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arXiv:2610.40052

Identifying Logical Operations from Syndrome Histories for qLDPC Codes

Yizhuo Tan, Shifan Xu, Yongshan Ding, Jakub Szefer

quant-phSubmitted Oct 29, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

Syndrome histories are normally used to detect and correct errors in fault-tolerant quantum computation, but they can also reveal information about the logical operations executed during computation. We study logical-operation identification using syndrome histories alone or together with other classical measurement outcomes. We develop an exact inference method and a scalable approximation that account for unknown circuit faults and ideal measurement randomness. We evaluate automorphisms, logical Pauli measurements, T injections, and logical circuits combining these instructions on \([[18,4,2]]\) and \([[72,12,6]]\) bivariate bicycle (BB) codes under multiple noise models and error rates. A single logical instruction can be identified with over 94\% accuracy under circuit-level noise with error rates between \(10^ {-5} \) and \(10^{-3}\). We also achieve over 99\% accuracy for automorphism-based logical circuits and 78.32\% for logical circuits combining all three instructions. We further analyze identification ambiguities arising from insufficient or indistinguishable measurement information, fault timing, and approximation inference. Together, these results extend the role of syndromes beyond error decoding to identifying executed logical operations and circuits during fault-tolerant quantum computation. They also suggest that access to syndrome histories and related classical measurement outcomes may unintentionally reveal information about the executed computation.

AI Summary

Can classical QEC records reveal which logical operations were executed, rather than just which errors occurred? This paper studies closed-candidate identification for bivariate bicycle codes, using error transport under automorphisms and operation-dependent correlations from gauging-based Pauli measurements and $T$ injection. Candidate likelihoods marginalize unknown faults and intrinsic measurement randomness. Exact inference uses hidden-state forward propagation or convolution of fault-induced parity violations; approximations truncate fault patterns and intermediate states. Simulations cover $[[18,4,2]]$ and $[[72,12,6]]$ codes, individual instructions, and short circuits under several stochastic noise models. On the 72-qubit code, ideal Bayes accuracy is 94.28% among 15 logical Pauli measurements and 97.85% among six injection axes. Under circuit noise, injection identification remains above 95% across the tested profiles. Pauli-measurement identification reaches 95.70% at nominal noise and 90.23% at high noise after inference refinement. BB stabilizer outcomes alone give 93.70% ideal Pauli-measurement accuracy. Automorphism circuits exceed 99% accuracy in tested low-noise settings when a known Pauli error is deliberately inserted as a marker; an eight-candidate circuit combining all three instruction types reaches 78.32% on the 18-qubit code under low noise with a marker. Limitations include overlapping candidate outcome distributions, dependence on syndrome-extraction cadence, and—for passive automorphism identification—the occurrence and timing of informative data faults. Approximation truncation can dominate failures. Inference assumes known candidates, compiled circuits, input state, and noise model, plus fault-free magic-state preparation. Real-time feasibility, hardware overhead, model mismatch, unknown-operation rejection, and security implications remain unestablished.

