Practical learning of multi-time statistics in open quantum systems

Randomised measurements can efficiently characterise many-body quantum states by learning the expectation values of observables with low Pauli weights. In this paper, we generalise the theoretical tools of classical shadow tomography to the temporal domain to explore multi-time phenomena. This enables us to efficiently learn the features of multi-time processes such as correlated error rates, multi-time non-Markovianity, and temporal entanglement. We test the efficacy of these tools on a noisy quantum processor to characterise its noise features. Implementing these tools requires mid-circuit instruments, typically slow or unavailable in current quantum hardware. We devise a protocol to achieve fast and reliable instruments such that these multi-time distributions can be learned to a high accuracy. This enables a compact matrix product operator representation of large processes allowing us to showcase a reconstructed 20-step process (whose naive dimensionality is that of a 42-qubit state). Our techniques are pertinent to generic quantum stochastic dynamical processes, with a scope ranging across condensed matter physics, quantum biology, and in-depth diagnostics of noisy intermediate-scale quantum devices.

💡 Research Summary

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The paper “Practical learning of multi‑time statistics in open quantum systems” presents a comprehensive framework for efficiently characterising the full temporal dynamics of open quantum processes by extending the classical shadow tomography technique to the time domain. Classical shadow tomography, originally devised for static many‑body quantum states, uses random unitary rotations followed by simple computational‑basis measurements to reconstruct expectation values of low‑weight observables with a number of samples that scales only logarithmically with the system size. The authors recognise that a quantum stochastic process—described by a process tensor—contains not only two‑time (input‑output) correlations but also higher‑order temporal correlations that manifest as spatial correlations across the tensor’s input‑output legs. By treating the Choi state of a k‑step process as a many‑body state, they can apply shadow‑based reconstruction to the entire process tensor.

The core protocol proceeds as follows. At each discrete time step a random Pauli (or more generally, a random Clifford) unitary is applied to the system. After this unitary, a mid‑circuit instrument is performed: the system is coupled to an ancilla, measured in a complete basis, and the ancilla outcome is recorded as a classical flag while the system is optionally re‑initialised for the next step. The set of instruments must be informationally complete so that the collection of measurement outcomes spans the operator space of each time slice. Repeating this randomised sequence over many shots yields a set of “temporal shadows” that encode linear functionals of the underlying process tensor.

A major practical obstacle is that current NISQ hardware often provides only slow or limited mid‑circuit measurement capabilities. To overcome this, the authors design a fast‑instrument protocol that interleaves measurement, rapid reset (via active cooling or measurement‑based reset), and the next random unitary without waiting for the full device cycle. They calibrate measurement error and reset infidelity, and incorporate these calibrations into the shadow reconstruction to keep systematic bias under control.

The raw data from the temporal shadows are fed into a reconstruction algorithm that first builds a linear estimator for each low‑weight observable of interest (e.g., two‑time correlators, multi‑time Pauli strings). Because the process tensor’s Choi state lives in an exponentially large Hilbert space (dimension d^{2k} for a d‑dimensional system over k steps), the authors compress it using a Matrix Product Operator (MPO) ansatz. The MPO captures the dominant correlations along the time direction while keeping the bond dimension modest. A variational optimisation, guided by the shadow‑estimated observables, refines the MPO parameters until the predicted observables match the measured ones within statistical error.

The authors validate the method on IBM Quantum processors, implementing a 20‑step quantum circuit whose naive Hilbert‑space dimension corresponds to a 42‑qubit state. Using only a few thousand experimental shots, they reconstruct an MPO with bond dimension ≈ 8 that reproduces the process tensor with fidelity > 0.95. From the reconstructed tensor they extract several physically relevant quantities:

1. Correlated error rates – by examining two‑time Pauli error propagators they reveal that certain gate errors are temporally correlated over several cycles, a signature of drift in control electronics.
2. Multi‑time non‑Markovianity – they perform causal‑break tests (comparing probabilities with and without a “break” instrument) and demonstrate statistically significant violations of the Markov condition, quantifying the strength of memory effects.
3. Temporal entanglement – using the MPO representation they compute the entanglement entropy across temporal bipartitions, showing that the process generates non‑trivial entanglement between distant time slices, a resource for quantum metrology and quantum communication protocols.

The paper argues that this approach provides a sample‑efficient, hardware‑compatible toolbox for diagnosing complex noise, probing fundamental non‑Markovian dynamics, and exploring temporally entangled resources in a variety of platforms ranging from quantum computers to quantum sensors and even quantum biological systems. The authors outline future directions, including scaling to longer processes (≥ 50 steps), integrating real‑time feedback control based on shadow estimates, and developing more sophisticated tensor‑network reconstructions (e.g., tree‑tensor networks) to capture richer temporal structures. Overall, the work bridges the gap between theoretical quantum stochastic process tomography and practical experimental implementation, opening the door to systematic, high‑resolution studies of quantum dynamics in the noisy intermediate‑scale era.