Coherent errors accumulate linearly rather than diffusively, making them invisible to Pauli-twirl-based char- acterization. It poses a central challenge for quantum error correction, yet its characterization typically requires multi-qubit process tomography whose cost grows exponentially with system size. We show that measurement- free error correction projects arbitrary few-qubit coherent noise onto a fixed single-logical-qubit effective chan- nel, enabling tomography at constant cost. Exploiting the polarization effects of bit-flip and phase-flip codes, we construct a concatenated effective-channel fingerprint and prove, by a parity-based contradiction argument, that the observation map is globally injective over a prescribed coherent-noise family; hence the underlying noise parameters are uniquely recovered without any sparsity or low-rank assumptions. The recovered coherent noise model can be fed into decoding algorithms; relative to independent noise priors, we expect this to reduce the logical error rate for surface-code decoders. The protocol offers a scalable route to coherent-noise identification whenever the noise admits a moderate-dimensional parameterization.
Coherent errors accumulate linearly rather than diffusively, making them invisible to Pauli-twirl-based char- acterization. It poses a central challenge for quantum error correction, yet its characterization typically requires multi-qubit process tomography whose cost grows exponentially with system size. We show that measurement- free error correction projects arbitrary few-qubit coherent noise onto a fixed single-logical-qubit effective chan- nel, enabling tomography at constant cost. Exploiting the polarization effects of bit-flip and phase-flip codes, we construct a concatenated effective-channel fingerprint and prove, by a parity-based contradiction argument, that the observation map is globally injective over a prescribed coherent-noise family; hence the underlying noise parameters are uniquely recovered without any sparsity or low-rank assumptions. The recovered coherent noise model can be fed into decoding algorithms; relative to independent noise priors, we expect this to reduce the logical error rate for surface-code decoders. The protocol offers a scalable route to coherent-noise identification whenever the noise admits a moderate-dimensional parameterization.
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.
Scalable characterization of quantum processors is crucial for mitigating noise and imperfections. While randomized measurement protocols enable efficient access to local observables, inferring a globally consistent description of multi-qubit processes remains challenging. Here we introduce a local-to-global reconstruction framework for one-dimensional multi-qubit states and channels. The method is efficient provided that correlations, as quantified by the conditional mutual information, decay exponentially. In particular, we prove that under this assumption, the required number of samples scales polynomially with the system size and the desired global reconstruction error. Our approach is based on combining local shadow tomography with locally optimal recovery maps obtained by convex optimization. We supplement these rigorous guarantees by studying the performance of the protocol numerically for a system evolving under a local Lindbladian and a noisy, shallow circuit. By employing a tensor networ representation, we reconstruct channels acting on up to 50 qubits and accurately recover global diagnostics such as the process fidelity, the Choi state purity, and Pauli-weight-resolved process matrix elements. Our work thus extends the powerful toolbox local shadow tomography to scalable channel characterization with access to global properties.
Scalable characterization of quantum processors is crucial for mitigating noise and imperfections. While randomized measurement protocols enable efficient access to local observables, inferring a globally consistent description of multi-qubit processes remains challenging. Here we introduce a local-to-global reconstruction framework for one-dimensional multi-qubit states and channels. The method is efficient provided that correlations, as quantified by the conditional mutual information, decay exponentially. In particular, we prove that under this assumption, the required number of samples scales polynomially with the system size and the inverse of the desired global reconstruction error, which we quantify by the trace-norm distance between the true and the reconstructed Choi states. Our approach is based on combining local shadow tomography with locally optimal recovery maps obtained by convex optimization. We supplement these rigorous guarantees by studying the performance of the protocol numerically for a system evolving under a local Lindbladian and a noisy, shallow circuit. By employing a tensor network representation, we reconstruct channels acting on up to 50 qubits and accurately recover global diagnostics such as the process fidelity, the Choi state purity, and Pauli-weight-resolved process matrix elements. Our work thus extends the powerful toolbox of local shadow tomography to scalable channel characterization with access to global properties.
Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a structure-agnostic learning framework for noisy n-qubit quantum circuits under~i.i.d.~single-qubit noise. We first prove that quantum states with unital noise channels admit an efficient learnable representation in the logarithmic-depth regime. We then extend this framework to quantum process tomography under constant noise, deriving a unified protocol that applies to both unital and non-unital noisy channels and retains efficient guarantees for logarithmic-depth circuits. This process-learning formulation is input-agnostic and imposes no distributional assumptions on the input quantum states. We further study a more general regime with arbitrary noise strength. In this setting, low-weight Pauli propagation induces a terminal truncation whose threshold depends logarithmically on the inverse accuracy, leading to quasi-polynomial complexity and near-unit success probability in the average case. In contrast to the preceding two results, this arbitrary-noise guarantee does not impose any restriction on the circuit depth, and therefore covers arbitrary-depth circuits, including both the noiseless limit (γ=0) and the strong-decoherence regime (γ=Θ(1)). Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.
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.
Coherent Hamiltonian drifts and systematic errors can pass through quantum error correction differently than stochastic Pauli noise: they generate phase components of the logical channel that are lost under full Pauli twirling. We consider the Lindbladian dynamics of n physical qubits propagated through an effective logical channel of a fixed encoding, recovery, and decoding scheme. The work is methodological in character: standard ideas of observability and minimal realization from linear systems theory are applied to the channel \[
ω
\mapsto
Φ_t(ω)
=
\cD e^{t\cL}\cE(ω), \] where \(\cE\) encodes the logical state and \(\cD\) is the chosen recovery map with final decoding. The minimal code-visible space \(\cK_{\rm code}\) is defined as the Krylov closure of the decodable logical observables with respect to \(\cL^\dagger\); its observable factorization yields a minimal realization of the logical channel. In this form, the Knill--Laflamme conditions serve as a short-time consistency check, Pauli noise on stabilizer codes reduces to a chain on syndrome-logical classes, and the choice of recovery map acts as a spectral filter on the visible modes. For quantum memory, the same language yields an estimate of the storage time in terms of the dominant visible eigenvalues of the logical channel of the correction cycle. Particular attention is paid to the spectral signatures of coherent noise and to how these signatures depend on the chosen decoder.