History-Conditioned Quantum Error Correction under Temporally Correlated Noise: Finite-Horizon Performance and Reproducible Simulations of a Three-Qubit Phase-Error Code
This preliminary research preprint presents a reproducible numerical study of history-conditioned quantum error correction under temporally correlated noise. We consider a three-qubit phase-error code subject to a synthetic Pauli noise model with a persistent hidden noise regime. A noisy classical monitor and the accumulated syndrome/measurement history are used to estimate that regime through standard Bayesian hidden-Markov filtering, and recovery operations are selected by a causal maximum-a-posteriori rule. The central question is whether accessible temporal information can extend the finite interval over which encoded quantum information remains reliable. In 100,000 matched synthetic trajectories, survival through 20 correction rounds increases from 0.74406 for a snapshot MAP decoder to 0.91455 for the history-conditioned decoder. Under a 90% no-logical-fault survival criterion, the empirical computational-memory window increases from 6 to 25 rounds, with a simultaneous 95% lower confidence band supporting 23 rounds for the history-conditioned policy. Control experiments show that the advantage disappears when temporal predictability is removed: when the hidden regime becomes independent from round to round, the history-conditioned and snapshot policies coincide. Additional monitor and model ablations demonstrate that the benefit depends on correction-relevant temporal correlations, informative side information, and adequate calibration. The work also formulates finite-horizon survival criteria and sufficient error-budget bounds without assuming independent logical faults, while explicitly separating classical filter stability from preservation of quantum information. This study does not claim a new Bayesian filtering method, superiority over existing temporal quantum-error-correction decoders, a scalable fault-tolerance threshold, or demonstrated room-temperature quantum-computing hardware. Rather, it provides a concrete operational specialization of the broader TrinityLabo history-first framework and a small, reproducible benchmark showing how temporal inference can condition quantum recovery under specified noise assumptions. Ambient-temperature operation remains a future experimental application requiring calibrated device-level noise models, fault-prone syndrome extraction, hardware latency analysis, and comparison against established temporal decoders.
This preliminary research preprint presents a reproducible numerical study of history-conditioned quantum error correction under temporally correlated noise. We consider a three-qubit phase-error code subject to a synthetic Pauli noise model with a persistent hidden noise regime. A noisy classical monitor and the accumulated syndrome/measurement history are used to estimate that regime through standard Bayesian hidden-Markov filtering, and recovery operations are selected by a causal maximum-a-posteriori rule. The central question is whether accessible temporal information can extend the finite interval over which encoded quantum information remains reliable. In 100,000 matched synthetic trajectories, survival through 20 correction rounds increases from 0.74406 for a snapshot MAP decoder to 0.91455 for the history-conditioned decoder. Under a 90% no-logical-fault survival criterion, the empirical computational-memory window increases from 6 to 25 rounds, with a simultaneous 95% lower confidence band supporting 23 rounds for the history-conditioned policy. Control experiments show that the advantage disappears when temporal predictability is removed: when the hidden regime becomes independent from round to round, the history-conditioned and snapshot policies coincide. Additional monitor and model ablations demonstrate that the benefit depends on correction-relevant temporal correlations, informative side information, and adequate calibration. The work also formulates finite-horizon survival criteria and sufficient error-budget bounds without assuming independent logical faults, while explicitly separating classical filter stability from preservation of quantum information. This study does not claim a new Bayesian filtering method, superiority over existing temporal quantum-error-correction decoders, a scalable fault-tolerance threshold, or demonstrated room-temperature quantum-computing hardware. Rather, it provides a concrete operational specialization of the broader TrinityLabo history-first framework and a small, reproducible benchmark showing how temporal inference can condition quantum recovery under specified noise assumptions. Ambient-temperature operation remains a future experimental application requiring calibrated device-level noise models, fault-prone syndrome extraction, hardware latency analysis, and comparison against established temporal decoders.
Abstract An intelligent quantum error-correction system should improve recovery using information available during computation, but syndrome observations do not necessarily identify the physical mechanism producing them. We analyze this tension through a precisely specified, reproducible model of a genuine [[7,1,3]] Steane CSS code subjected to hidden, temporally persistent, correlated Pauli noise. The simulation exhaustively enumerates all 47 = 16,384 physical Pauli patterns and the 64 syndrome classes. A causal hidden-Markov filter uses only the observed syndrome history to update a posterior over two correlated-error environments; a logical-class maximum-a-posteriori decision then conditions on this posterior. A learned logistic temporal model provides a separate data-driven comparator. On 220,000 locked, sequentially correlated instances, the stationary noise-aware decoder fails in 2.7923% of cases versus 2.5218% for the hidden-Markov decoder, an observed relative reduction of 9.69%. A moving-block bootstrap yields a paired absolute difference of 0.2705 percentage points (approximate 95% interval [0.2354,0.3064] points). A state-oracle decoder achieves 1.9809%; deliberate model misspecification increases the hidden-Markov failure rate to 2.8809%. We derive an information theoretic bound on the value of history, prove a decoder-regret bound controlled by posterior-state error, and give an explicit syndrome-aliasing construction on the same code: two physically different fault channels induce identical entire syndrome-history laws while requiring incompatible recoveries, forcing a strictly positive failure probability for every syndrome-only controller. These mathematical facts arise from established Bayesian decision and indistinguishability principles; we make no claim to a new fault-tolerance theorem. This is an original, small-code model study, not a circuit-level, real-time, scaling, or hardware result. It identifies both an opportunity and a non-removable information limit for intelligent QEC.
