Exact identification of unknown unitary processes

TL;DR

This paper develops an optimal, zero-error quantum protocol for identifying faulty devices applying different unknown unitaries, independent of total device count, with practical advantages like independent testing.

Contribution

It introduces a representation-theoretic model and a simple ancillary-based protocol that achieves optimal fault identification in quantum devices, extending to arbitrary anomalies and dimensions.

Findings

Derived the optimal success probability for single- and two-anomaly cases

Presented a protocol using ancillary systems that achieves optimal success probability

Extended analysis to arbitrary number of anomalies and system dimensions

Abstract

The accurate identification of faulty hardware is a fundamental requirement for reliable quantum information processing. We address this problem in a quantum setting, where a series of $n$ devices is intended to apply the same unitary operation, but $k$ malfunctioning devices among them apply a different, unknown unitary action. Under the assumption of complete ignorance regarding the specific unitary transformation applied, we model our hypotheses using representation-theoretic tools and study the zero-error protocol for identifying these faulty devices. We derive the optimal success probability for the single- and two-anomaly scenarios, demonstrating that it is independent of the total number of devices in the series. Furthermore, we present a simple protocol that makes use of ancillary systems that achieves this optimal limit. Notably, this protocol offers significant operational advantages, such as allowing us to test each device independently. Finally, we extend our analysis to the general scenario in which both the number of anomalies and the local dimension of the systems are arbitrary, evaluating our protocol's performance and conjecturing its global optimality in the general case.