# Techniques: Localized statistics decoding for quantum low-density parity-check codes

## What the paper contributes (one paragraph)
The paper introduces localized statistics decoding (LSD), a reliability-guided inversion decoder for arbitrary quantum low-density parity-check (QLDPC) codes that addresses the runtime limitations of the gold-standard BP+OSD decoder. Instead of inverting the whole decoding graph, LSD exploits the fact that sub-threshold errors form small disjoint clusters, factorises the decoding problem into independent local sub-systems, and solves them concurrently. The clusters are built with a weighted, reliability-based growth strategy (guided by belief propagation soft information) until each is valid. The efficiency rests on a new linear algebra routine, on-the-fly elimination, that maintains a PLU factorisation of each cluster and merges clusters without recomputing row operations, enabling parallel execution. A higher-order reprocessing variant (LSD-mu) is also proposed. Numerical simulations on surface, hypergraph product, and bivariate bicycle codes show BP+LSD matches BP+OSD performance while reducing sub-threshold runtime complexity from cubic in the full matrix to cubic in the maximum cluster size.

## Techniques offered
- **localized statistics decoding (LSD)** - A parallel, reliability-guided inversion decoder for arbitrary QLDPC codes that factorises the decoding graph into independent error clusters and inverts each cluster sub-matrix concurrently, supplying a general QLDPC decoder with sub-threshold runtime depending on cluster size rather than full graph size.
  - guarantee: statistical bound; parallel worst-case time complexity scales as the cube of the maximum cluster size, contrasting with the cube of the full matrix size for OSD, with logical decoding performance on par with BP+OSD in numerical simulation.
  - quote: "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."
- **on-the-fly elimination** - A parallel matrix-factorisation routine that maintains a PLU decomposition over GF(2) and incrementally eliminates only newly added columns when clusters grow or merge, supplying a reusable parallel linear-algebra primitive for solving sparse linear systems without re-running Gaussian elimination.
  - guarantee: static proof of correctness for the factorisation plus a parallel-depth bound; an iteration of the parallel LSD loop has depth O(polylog(n) + kappa^3) where kappa is the maximum cluster size, and merging two clusters reduces to adding a single fault node to one cluster.
  - quote: "The efficiency of the LSD algorithm is made possible by a new linear algebra routine, which we call \textit{on-the-fly elimination}, that transforms the serial process of Gaussian elimination into a parallel one."
- **belief propagation plus LSD post-processing (BP+LSD)** - The combined decoder in which belief propagation is run first as a pre-decoder and LSD is invoked as a post-processor using BP soft information, supplying an open-source drop-in replacement for BP+OSD that needs far fewer BP iterations.
  - guarantee: statistical bound; matches BP+OSD threshold (around p approximately 0.7% for surface codes) and logical error rates using only 30 BP iterations, and outperforms BP+SSF on hypergraph product codes by almost two orders of magnitude at p approximately 0.1%.
  - quote: "Our main numerical finding is that BP+LSD can decode QLDPC codes with performance on par with BP+OSD."
- **higher-order LSD (LSD-mu)** - A local higher-order reprocessing routine for LSD, analogous to higher-order OSD, that searches low-weight fault configurations within individual clusters to improve accuracy, supplying parallel reprocessing with a reduced search space and smaller matrix-vector multiplications.
  - guarantee: statistical bound; numerically achieves decoding improvements on par with global BP+OSD-w reprocessing while operating locally per cluster.
  - quote: "Inspired by higher-order OSD, we propose a higher-order reprocessing method for LSD, which we refer to as LSD$-\mu$."

## Where it could apply
- Target primitive(s): syndrome decoding for fault-tolerant quantum error correction (real-time circuit-level syndrome decoding); not VQE/QAOA/QPE/sampling.
- Target application group(s): error_correction; platforms_tooling (real-time decoder co-processors on FPGA/ASIC/GPU).
- Code family / hardware assumptions: CSS QLDPC codes including surface codes, hypergraph product codes, and bivariate bicycle codes; circuit-level depolarizing noise; targets neutral atom and bi-layer superconducting QLDPC architectures and specialised classical decoding hardware.

## Caveats
The fully parallel implementation is not yet benchmarked; only a serial C++/Python implementation (in the LDPCv2 package) is evaluated, and the authors state real-time hardware benchmarking on GPUs or FPGAs is left to future work, so it is not yet a real-time decoder. Erasure decoding and parallel-window decoding are conjectured or sketched, not demonstrated. The decoder targets minimum-weight rather than maximum-likelihood decoding. No paywall or access limitation; full LaTeX source was available.