A controller-integrated realization is reported for an exact closed-formula maximum-likelihood decoder acting on the released distance-three, logical-X, ten-round Willow surface-code memory-readout circuit. A newly frozen same-shot benchmark of 40,000 Stim-generated shots (seed 20260929) gives 197 logical failures for exact arithmetic, 218 for belief propagation plus ordered-statistics decoding, 245 for correlated matching, and 290 for plain minimum-weight perfect matching. Quantum Transfer Level (QTL) is used as a proprietary black-box framework to integrate the decoder while preserving the quantum schedule. The architecture contains the unchanged quantum memory circuit, a 91-exclusive-OR detector/relation shell, and a nonlinear decision cell. The shell carries an eight-bit measurement record through the repeated-round region, four bits near terminal readout, and zero after terminal consumption; these widths do not bound decoder state. Historical bounded-arithmetic measurements from the original realization run report a 9-bit minifloat decision cell with an eight-entry, 136-bit carried record, 25,124 coefficients before sharing, 7,646 after seven bulk banks were verified identical at the time, a maximum workload of 373 multiply-accumulate operations in one cycle, and doubled-noise counts of 313, 292, and 286 failures for retained-record capacities 8, 16, and 32. The final minifloat executable and coefficient-bank data were not preserved, so these values are not independently reproduced by the current public release. Integration adds no quantum operation or SI1000 noise site. The 9,089 physical single-fault alternatives induce 1,005 distinct detector-error-model mechanisms; exact arithmetic decodes all mechanisms correctly, while the original bounded-arithmetic run reported the same zero-failure result for minifloat arithmetic. The result is zero-quantum-overhead integration of a bounded classical decoder, not elimination of the decoder itself.
A controller-integrated realization is reported for an exact closed-formula maximum-likelihood decoder acting on the released distance-three, logical-X, ten-round Willow surface-code memory-readout circuit. A newly frozen same-shot benchmark of 40,000 Stim-generated shots (seed 20260929) gives 197 logical failures for exact arithmetic, 218 for belief propagation plus ordered-statistics decoding, 245 for correlated matching, and 290 for plain minimum-weight perfect matching. Quantum Transfer Level (QTL) is used as a proprietary black-box framework to integrate the decoder while preserving the quantum schedule. The architecture contains the unchanged quantum memory circuit, a 91-exclusive-OR detector/relation shell, and a nonlinear decision cell. The shell carries an eight-bit measurement record through the repeated-round region, four bits near terminal readout, and zero after terminal consumption; these widths do not bound decoder state. Historical bounded-arithmetic measurements from the original realization run report a 9-bit minifloat decision cell with an eight-entry, 136-bit carried record, 25,124 coefficients before sharing, 7,646 after seven bulk banks were verified identical at the time, a maximum workload of 373 multiply-accumulate operations in one cycle, and doubled-noise counts of 313, 292, and 286 failures for retained-record capacities 8, 16, and 32. The final minifloat executable and coefficient-bank data were not preserved, so these values are not independently reproduced by the current public release. Integration adds no quantum operation or SI1000 noise site. The 9,089 physical single-fault alternatives induce 1,005 distinct detector-error-model mechanisms; exact arithmetic decodes all mechanisms correctly, while the original bounded-arithmetic run reported the same zero-failure result for minifloat arithmetic. The result is zero-quantum-overhead integration of a bounded classical decoder, not elimination of the decoder itself.
Quantum Transfer Level (QTL) is applied as a contract-first design and verification framework to the Willow surface-code memory. This manuscript reports a milestone rather than a completed physical optimization and makes a central distinction between hardware width and future-distinguishable boundary-record width. Under the frozen logical-X and logical-Z eigenstate functional contract, initialization and cycle-boundary transformations reduce exactly, ideal data-qubit leakage removal closes side-locally, repeated ideal syndrome cycles collapse to one quantum extraction followed by classical repetitions, and the complete future response is a sparse affine transition family. After the first ideal extraction, no branch-dependent coherent record is required; the full ideal benchmark is generated by parity and exclusive OR (XOR) logic with 48 fresh unbiased bits. Separately, a fixed-basis stabilizer theorem gives a raw distance-seven Pauli-frame boundary record of 48 syndrome coordinates plus one logical-flip coordinate, with a one-bit conditioned residual only when the exact syndrome is already past context. A structural stochastic-fault extension yields conditioned weighted-history partitions, a binary group algebra, and Walsh-character attenuation. In the canonical code-capacity data-Pauli model, the logical-difference distance equals the code distance and the minimum branch-competition barrier is (d + 1)/2; this statement is not promoted to circuit-level superconducting-inspired 1000-ns (SI1000) noise, where hook and correlated mechanisms can reduce effective fault distance. A published distance-three/distance-five constant-Λ extrapolation agrees with the published distance-seven mean at the percent level, as expected from standard below-threshold scaling, and is reported only as a consistency check. At the milestone boundary, exactclosure-complete SI1000 cut certification and the remaining physical QTL optimization remained open.
