Efficient learning of logical noise from syndrome data

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.

💡 Research Summary

Characterizing errors in fault‑tolerant quantum processors is a pressing challenge because logical error rates are dramatically suppressed relative to the underlying physical noise, making direct logical benchmarking prohibitively sample‑intensive. Building on the insight of Wagner et al. that, for phenomenological Pauli noise, the logical channel can be inferred from syndrome measurements taken during error correction, this work extends the idea to realistic circuit‑level noise models that include gate, measurement, and idle errors.

The authors first formalize a unified code‑theoretic framework using the spacetime Pauli group, which they treat as a Boolean group A equipped with a symplectic inner product. Within this algebra they identify a gauge subgroup G (encoding the code’s symmetry) and a measurement subgroup M (the stabilizers actually measured). Logical operators are defined up to multiplication by elements of G, leading to an equivalence relation a ∼_G b. By averaging the physical Pauli fault distribution Λ over G they define an “effective distribution” Λ_eff that contains precisely the information that can influence logical outcomes.

Two central learnability questions are addressed: (1) which Pauli eigenvalues λ_a (the Fourier coefficients of Λ) are recoverable from syndrome data alone, and (2) which of them are recoverable up to logical equivalence. The authors prove that a necessary and sufficient condition for full recoverability is that the measurement subgroup M generates the full character space of A; equivalently, the set of measured stabilizers must span the entire symplectic dual of the gauge group. When this condition fails, only the components of Λ that lie in the span of M are identifiable, while the remaining degrees of freedom are fundamentally unlearnable from syndrome data. Up to logical equivalence, the learnable parameters are precisely those that survive the quotient by G; any two Pauli errors that differ by a gauge element produce identical syndrome statistics and therefore cannot be distinguished.

To turn these information‑theoretic statements into a practical algorithm, the paper adopts a Fourier‑based parametrization of any Pauli channel:
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