The measured logical error rate of a quantum low-density parity-check (QLDPC) code depends on the decoding algorithm. We develop a framework that certifies non-degenerate maximum-likelihood (ML) decoding outcomes on individual sampled shots under circuit-level noise, providing a common reference for comparing decoders and codes. Upper and lower bounds on the minimum correction cost in the true and competing logical classes establish success, failure, or an undecided outcome. Branch-and-bound search refines these intervals while retaining valid bounds under resource limits. Counting the outcomes bounds the empirical ML logical error rate, with interval width equal to the undecided fraction. Results for two bivariate bicycle codes show that best-first ordered statistics decoding (BF-OSD) tracks the non-degenerate ML benchmark closely while leaving some ML-correctable shots.
The measured logical error rate of a quantum low-density parity-check (QLDPC) code depends on the decoding algorithm. We develop a framework that certifies non-degenerate maximum-likelihood (ML) decoding outcomes on individual sampled shots under circuit-level noise, providing a common reference for comparing decoders and codes. Upper and lower bounds on the minimum correction cost in the true and competing logical classes establish success, failure, or an undecided outcome. Branch-and-bound search refines these intervals while retaining valid bounds under resource limits. Counting the outcomes bounds the empirical ML logical error rate, with interval width equal to the undecided fraction. Results for two bivariate bicycle codes show that best-first ordered statistics decoding (BF-OSD) tracks the non-degenerate ML benchmark closely while leaving some ML-correctable shots.
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.
Belief Propagation (BP) followed by Ordered Statistics Decoding (OSD) has emerged as the gold standard for decoding quantum low-density parity-check (QLDPC) codes. Recent advancements in this field have proposed new methods and algorithms to lower the complexity of this standard pipeline. Because of code degeneracy, and more in general because multiple distinct error patterns can produce the same syndrome, OSD is inherently a list-decoding technique; that is, it enumerates a set of syndrome-consistent candidates and returns the most probable one. In this work, we propose a variant of OSD, which we call Best-First OSD (BF-OSD), that explores the error-candidate space more efficiently by traversing it in order of decreasing likelihood, rather than by brute-force enumeration of a pre-selected subset. In addition, we depart from the conventional BP+OSD cascade: instead of conditioning the OSD invocation on BP convergence, we invoke OSD after a fixed, small number of BP iterations. This design choice is motivated by the full circuit-level noise regime, in which BP is particularly unreliable. Monte Carlo simulations of a family of Bivariate Bicycle (BB) codes under full circuit-level noise show that BF-OSD matches the performance of the BP+OSD baseline while exploring the solution space with 1/100th of the query budget.
Quantum decoding succeeds when a correction matches the logical class of the error, whereas ordered-statistics decoding (OSD) searches a finite list of physical corrections. We connect these spaces through a rank-k logical-signature map Mlog:F2q→F2k. For fixed, error-independent orderings it yields the exact logical-signature distribution, the OSD-0 logical error rate, and the logical-class coverage of any fixed list. For minimum-cost list decoding we prove PL=Pmiss+Psel, separating absent-class from wrong-selection failures. Hypergraph-product and bivariate-bicycle codes are consistent with the coverage predictions. On the [[144,12,12]] gross code, 672 coverage-aware candidates outperform 899 combination-sweep candidates (paired McNemar p=0.013), while two lists with identical logical coverage give PL=11.30% and 3.78%; the signature-based coverage compression persists across 30 random column orderings. Logical coverage and within-class representative selection are therefore complementary resources that candidate count alone does not capture. The framework guides OSD list construction and compression by identifying coverage requirements and distinguishing coverage losses from failures of representative selection.
Quantum error correction is the building block for constructing fault-tolerant quantum processors that can operate reliably even if its constituting elements are corrupted by decoherence. In this context, real-time decoding is a necessity for implementing arbitrary quantum computations on the logical level. In this work, we present a new decoder for Quantum Low Density Parity Check (QLDPC) codes, named the closed-branch decoder, with a worst-case complexity loosely upper bounded by O(nmaxgrmaxbr), where maxgr and maxbr are tunable parameters that pose the accuracy versus speed trade-off of decoding algorithms. For the best precision, the maxgrmaxbr product increases exponentially as ∝djd, where d indicates the distance of the code and j indicates the average row weight of its parity check matrix. Nevertheless, we numerically show that considering small values that are polynomials of the code distance are enough for good error correction performance. The decoder is described to great extent and compared with the Belief Propagation Ordered Statistics Decoder (BPOSD) operating over data qubit, phenomenological and circuit-level noise models for the class of Bivariate Bicycle (BB) codes. The results showcase a promising performance of the decoder, obtaining similar results with much lower complexity than BPOSD when considering the smallest distance codes, but experiencing some logical error probability degradation for the larger ones. Ultimately, the performance and complexity of the decoder depends on the product maxgrmaxbr, which can be considered taking into account benefiting one of the two aspects at the expense of the other.
Accurate decoding of quantum low-density parity-check (qLDPC) codes often relies on expensive post-processing search, although decoding difficulty varies strongly between syndromes. We find that the benefit of deeper post-processing search is highly concentrated in a small subset of decoding instances, and that these instances can be identified directly from the decoder itself. To this end, we introduce an adaptive decoder based on belief propagation (BP) and ordered-statistics decoding (OSD), using the disagreement between the BP hard decision and the syndrome-consistent order-zero OSD solution as an internal risk signal to determine where deeper search is required. On a [[144,12,12]] bivariate bicycle code under circuit-level depolarising noise, encoding 12 logical qubits in 144 data qubits with distance 12, escalating only the highest-risk 20% of instances recovers 85% to 92% of the improvement in logical error rate available from a full one-free-variable sweep, while reducing the mean serial decoding cost by a factor of 3.6 relative to applying the same truncated search to every instance. At matched budget, disagreement-guided routing also outperforms routing based on the BP syndrome residual, syndrome weight and random selection. The same behaviour reappears on a structurally distinct radial qLDPC code obtained from a lifted-product construction, where escalating 20% of instances recovers 87% to 94% of the available gain. A Z-memory experiment on the Quantinuum H2 trapped-ion processor, implementing both X-type and Z-type checks, further shows that the signal remains predictive under device noise. These results show that decoder-internal disagreement can expose where additional decoding effort is valuable, allowing classical computation to be concentrated on the instances most likely to benefit from it.
Accurate and efficient decoding is a crucial component for achieving fault-tolerant quantum computing. Realistic circuit-level noise introduces temporal correlations and degeneracy, making optimal (maximum-likelihood) decoding computationally intractable in general. As a result, practical decoders rely on heuristic approximations, and it is generally difficult to quantify how suboptimal they are, as this strongly depends on the code and noise model considered. In this work, we study the accuracy of practical decoding algorithms under circuit-level noise by comparing them against a maximum likelihood decoding benchmark. Our approach propagates the density matrix through the full memory experiment and computes the optimal decoding decision for each syndrome history. We introduce pruning techniques with rigorous bounds, allowing us to access larger numbers of syndrome-extraction rounds. We apply this framework to small instances of the repetition code and a cellular automaton code, and benchmark minimum-weight perfect matching (MWPM), belief propagation with ordered statistics decoding (BP+OSD), Tesseract, and Planar decoders against optimal decoding. While standard decoders remain close to optimal for the repetition code, we find significant deviations for the cellular automaton code, with BP+OSD deteriorating already in experimentally relevant noise regimes. Moreover, the pruning method developed here highlights that, at low physical error rates, only a narrow fraction of syndrome histories contributes significantly to the logical error rate.