Surface-code continuous-time Markov chain (CTMC) models with boundary injection and sink absorption provide a continuous-time view of defect transport and logical-parity flips, but policy-gated injection introduces a normalization ambiguity: should flips be normalized by attempted or accepted injections? Standard discrete-round analyses and naive "flips per attempt" ratios do not separate throughput control from per-accepted-attempt maliciousness. We establish an exact renewal identity: under empty-manifold gating (accept only when the syndrome is empty), accepted injections launch disjoint excursions, and the hazard-normalized estimator peff,acc = N flip /A acc is unbiased for the odd-parity excursion probability under the accepted-start distribution (Theorem 1). A representative CTMC dataset at L = 11, η = 0.9 shows that peff,acc remains stable (within statistical error) even when the acceptance fraction varies by more than a factor of two, while "per attempted injection" ratios change proportionally (values reported in companion application paper). We provide portable bridges to practice: mapping sustaining flip rates to per-cycle logical error probability, a decoder-integration pipeline from CTMC logs to detection events, and a reduced-model exact cross-check. We do not claim an optimal admission policy, nor do we address non-renewal (overlap) regimes in this work.
Surface-code continuous-time Markov chain (CTMC) models with boundary injection and sink absorption provide a continuous-time view of defect transport and logical-parity flips, but policy-gated injection introduces a normalization ambiguity: should flips be normalized by attempted or accepted injections? Standard discrete-round analyses and naive "flips per attempt" ratios do not separate throughput control from per-accepted-attempt maliciousness. We establish an exact renewal identity: under empty-manifold gating (accept only when the syndrome is empty), accepted injections launch disjoint excursions, and the hazard-normalized estimator peff,acc = N flip /A acc is unbiased for the odd-parity excursion probability under the accepted-start distribution (Theorem 1). A representative CTMC dataset at L = 11, η = 0.9 shows that peff,acc remains stable (within statistical error) even when the acceptance fraction varies by more than a factor of two, while "per attempted injection" ratios change proportionally (values reported in companion application paper). We provide portable bridges to practice: mapping sustaining flip rates to per-cycle logical error probability, a decoder-integration pipeline from CTMC logs to detection events, and a reduced-model exact cross-check. We do not claim an optimal admission policy, nor do we address non-renewal (overlap) regimes in this work.
We propose a new decoder for "matchable'' qLDPC codes that uses a Markov Chain Monte Carlo algorithm - called the worm algorithm - to approximately compute the probabilities of logical error classes given a syndrome. The algorithm hence performs (approximate) optimal decoding, and we expect it to be computationally efficient in certain settings. The algorithm is applicable to decoding random errors for the surface code, the honeycomb Floquet code, and hyperbolic surface codes with constant rate, in all cases with and without measurement errors. The efficiency of the decoder hinges on the mixing time of the underlying Markov chain. We give a rigorous mixing time guarantee in terms of a quantity that we call the defect susceptibility. We connect this quantity to the notion of disorder operators in statistical mechanics and use this to argue (non-rigorously) that the algorithm is efficient for typical errors in the entire decodable phase. We also demonstrate the effectiveness of the worm decoder numerically by applying it to the surface code with measurement errors as well as a family of hyperbolic surface codes. For most codes, the matchability condition restricts direct application of our decoder to noise models with independent bit-flip, phase-flip, and measurement errors. However, our decoder returns soft information which makes it useful also in heuristic "correlated decoding'' schemes which work beyond this simple setting. We demonstrate this by simulating decoding of the surface code under depolarizing noise, and we find that the threshold for "correlated worm decoding'' is substantially higher than for both minimum-weight perfect matching and for correlated matching.
Periodic admission control (AC) can change when accepted events occur within a cycle even when long-run averages (e.g. total accepted injections per unit time) are unchanged. This "temporal bias" matters whenever logical-flip risk is phase-dependent or when analysts condition on accepted events without correcting for exposure. We formalize an exposure-proportional null for phase-resolved logical flips and propose two lightweight diagnostics: (i) a Poisson sinusoid regression on log-risk with an exposure offset, and (ii) an exact window-enrichment test (binomial conditional on the total number of flips), with a scan-corrected [5] Monte Carlo p-value when the phase offset is unknown. Applied to a representative L=11 run (η=0.9, γ=0.002, period 10, 60 phase bins, T max =10 7) we record N = 466,947 accepted injections and Y = 87 logical-Z flips (p 0 ≈ 1.863 × 10 −4 per accepted injection) and find no evidence of phase modulation at current statistics (sinusoid p ≈ 0.466; scan-corrected window p ≈ 0.694). We summarize power benchmarks (e.g. ≳ 700 flips to rule out ≥ 30% peak/trough modulation) and recommend reporting rules that prevent spurious phase effects in gated simulations and hardware-aligned experiments.
