Abstract Quantum low-density parity-check (qLDPC) memories promise lower-overhead fault tolerance, but reproducible finite-instance benchmarking has been hin- dered by the absence of explicit decoder representations and data provenance. We introduce adversarial channel equilibria (ACE), a bounded-Pauli channel game that organises fidelity guarantees, certificates and decoder benchmarks for bivariate-bicycle (BB) memories. Applying Sion’s minimax theorem to compact convex sets of completely positive trace-preserving maps, we prove existence of a saddle-point equilibrium and derive a closed-form fidelity floor. We provide the first SHA-256-anchored three-code benchmark comparing BP+OSD against two graphified baselines—DEM-chain MWPM and UF-BFS—on BB72, BB144 and BB288, using 10,000 trials per point and deterministic seeds. BP+OSD achieves risk ratios of 595× to 67.8× over DEM-chain MWPM on BB72 across tested error rates; at larger code sizes BP+OSD failure counts are zero while the graphi- fied baselines fail at rates above 50%. Repeated-cycle BP+OSD remains near zero through the tested range. A BB72 certificate verifies commutation, dimension, distance lower bound and radius-two correctability. Hardware-noise simulations are reported as model sensitivities only. All claims are mapped to verifiable data, code and SHA-256 hash artefacts archived at doi:10.5281/zenodo.20588303.
Abstract Quantum low-density parity-check (qLDPC) memories promise lower-overhead fault tolerance, but reproducible finite-instance benchmarking has been hin- dered by the absence of explicit decoder representations and data provenance. We introduce adversarial channel equilibria (ACE), a bounded-Pauli channel game that organises fidelity guarantees, certificates and decoder benchmarks for bivariate-bicycle (BB) memories. Applying Sion’s minimax theorem to compact convex sets of completely positive trace-preserving maps, we prove existence of a saddle-point equilibrium and derive a closed-form fidelity floor. We provide the first SHA-256-anchored three-code benchmark comparing BP+OSD against two graphified baselines—DEM-chain MWPM and UF-BFS—on BB72, BB144 and BB288, using 10,000 trials per point and deterministic seeds. BP+OSD achieves risk ratios of 595× to 67.8× over DEM-chain MWPM on BB72 across tested error rates; at larger code sizes BP+OSD failure counts are zero while the graphi- fied baselines fail at rates above 50%. Repeated-cycle BP+OSD remains near zero through the tested range. A BB72 certificate verifies commutation, dimension, distance lower bound and radius-two correctability. Hardware-noise simulations are reported as model sensitivities only. All claims are mapped to verifiable data, code and SHA-256 hash artefacts archived at doi:10.5281/zenodo.20588303.
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.
Fair threshold estimation for bivariate bicycle (BB) codes on the quantum erasure channel runs into two recurring problems: decoder-baseline unfairness and the conflation of finite-size pseudo-thresholds with true asymptotic thresholds. We run both uninformed and \emph{erasure-aware} minimum-weight perfect matching (MWPM) toric code baselines alongside BP-OSD decoding of BB codes. With standard depolarizing-weight MWPM and no erasure information, performance matches random guessing on the erasure channel in our tested regime -- so prior work that compares against this baseline is really comparing decoders, not codes. Using 200{,}000 shots per point and bootstrap confidence intervals, we sweep five BB code sizes from N=144 to N=1296. Pseudo-thresholds (WER = 0.10) run from p∗=0.370 to 0.471; finite-size scaling (FSS) gives an asymptotic threshold p∞∗≈0.488, within 2.4\% of the zero-rate limit and without maximum-likelihood decoding. On the fair baseline, BB at N=1296 has a modest edge in threshold over the toric code at twice the qubit count, and a 12× lower normalized overhead -- the latter is where the practical advantage sits. All runs are reproducible from recorded seeds and package versions.
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.
Erasure qubits reduce overhead in fault-tolerant quantum error correction (QEC) by converting dominant faults into detectable errors known as erasures. They have demonstrated notable improvements in thresholds and scaling in surface and Floquet code memories. In this work, we use erasure qubits on Bivariate Bicycle (BB) codes from the quantum low-density parity-check (QLDPC) regime. Owing to their sparse structure and favorable rate-distance trade-offs, BB codes are practical candidates for QEC. We introduce BiBiEQ, a novel framework that compiles a given BB code into an erasure-aware memory circuit C_E. This erasure circuit C_E comprises erasure checks (ECs), resets, and erasures spread over a user-specified erasure check schedule (2EC, 4EC). BiBiEQ converts this erasure circuit C_E into the stabilizer circuit C for general-purpose decoding. BiBiEQ provides two engines for this conversion, BiBiEQ-Exact and BiBiEQ-Approx. BiBiEQ-Exact preserves the joint-erasure correlations and serves as our accuracy benchmark, while BiBiEQ-Approx uses an independence approximation to accelerate large sweeps and expose accuracy-throughput trade-offs. Using BiBiEQ, we decode the stabilizer circuits to get a per-round logical error rate (LER) for the BB codes and quantify the effect of the EC schedules on the correctable operating region below the pseudo-threshold. The 4EC schedule keeps the accuracy of both engines close to one another, making BiBiEQ-Approx a reliable proxy for BiBiEQ-Exact for faster sweeps. Below the pseudo-threshold, the code distance (d) hop from distance (d) 6 to 10 yields a drop in LER by 10-17x larger than distance (d) 10 to 12, showing that most gains are realized by d=10.
Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.
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.