Quantum Advantage in Learning Mixed Unitary Channels

TL;DR

This paper investigates the quantum resource requirements for learning mixed unitary channels, revealing that the sample complexity depends on the channel's rank, system dimension, and ancilla, with practical implications for efficiently learning such channels.

Contribution

It provides a precise analysis of the sample complexity for learning mixed unitary channels, emphasizing the importance of ancilla and channel rank, and demonstrates the ease of learning random mixed unitary channels.

Findings

Sample complexity scales as r/(d*ε^2)

Ancilla is a critical resource for learning

Random mixed unitary channels are easy to learn

Abstract

We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as $\frac{r}{d\varepsilon^2}$, where $r$ is the rank of the channel (i.e.\ the number of different unitaries), $d$ is the dimension of the system, and $\varepsilon^2$ is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that $r$ is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.