We present a contract-preserving, family-aware comparison of decoder behavior across four syndrome-generation stacks (PennyLane, Qiskit, Cirq, and a LiDMaS+ reference) under a fixed replay interface. Request streams from repeated-round surface-code and digitized-GKP circuits are replayed through BP, MWPM, and UF with matched controls. The unified matrix spans 24 cells and achieves line-level integrity: 24000 request lines, 24000 response lines, response ratio =1.0 in every cell, zero parse failures, and zero decoder-name mismatches. Fifteen warning-no-syndrome events occur only in GKP-Cirq rows. Within-family ordering is stable in both families (BP<MWPM<UF); source-averaged flip counts are 2.435, 4.768, and 5.870 for surface and 1.595, 2.908, and 3.681 for GKP. Relative to MWPM, BP reduces mean intervention volume by 48.9% in surface and 45.1% in GKP. Source-vs-reference effects are family dependent, with larger coherent shifts in GKP. Source-bootstrap ranks remain unchanged, and hidden-truth sidecars add an outer-code logical-parity error-rate check in which UF has the largest source-mean rate in both families. The comparison is stack-aware, contract-verified, and avoids raw cross-family threshold-equivalence claims.
We formulate stabilizer-code error correction as a code- and recovery-dependent transformation of the noise channel. For a fixed recovery map, syndrome projection produces a block-diagonal logical channel whose diagonal Pauli weights follow from code-induced error classes, with no physical Pauli twirl; concatenation is then iterated redistribution, with the dominant logical error guiding the next code. The bit-flip and phase-flip three-qubit codes with standard recoveries supply a closed-form benchmark, and logically inequivalent recoveries appear as a design degree of freedom for shaping the channel presented to the next decoder. For amplitude damping, the three-qubit rules select C2(σ^y)C1(σ^x) and give fl=1−O(γ2), with an ideal-gate crossover at γc≈0.30 (f=1−1.72755γ2) and a favorable gate-count comparison with the five-qubit code under the same assumptions. Correlated noise requires joint class statistics, whereas coherent noise retains intra-class coherences that make the same transformation usable in reverse as a tomographic fingerprint; there the companion work proves local identifiability from a rank certificate evaluated inside the parameter domain, classifies the sole permutation-symmetry ambiguity as structural at any chain length, and closes it with one two-qubit marginal channel. The construction remains an analytic benchmark, not a universal prescription for an inner code beneath a surface code.
We develop a physics-informed graph neural network (GNN) decoder for the surface code that solves a discrete Poisson equation on the syndrome graph, with the syndrome as the charge source. We compare four readout architectures for extracting the logical-error probability: a potential-based readout that maps the Poisson field through a multilayer perceptron, two current-based readouts under single- and two-sink Dirichlet boundary conditions, and a diffusion-based variant. Comparing these, we show that the solver's edge current is a pure gradient flow whose harmonic (circulating) part vanishes identically. The logical signal therefore cannot be read as a component of the current itself; it is instead a topological pairing between the syndrome and a boundary-fixed harmonic coordinate that distinguishes the two code boundaries linked by the logical operator. We prove that this pairing is evaluated exactly and in closed form, with no learned readout parameters, as the net current drained between the two boundary sinks. On the rotated surface code under circuit-level depolarising noise, this single closed-form scalar matches the best full-field readout and, at larger code distance, significantly exceeds the single-sink current pool, so that isolating the pairing helps more, not less, as the field grows larger and sparser. The decoder is not intended to surpass minimum-weight perfect matching, near-optimal for this noise model; its contribution is an interpretable characterisation of the logical signal itself.
Decoders for quantum error correction (QEC) experiments rely on detector error models (DEMs), which encode, for each error, its probability and the detectors and logical observables it flips. Here we show that estimating DEM event probabilities from experimental syndromes is feasible, avoids independent device benchmarking, and produces useful decoder priors for estimating and reducing decoded logical error probabilities. We evaluate our methods using open-source data from surface-code memory experiments performed on Google's Willow chip, and we carry out analogous surface-code experiments on IBM's \texttt{ibm\_miami} processor. Despite the different physical error scales of the Google and IBM devices, in both cases our estimated DEMs improve logical error probabilities relative to baseline device-informed DEMs, typically at the 5%−10% level and with larger gains in some IBM cases, without additional calibration circuits, decoder fine-tuning, or supervised fitting to logical outcomes.