Online learning of quantum processes

TL;DR

This paper explores online learning methods for quantum processes, demonstrating feasibility for certain classes of channels, providing shadow tomography techniques, and establishing theoretical bounds for learning quantum channels and non-Markovian processes.

Contribution

It introduces the first online learning algorithms for specific quantum channels and non-Markovian processes, with bounds and shadow tomography methods, advancing quantum process learning.

Findings

Online learning feasible for bounded gate complexity channels and Pauli channels.

Provided sample-efficient shadow tomography for Pauli channels.

Extended online learning techniques to non-Markovian multi-time quantum processes.

Abstract

Among recent insights into learning quantum states, online learning and shadow tomography procedures are notable for their ability to accurately predict expectation values even of adaptively chosen observables. In contrast to the state case, quantum process learning tasks with a similarly adaptive nature have received little attention. In this work, we investigate online learning tasks for quantum processes. Whereas online learning is infeasible for general quantum channels, we show that channels of bounded gate complexity as well as Pauli channels can be online learned in the regret and mistake-bounded models of online learning. In fact, we can online learn probabilistic mixtures of any exponentially large set of known channels. We also provide a provably sample-efficient shadow tomography procedure for Pauli channels. Our results extend beyond quantum channels to non-Markovian multi-time processes, with favorable regret and mistake bounds, as well as a shadow tomography procedure. We complement our online learning upper bounds with mistake as well as computational lower bounds. On the technical side, we make use of the multiplicative weights update algorithm, classical adaptive data analysis, and Bell sampling, as well as tools from the theory of quantum combs for multi-time quantum processes. Our work initiates a study of online learning for classes of quantum channels and, more generally, non-Markovian quantum processes. Given the importance of online learning for state shadow tomography, this may serve as a step towards quantum channel variants of adaptive shadow tomography.