Little sign of AI-generated textIncrementalMissing comparisonStrong assumptions

Error found: The exact automorphism evaluation appears to mix strict unique identification with ordinary MAP accuracy: it reports zero success for settings with unresolved ties despite separately reporting nonzero Bayes accuracy for corresponding ideal settings, without explaining the metric or tie-handling change.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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Syndrome histories are normally used to detect and correct errors in fault-tolerant quantum computation, but they can also reveal information about the logical operations executed during computation. We study logical-operation identification using syndrome histories alone or together with other classical measurement outcomes. We develop an exact inference method and a scalable approximation that account for unknown circuit faults and ideal measurement randomness. We evaluate automorphisms, logical Pauli measurements, T injections, and logical circuits combining these instructions on ([[18,4,2]]) and ([[72,12,6]]) bivariate bicycle (BB) codes under multiple noise models and error rates. A single logical instruction can be identified with over 94% accuracy under circuit-level noise with error rates between (10^ {-5} ) and (10^{-3}). We also achieve over 99% accuracy for automorphism-based logical circuits and 78.32% for logical circuits combining all three instructions. We further analyze identification ambiguities arising from insufficient or indistinguishable measurement information, fault timing, and approximation inference. Together, these results extend the role of syndromes beyond error decoding to identifying executed logical operations and circuits during fault-tolerant quantum computation. They also suggest that access to syndrome histories and related classical measurement outcomes may unintentionally reveal information about the executed computation.
Can classical QEC records reveal which logical operations were executed, rather than just which errors occurred? This paper studies closed-candidate identification for bivariate bicycle codes, using error transport under automorphisms and operation-dependent correlations from gauging-based Pauli measurements and TTT injection.
Candidate likelihoods marginalize unknown faults and intrinsic measurement randomness. Exact inference uses hidden-state forward propagation or convolution of fault-induced parity violations; approximations truncate fault patterns and intermediate states. Simulations cover [[18,4,2]][[18,4,2]][[18,4,2]] and [[72,12,6]][[72,12,6]][[72,12,6]] codes, individual instructions, and short circuits under several stochastic noise models.
On the 72-qubit code, ideal Bayes accuracy is 94.28% among 15 logical Pauli measurements and 97.85% among six injection axes. Under circuit noise, injection identification remains above 95% across the tested profiles. Pauli-measurement identification reaches 95.70% at nominal noise and 90.23% at high noise after inference refinement. BB stabilizer outcomes alone give 93.70% ideal Pauli-measurement accuracy. Automorphism circuits exceed 99% accuracy in tested low-noise settings when a known Pauli error is deliberately inserted as a marker; an eight-candidate circuit combining all three instruction types reaches 78.32% on the 18-qubit code under low noise with a marker.
Limitations include overlapping candidate outcome distributions, dependence on syndrome-extraction cadence, and—for passive automorphism identification—the occurrence and timing of informative data faults. Approximation truncation can dominate failures. Inference assumes known candidates, compiled circuits, input state, and noise model, plus fault-free magic-state preparation. Real-time feasibility, hardware overhead, model mismatch, unknown-operation rejection, and security implications remain unestablished.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉ for distance-ddd codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed TTT gate on the [[16,4,4]][[16,4,4]][[16,4,4]] BB code, TTT-type rotations on the [[64,8,8]][[64,8,8]][[64,8,8]] BB code, and CCZ-, CS-, and TTT-type operations on the [[36,6,6]][[36,6,6]][[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields Cr{}^rrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.
Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a k/2k/2k/2-dimensional annihilator subspace and a k/2k/2k/2-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.
The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.
Characterizing how quantum error correction circuits behave under realistic hardware noise is essential for testing the premises that enable scalable fault tolerance. Logical error rates conditioned on syndrome outcomes are needed to enable noise-aware decoding and validate threshold-relevant assumptions. We introduce a protocol to directly estimate the logical Pauli channels (and pure errors) associated with detector regions formed of two or more syndrome extraction gadgets, conditioned on observing a particular parity in the syndrome outcomes. The method is SPAM-robust and most suitable for flag-based syndrome measurement schemes. For classical processing of the experimental data we implement a Bayesian modelling approach. We validate this new protocol on a small error-detecting code using Quantinuum H1-1, a trapped-ion device, and demonstrate that several noise diagnostic tests for fault tolerance improve significantly when using noise tailoring and mitigation strategies, such as swapped measurements for leakage protection, and Pauli frame randomization.
Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. Numerous protocols exist for benchmarking devices before running actual algorithms. In this work, we show that both physical and logical errors of fault-tolerant circuits can even be characterized in-situ using syndrome data. To achieve this, we map general fault-tolerant Clifford circuits to subsystem codes using the spacetime code formalism and develop a scheme for estimating Pauli noise in Clifford circuits using syndrome data. We give necessary and sufficient conditions for the learnability of physical and logical noise from given syndrome data, and show that we can accurately predict logical fidelities from the same data. Importantly, our approach requires only a polynomial sample size, even when the logical error rate is exponentially suppressed by the code distance, and thus gives an exponential advantage against methods that use only logical data such as direct fidelity estimation. We demonstrate the practical applicability of our methods in various scenarios using synthetic data as well as the experimental data from a recent demonstration of fault-tolerant circuits by Bluvstein et al. [Nature 626, 7997 (2024)]. Our methods provide an efficient, in-situ way of characterizing a fault-tolerant quantum computer to help gate calibration, improve decoding accuracy, and verify logical circuits.