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Quantum Maxwell Erasure Decoder for qLDPC codes

Bruno Costa Alves Freire, François-Marie Le Régent, Anthony Leverrier

TL;DR

The paper tackles fast decoding for CSS qLDPC codes on the quantum erasure channel by introducing a quantum Maxwell erasure decoder that extends peeling with bounded symbolic guesses tracked as affine forms. The method provides a tunable tradeoff between linear-time decoding and ML performance through a budget parameter $G_{ ext{max}}$, with two CSS components decoded independently. The authors establish linear-time scaling for fixed budgets and show that, asymptotically, a sufficiently large budget matches ML performance at the leading exponent, supported by empirical results on bivariate bicycle and quantum Tanner codes. The approach offers a scalable alternative to ML decoding with competitive performance relative to cluster decoding and suggests promising extensions to broader stabilizer codes and noise models.

Abstract

We introduce a quantum Maxwell erasure decoder for CSS quantum low-density parity-check (qLDPC) codes that extends peeling with bounded guessing. Guesses are tracked symbolically and can be eliminated by restrictive checks, giving a tunable tradeoff between complexity and performance via a guessing budget: an unconstrained budget recovers Maximum-Likelihood (ML) performance, while a constant budget yields linear-time decoding and approximates ML. We provide theoretical guarantees on asymptotic performance and demonstrate strong performance on bivariate bicycle and quantum Tanner codes.

Quantum Maxwell Erasure Decoder for qLDPC codes

TL;DR

The paper tackles fast decoding for CSS qLDPC codes on the quantum erasure channel by introducing a quantum Maxwell erasure decoder that extends peeling with bounded symbolic guesses tracked as affine forms. The method provides a tunable tradeoff between linear-time decoding and ML performance through a budget parameter , with two CSS components decoded independently. The authors establish linear-time scaling for fixed budgets and show that, asymptotically, a sufficiently large budget matches ML performance at the leading exponent, supported by empirical results on bivariate bicycle and quantum Tanner codes. The approach offers a scalable alternative to ML decoding with competitive performance relative to cluster decoding and suggests promising extensions to broader stabilizer codes and noise models.

Abstract

We introduce a quantum Maxwell erasure decoder for CSS quantum low-density parity-check (qLDPC) codes that extends peeling with bounded guessing. Guesses are tracked symbolically and can be eliminated by restrictive checks, giving a tunable tradeoff between complexity and performance via a guessing budget: an unconstrained budget recovers Maximum-Likelihood (ML) performance, while a constant budget yields linear-time decoding and approximates ML. We provide theoretical guarantees on asymptotic performance and demonstrate strong performance on bivariate bicycle and quantum Tanner codes.
Paper Structure (10 sections, 6 theorems, 11 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 6 theorems, 11 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Quantum Maxwell decoder
  3. Theoretical properties
  4. Notation and data structure model
  5. Complexity analysis
  6. Performance in the asymptotic regime
  7. Numerical results
  8. Conclusion
  9. Pruning and redundant checks
  10. Comparing guessing strategies

Key Result

Theorem 1

Algorithm algo:maxwellpeel runs in $O\left(e d_v d_c G_{\max}^2\right)$ bit operations. In particular, for fixed $G_{\max}$ and bounded degrees, the runtime is $O(e)$.

Figures (3)

  • Figure 1: Logical error rate $p_L$ versus erasure rate $\epsilon$ for two BB codes (top) and two quantum Tanner codes (bottom). We compare peeling (black), ML decoding (purple), the quantum Maxwell decoder with $1 \leq G_{\max} \leq 6$ (with score-based pivot selection) and the cluster decoder with cluster cutoff $C \in \{20, 100\}$cluster_yao_gokduman_pfister.
  • Figure 2: Impact of pruning and parity-check regularization on the $[[576,24,\leq 24]]$ quantum Tanner code. Rows: pruning off/on. Columns: original vs regularized parity-check matrices. Curves show peeling, ML, and quantum Maxwell for $1\leq G_{\max} \leq 6$.
  • Figure 3: Random vs score-based pivot selection on the $[[576,24,\leq 24]]$ quantum Tanner code. Solid/dashed curves indicate pruning depth $D \in \{0,1\}$. Score-based selection significantly lowers the error floor at small $\epsilon$ for modest $G_{\max}$.

Theorems & Definitions (9)

  • Theorem 1: Runtime of symbolic MAXWELLPEEL under PRP
  • Lemma 1
  • Definition 1
  • Definition 2: Stopping sets and stopping distance
  • Definition 3: Distribution gaps
  • Theorem 2: Budget exhaustion implies large erasure
  • Corollary 1: Divisibility of the gap polynomial
  • Corollary 2: Matching the ML exponent for CSS codes
  • Proposition 1