methods like Relay most promising despite some limitations.

Contribution

It introduces FPGA-adapted versions of three quantum LDPC decoders and analyzes their performance, highlighting the advantages of message passing algorithms for real-time decoding.

Findings

Relay decoder is the most promising for real-time decoding.

FPGA-adapted clustering and filtered OSD are less accurate and slower.

Message passing remains the most viable approach for FPGA-based quantum LDPC decoding.

Abstract

Real-time decoding is crucial for fault-tolerant quantum computing but likely requires specialized hardware such as field-programmable gate arrays (FPGAs), whose parallelism can alter relative algorithmic performance. We analyze FPGA-tailored versions of three decoder classes for quantum low-density parity-check (qLDPC) codes: message passing, ordered statistics, and clustering. For message passing, we analyze the recently introduced Relay decoder and its FPGA implementation; for ordered statistics decoding (OSD), we introduce a filtered variant that concentrates computation on high-likelihood fault locations; and for clustering, we design an FPGA-adapted generalized union-find decoder. We design a systolic algorithm for Gaussian elimination on rank-deficient systems that runs in linear parallel time, enabling fast validity checks and local corrections in clustering and eliminating costly full-rank inversion in filtered-OSD. Despite these improvements, both remain far slower and less accurate than Relay, suggesting message passing is the most viable route to real-time qLDPC decoding.