In realistic hardware for quantum computation that possesses fault-tolerance, non-stationary noise and stochastic drift lead to logical failure from the temporal accumulation of errors, not from independent events. Static decoding and fixed calibration techniques are structurally incompatible with this situation because they do not take into account temporal correlations between errors or control-induced back-action of errors. These effects motivate control policies that must track noise evolution across correction cycles, rather than respond to individual syndromes in isolation.
We treat fault-tolerant quantum computation as a stochastic control problem, modelled using reduced quantum dynamics in which Pauli error processes are governed by latent noise parameters that vary temporally. From this perspective, logical failure arises through the accumulation of a hazard variable, and the corresponding control objective depends on the full history of observations.
Operating under these conditions, a Chronological Deep Q-Network (Ch-DQN) maintains an internal belief state that tracks both noise evolution and accumulated hazard. During training, backward refinement of trajectories is used to sample slowly drifting modes of operation, while runtime inference remains strictly causal. A fractional meta-update stabilizes learning in the presence of non-stationary, control-coupled dynamics.
Through multi-distance simulations that incorporate stochastic drift and feedback from decision-making, Ch-DQN suppresses hazard accumulation and extends logical survival time relative to static and recurrent baselines. Error correction in this regime is therefore no longer a static decoding task, but a control process whose success is determined over time by the underlying noise dynamics.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent X rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching 1/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below 1/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's X distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.
The performance of a given quantum error correction (QEC) code depends upon the noise model that is assumed. Independent Pauli noise, applied after each quantum operation, is a simplistic noise model that is easy to simulate and understand in the context of stabilizer codes. Although such a noise model is artificial, it is equivalent to independent, random, unbiased qubit rotations. What about spatially or temporally correlated qubit rotations? Such a noise model is applicable to global operations (e.g., NMR or ESR), common control sources (e.g., lasers), or slow drift (e.g., charge or magnetic noise) in various qubit technologies. In the worst case, such errors can combine constructively and result in a post-correction failure rate that increases with the number of error correction cycles. However, we show that this worst case does not generally arise unless taking active corrective actions while performing QEC. That is, by employing virtual Pauli frame updates ("passive" error correction) rather than physical corrections ("active" error correction), coherent errors do not compound appreciably. Starting in a random Pauli frame is also advantageous. In fact, through perturbation theory arguments and supporting numerical simulations, we show that the logical qubit performance beyond distance 3 for correlated single-qubit Hamiltonian noise models (i.e., global errant qubit rotations), when employing these "lazy" strategies, essentially matches the performance of Pauli noise model with the same process fidelity (fidelity after one application). In a more general circuit model of noise, correlations may add constructively within syndrome extraction rounds but Pauli frame randomization from passive error correction mitigates this effect across multiple rounds.
Noise is the fundamental challenge to quantum computation. Large-scale quantum computers will overcome this challenge with fault-tolerant architectures that leverage quantum error correction. This thesis introduces and investigates a scalable noise characterisation protocol suited to this key context. First, we introduce this protocol by building upon averaged circuit eigenvalue sampling (ACES), a framework for noise characterisation experiments that simultaneously estimates the Pauli error probabilities of all gates in a Clifford circuit and captures averaged spatial correlations between gates implemented simultaneously in the layers of the circuit. We demonstrate the scalability and performance of our protocol through circuit-level numerical simulations of the entire noise characterisation procedure. Then we demonstrate in circuit-level numerical simulation that this protocol is practically capable of calibrating a fast correlated matching decoder, enabling noise-aware decoding. We find that noise-aware decoding increases the error suppression factor of the code, leading to reductions in the logical error rate that increase exponentially with the code distance. Finally, we present several results obtained from experimentally implementing this protocol. We use the noise characterisation results to design an improved syndrome extraction circuit for a heavy hexagon memory that is adapted to the noise characteristics of the quantum device. The model error of our noise characterisation results suggests that the circuit-level Pauli noise model estimated by ACES can describe the essential features of quantum noise. We also operate a heavy hexagon memory and use noise estimates to predict performance and inform decoding. These results demonstrate that Pauli noise estimates can calibrate decoders to enable noise-aware decoding and inform the co-design of quantum error correcting codes, decoders, fault-tolerant circuits, and quantum devices.
Quantum error-correcting decoders provide both corrections and confidence estimates for the logical-error risk of individual measurement records. Such confidence can guide postselection, but it depends on the accuracy of the decoder noise model. When the physical device drifts away from that model, confidence measures such as the logical gap can become miscalibrated. We show that this mismatch can itself provide useful information. On IBM quantum hardware, syndrome statistics drift across consecutive acquisition windows, and the resulting detector histories differ substantially from those predicted by the stationary detector error model used for decoding. We observe a similar hardware-model mismatch in independent Google surface-code memory data. We exploit this effect by assigning additional rejection weight to detector histories that are unlikely under the assumed model. For minimum-weight perfect matching (MWPM), the minimum weight W provides an inexpensive proxy for syndrome surprisal. On held-out IBM repetition-code memories, likelihood-aware postselection reduces the retained logical-error probability (LEP) by 10.6% at d=7 and 28.8% at d=9, both at 15% rejection. On Google ten-round d=5 XZZX surface-code memory data, it reduces retained LEP by 75.4% at 90% rejection. Applied to Google's magic-state cultivation experiment with the Tesseract decoder, it reduces tomography infidelity by 34.7% with 30% additional grafting-stage rejection, or increases accepted yield by 30.4% at a target fidelity of 97.0%. Burst-noise simulations show that the advantage persists as code distance and syndrome-extraction volume increase. The method reuses quantities already produced during logical-gap decoding and requires only an additional fitting step.