Practical learning of multi-time statistics in open quantum systems

TL;DR

This paper extends classical shadow tomography to the temporal domain, enabling efficient learning of multi-time quantum processes and their features on noisy quantum hardware.

Contribution

It introduces a novel framework for learning multi-time quantum phenomena, including non-Markovianity and temporal entanglement, with a protocol for fast, reliable measurements.

Findings

Successfully reconstructed a 20-step process with high accuracy

Demonstrated the approach on a noisy quantum processor

Achieved a compact matrix product operator representation

Abstract

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.