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Stabilizer-Assisted Inactivation Decoding of Quantum Error-Correcting Codes with Erasures

Giulio Pech, Mert Gökduman, Hanwen Yao, Henry D. Pfister

TL;DR

A reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures is developed, and it is shown that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes.

Abstract

In this work, we develop a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures. Our decoder combines classical inactivation decoding, which integrates peeling with symbolic guessing, with a new dual peeling procedure. In the dual peeling stage, we perform row operations on the stabilizer matrix to efficiently reveal stabilizer generators and their linear combinations whose support lies entirely on the erased set. Each such stabilizer identified allows us to freely fix a bit in its support without affecting the logical state of the decoded result. This removes one degree of freedom that would otherwise require a symbolic guess, reducing the number of inactivated variables and decreasing the size of the final linear system that must be solved. We further show that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes. Simulations across several QLDPC code families confirm that our decoder matches ML logical failure performance while significantly reducing the complexity of inactivation decoding, including more than a 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates.

Stabilizer-Assisted Inactivation Decoding of Quantum Error-Correcting Codes with Erasures

TL;DR

A reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures is developed, and it is shown that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes.

Abstract

In this work, we develop a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures. Our decoder combines classical inactivation decoding, which integrates peeling with symbolic guessing, with a new dual peeling procedure. In the dual peeling stage, we perform row operations on the stabilizer matrix to efficiently reveal stabilizer generators and their linear combinations whose support lies entirely on the erased set. Each such stabilizer identified allows us to freely fix a bit in its support without affecting the logical state of the decoded result. This removes one degree of freedom that would otherwise require a symbolic guess, reducing the number of inactivated variables and decreasing the size of the final linear system that must be solved. We further show that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes. Simulations across several QLDPC code families confirm that our decoder matches ML logical failure performance while significantly reducing the complexity of inactivation decoding, including more than a 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates.
Paper Structure (23 sections, 3 theorems, 15 equations, 3 figures)

This paper contains 23 sections, 3 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Erasure Correction
  4. Stabilizer Formalism
  5. CSS Codes and Quantum LDPC Codes
  6. Syndrome Decoding on the Quantum Erasure Channel
  7. Inactivation Decoding
  8. Quantum Decoding
  9. Stage 1: Dual peeling and primal peeling
  10. Stage 2: Inactivation decoding
  11. Relation to ML decoding on surface codes
  12. Numerical Results
  13. Logical Error Rates and ML Optimality
  14. Effect of Dual Peeling
  15. Conclusion and Future Work
  16. ...and 8 more sections

Key Result

Lemma 1

Fix ${\cal E}$ and ${\cal K}={\cal E}^c$. After running dual peeling on $(H_X,{\cal K})$ to exhaustion, re-running rule-2 dual peeling after any amount of primal peeling cannot produce a new fully erased $X$-stabilizer.

Figures (3)

  • Figure 1: Block structure of the augmented erased submatrix $H_{{\cal E}}$ during inactivation decoding. The $k$ erased columns are split into $\ell_r$ active columns processed by peeling (left) and $\ell_x$ inactivated/guessed columns (gray band, right). The last column $s$ is the syndrome, used to create the augmented matrix to solve \ref{['eq:syndrome-system']}. Blocks $C$ and $D$ contain the coefficients of the $\ell_x$ guessed variables in these two sets of constraints, $C$ to solve for the inactive ones, and $D$ to determine the active bits given the inactive ones.
  • Figure 2: Decoder comparison across code families on the quantum erasure channel: logical failure rate versus erasure rate $p$ for peeling, inactivation, and the two stabilizer-assisted variants.
  • Figure 3: Comparison of the expected size of the inverted matrix for inactivation decoding, with and without stabilizer peeling, versus erasure rate $p$.

Theorems & Definitions (8)

  • Lemma 1: No gain from repeating dual peeling after primal
  • proof
  • Definition 1: Surface code
  • Lemma 2: Known-column weight-2 elimination identifies all fully-erased $X$-stabilizers on a surface code
  • proof
  • Theorem 1: Known-column weight-2 elimination followed by primal peeling is ML on surface codes over erasure
  • proof
  • proof