Impulse Decoding of Quantum LDPC Codes: Equivalence of Degeneracy and Code-Shortening

Abstract: Quantum error correction is essential for building scalable quantum computers. Within the stabilizer formalism, the Calderbank-Shor-Steane framework constructs quantum codes from pairs of classical linear codes. A distinctive feature in this setting is degeneracy, where multiple equivalent error estimates exist-a phenomenon that has no classical counterpart, and the lack of a meaningful classical coding-theoretic interpretation of which has remained a gap in the literature. In this paper, we demonstrate that degeneracy is closely related to the classical operation of shortening of a linear block code. Interestingly, the shortening here takes place at the decoder rather than at the encoder. Leveraging this insight, we present a parallel decoding scheme for quantum low-density parity-check codes, which we term impulse decoding, that significantly outperforms belief propagation with ordered statistics decoding, as well as several other existing techniques, under both code-capacity and circuit-level noise, with significantly lesser complexity. We then present another algorithm based on decoding of residual errors, which when combined with impulse decoding achieves further performance improvement under circuit-level noise.

• The paper establishes a rigorous equivalence between quantum degeneracy and classical code-shortening, enabling a reduced decoding search space.
• The impulse decoding protocol leverages parallel BP decoding with variable node shortening, achieving up to an order-of-magnitude improvement in failure rates.
• Numerical results demonstrate reduced latency and computational overhead, validating the scalability of the decoder in circuit-level noise scenarios.

Impulse Decoding of Quantum LDPC Codes: Degeneracy–Shortening Equivalence and Decoder Design

Introduction

Quantum error correction is a foundational element for scalable quantum computation, where the inherent fragility of quantum information necessitates robust code structures and efficient decoding algorithms. Within the stabilizer formalism, the Calderbank-Shor-Steane (CSS) family enables the construction of quantum error-correcting codes (QECCs) from classical codes, acting as a bridge between quantum and classical coding theory. A key quantum phenomenon—degeneracy—results in multiple valid error estimates for the same syndrome, lacking an analog in classical coding and posing both a theoretical and practical challenge. This paper establishes a rigorous connection between degeneracy and classical code-shortening, proposing a decoding strategy—impulse decoding—that exploits this equivalence for quantum low-density parity-check (QLDPC) codes (2606.18240).

Degeneracy as Code-Shortening at the Decoder

The authors formalize the relationship between quantum degeneracy and classical code-shortening operations. In classical codes, shortening is an encoder-side operation for producing subcodes with fixed values on certain positions. In quantum stabilizer codes, however, degeneracy allows the decoder to restrict its search space to shortened cosets post hoc, based on syndrome information.

For CSS codes, this means that, for any given qubit coordinate, the decoder can assume the error is either 0 or 1 at that location and decode within the corresponding shortened subcode. This equivalence is central: instead of exhaustively exploring all degenerate errors, the decoder can leverage code-shortening to reduce its search space and facilitate more efficient convergence, a feature unattainable in classical syndrome decoding.

Qualitative analysis shows that shortening the code to either 0 or 1 halves the space of degenerate errors compared to standard decoding, with empirical results demonstrating that shortening to 1 yields notably superior performance and reduced latency.

Impulse Decoding: Design and Algorithmic Framework

Building on the degeneracy–shortening equivalence, the paper introduces impulse decoding—a parallelized BP-based decoding protocol. The scheme involves:

• Performing standard BP decoding on the Tanner graph associated with the code.
• If BP fails, n parallel decoders are instantiated, each shortening a different variable node to 1 by setting its channel LLR to $-\infty$.
• Each parallel decoder executes BP. The final error estimate is selected among converged decoders using either the minimum-weight criterion or the first-convergence criterion.

This approach avoids modifications to the Tanner graph and requires only simple alterations to the BP initialization. For codes with a large number of variable nodes (such as those induced by circuit-level noise models), reliability-based selection (using BP output LLR magnitudes) guides impulse decoding to focus on the least reliable nodes, optimizing resource usage.

Furthermore, a residual-error impulse decoding algorithm is introduced, performing sequential residual error correction per decoder, in practice requiring fewer parallel decoding instances while offering further improvement under circuit-level noise.

Numerical Results and Comparative Performance

Simulation results for BB and LP code families showcase impulse decoding's state-of-the-art performance. For the [[288, 12, 18]] BB code, impulse decoding achieves up to an order-of-magnitude improvement in failure rate over BP-OSD decoding in the low physical error rate regime. Notably, shortening to 1 significantly outperforms shortening to 0, contradicting conventional intuition; this is ascribed to the expanded exploration of degenerate error space and more favorable decoder dynamics.

The performance gap between minimum-weight and first-convergence criteria is most significant at low error rates, attributable to increased logical error susceptibility when converging to degenerate errors. Latency analysis demonstrates exponential reduction in the number of variable nodes needed for convergence under the first-convergence criterion—a critical factor for hardware implementations.

For circuit-level noise, reliability-based impulse decoding reduces decoding complexity. The residual-error-based variant surpasses reliability-based impulse decoding under analogous resource allocation (N=20, R=6), with complexity scaling sublinearly with the number of decoder instances.

Algorithmic complexity is favorably compared to recent decoders (restart_belief, beam_search, relayBP), with impulse decoding yielding comparable or improved performance at lower computational and parallelization overhead.

Implications and Future Directions

The explicit equivalence between quantum degeneracy and classical code-shortening at the decoder not only grounds heuristic strategies used in QLDPC decoding literature in rigorous coding theory, but also suggests systematic methods for exploiting degeneracy to enhance both performance and latency.

Practically, impulse decoding offers a scalable, parallelizable strategy amenable to hardware implementation, with minimal additional complexity vis-à-vis standard BP decoding. The decoder's adaptability to circuit-level noise environments strengthens its relevance for fault-tolerant quantum computation.

Theoretically, the degeneracy–shortening equivalence clarifies the role of logical operator structure in determining minimum-distance and error correction capability in stabilizer codes. Future progress may focus on optimal node selection strategies for shortening, message-passing schedule optimization, handling correlated errors, and extending impulse decoding to emerging quantum code constructions.

Conclusion

This work rigorously connects quantum degeneracy to classical code-shortening, proposing impulse decoding—a principled, parallel decoding protocol for QLDPC codes. Strong numerical results validate its effectiveness and reduced complexity under both code-capacity and circuit-level noise. The degeneracy–shortening equivalence provides a foundation for future research in efficient quantum decoding and advances the theoretical understanding of stabilizer codes (2606.18240).