Quantum Transfer Level (QTL) optimization is continued from a certified class-level handoff into released Google Willow superconducting-inspired 1000-ns-round (SI1000) surface-code circuits. The post-handoff objective is an instance-level relation reduction followed by cross-distance and cross-basis testing. The initial target is distance-3 logical-X r10. At each physical cut, emitted detector rows A and a minimal future basis C are retained jointly. Because rowspan[A; C] = rowspan[A; Bd; Bo],equality of [A; C] signatures is exactly equivalent to equality of full detector-plus-logical signatures; the equivalence classes therefore coincide with those of a circuit-derived detector error model. A legal pair with probabilities 0.002 and 0.005 is fused to 0.00698, and direct detector-record incidence independently confirms the shared physical signature. Across all 54 inter-cycle cuts at distances 3, 5, and 7 in both logical bases, the raw conditional quotient is d^2 = (d^2 − 1) + 1, so the one-bit SI1000promotion criterion of the companion analysis fails under the released source-level contract. Under a declared source-channel-preserving factorization, exact binary crossing reduction ranges from 78.24% to 82.81% over all cuts (79.49%–82.81% for repeated-cycle cuts 2–9). Factorization-independent distinct nonzero full detector-plus-logical signatures are also reported. An exact Pauli-refactorization audit confirms that depolarizing channels admit independent binary decompositions, while the source-channel-preserving form is retained as the input to subsequent native lowering. No reduction of hardware qubits, native gates, cycle time, leakage resources, or measured logical-error rate is claimed.
Future-Observable Error Correction (FOEC) treats error correction as preservation of the distinctions about the past that a declared future can still observe. A distance-three Willow surface-code extraction circuit is extended here by a constructed single-logical-qubit Clifford contract. The released extraction body is retained, detector information is internalized as local relations, and probability over the logical Pauli frame is represented by three nontrivial logical-Walsh character fields rather than by a per-round physical correction. Twenty certified bounded parents attach the relation boundary to those fields. All 24 projective single-qubit Clifford elements reduce to six exact frame actions and act only by permutation of the three fields, with zero mismatch over 288 four-sector coordinate checks. The identity continuation reproduces a frozen 40,000-record exact closed-formula benchmark with zero decision mismatches and the same 197 logical failures. Scored point-of-use validation uses only truth-bearing logical-X and logical-Z memory records; therefore those scored tests consume one frame bit at a time and do not directly score the joint x–z character. Across twelve frame-action/truth-basis classes and 49,152 representative records, the FOEC lane path and an independently reconstructed two-sector maximum-likelihood reference give zero decision mismatches and identical failure sets. Exact character algebra gives K = G0 − G1, so the exact logical-Walsh and two-sector functionals coincide for every detector history in the frozen models; numerical residuals of 1.94 × 10^−14 and 1.12 × 10^−14 audit the independent implementations rather than supply the proof. Negative controls detect both corrupted relation wiring and omitted Clifford-frame propagation. The result establishes a complete error-correction path for the constructed frame-level continuing-Clifford contract, not a source-authentic noisy logical-gate experiment or a scored joint-frame readout.
Quantum error-correction decoders are typically benchmarked against synthetic circuit-level noise, under the assumption that a decoder's ranking under such noise transfers to hardware and improves as the noise model becomes more realistic. The Willow processor, the first to operate below the surface-code threshold, allows us to test this assumption. We rank a panel of six decoders using a four-rung ladder of noise models with increasing fidelity, evaluated against real data across three code distances, two bases, and fifteen round counts. Rank agreement with hardware appears once the noise model gives each operation type its own error rate. Calibrating the model to the device improves absolute error rates but not rank agreement. We additionally provide the first independent evaluation of NVIDIA's Ising pre-decoder on hardware, at code distances below its training receptive field and via a mapping onto the lattice on which it was trained. Under these conditions, it holds no accuracy-latency advantage: another panel decoder matches or improves on it in both per-cycle error rate and decode latency in 278 of the 280 evaluations. We release the full pipeline and the per-shot outcome of every evaluation, so future decoders and devices can be compared.
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.
Future-observable equivalence is tested across released Willow surface-code memories at distances d = 3, 5, 7. The principle states that a computation need retain only distinctions that its declared future can still observe. Three consequences are examined. First, exhaustive propagation of every physical single-fault alternative in the released superconducting-inspired SI1000 circuit-noise models finds no detector history with opposite terminal logical meanings and no zero-syndrome single faultthat flips the answer, across 9,089, 25,840, and 53,038 alternatives at distances 3, 5, and 7. Second, an exhaustive released-circuit data-error slice probes the first weight beyond the guaranteed correction radius. All protected layers decode without failure; at weights 2, 3, and 4, exact logical-class weighting fails on 18/36, 292/2300, and 4606/211876 subsets, respectively. Plain minimum-weight matching fails on exactly the same subsets, each paired with an opposite-logical lower-weight history on a minimum-weight logical operator. Third, the optimized distance-three decision architecture is transferred directly to distance five and seven without rediscovering intermediate synthesis stages. The same bounded local stochastic primitive remains sufficient: runtime parent counts are 20, 72, and 156, while maximum local parent width remains six and maximum atom count remains 16. The transferred sparse key/value record has structural footprint B(d, K) = K[(d^2 − 1) + 9] = K(d^2 + 8) bits for retained-record capacity K, rather than scaling with the 2^(d^2−1) raw relation-key space. For the inherited nominal K = 8 instance, the corresponding footprints are 136, 264, and 456 bits at d = 3, 5, 7; whether eight entries retain full circuit-noise accuracy at d = 5 or d = 7 remains an open benchmark. No global state vector or dense exponential decision table is constructed