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arXiv:2601.14236 · IEEE International Symposium on Information Theory (ISIT)

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

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

Published Jun 1, 2026• 1 citation
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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.

DOI
10.1109/isit62367.2026.11653654

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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.
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 low density parity-check (QLDPC) codes represent an attractive candidate for achieving fault-tolerant quantum computation. Erasure decoding of QLDPC codes has gained traction recently as many physical systems such as neutral-atom systems, photonic systems, trapped-ion systems etc. suffer from qubit erasures. Techniques for erasure decoding of QLDPC codes via belief propagation (BP) have been proposed in the literature, such as BP with guided decimation (BP-GD), where decimation refers to sequentially fixing hard values for the variable nodes. We provide an alternative decimation-based BP algorithm, which we term the BP with degree-based decimation (BP-DD) algorithm, that intelligently chooses a check node based on its degree and subsequently decimates a variable node in its neighborhood. We apply the BP-DD algorithm to two families of QLDPC codes, namely hypergraph product codes and lifted product codes, and show improved logical error performance over the BP-GD algorithm, without incurring additional complexity. Our simulations show that the BP-DD algorithm provides a versatile solution for erasure decoding of QLDPC codes, in contrast to some techniques in the literature that are applicable to only specific classes of QLDPC codes.
Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to the low operational depth of their syndrome-measurement circuits. Numerous studies advocate the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor when decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of thehypergraph-product(HGP) andlifted-product(LP) codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets (TS), which correspond to stabilizer-induced subgraphs in the Tanner graphs, and (2) classical trapping sets (TS), which originate from the classical codes used in the construction of HGP and LP codes. The dynamics of stabilizer-induced TSs are examined, and a straightforward modification of iterative decoders is proposed to circumvent these TSs. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent the classical TSs in both HGP and LP codes by deriving them from decoders capable of avoiding the TSs in the parent classical LDPC codes. When decoders that can avoid stabilizer-induced TSs are run in parallel with those that can mitigate the effect of classical TSs, the logical error rate improves significantly in the error-floor region.
We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the \emph{circuit-level} noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (\emph{hook errors}) create \emph{stabilizer-induced} trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding \emph{stabilizer-induced} TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of \emph{stabilizer-induced} TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a \emph{phenomenological} Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on LP codes show a significant improvement in logical error rate relative to BP+OSD. Moreover, the improved error correction relies on low-complexity finite-state iterative decoders, targeted for fast and low-latency FPGA implementation.
We propose a decoding algorithm for color codes over the quantum erasure channel, which is linear-time maximum likelihood (ML) when the set of erased qubits satisfies a certain condition called trimmability. Two methods are proposed for general erasure sets, either by extending the erasure set to make it trimmable, or by inactivating some vertices. The former is linear time but not ML, while the latter is ML but not linear time. Numerical results are provided to assess the error correction performance and the complexity of both methods.
We propose a post-processing method for message-passing (MP) decoding of CSS quantum LDPC codes, called stabilizer-inactivation (SI). It relies on inactivating a set of qubits, supporting a check in the dual code, and then running the MP decoding again. This allows MP decoding to converge outside the inactivated set of qubits, while the error on these is determined by solving a small, constant size, linear system. Compared to the state of the art post-processing method based on ordered statistics decoding (OSD), we show through numerical simulations that MP-SI outperforms MP-OSD for different quantum LDPC code constructions, different MP decoding algorithms, and different MP scheduling strategies, while having a significantly reduced complexity.