Query Complexity of Classical and Quantum Channel Discrimination
Theshani Nuradha, Mark M. Wilde
TL;DR
The paper develops a nonasymptotic notion of query complexity for quantum channel discrimination, defining the minimum number of channel uses required to achieve a target error probability across symmetric binary, asymmetric binary, and M-ary settings. It derives tight, information-theoretic bounds that express how the query complexity scales with the error threshold and key channel fidelities and divergences, including $F_H$, $\widehat{F}$, $d_{\widehat{F}}$, and the Cs divergences. The authors obtain precise results for discriminating two classical channels and two classical--quantum channels, and provide improved sample-complexity bounds that feed into sharper query-complexity characterizations, with alternative formulations via $Q_s$ and amortized divergences. The findings offer nonasymptotic guidance for designing channel-discrimination protocols, with implications for quantum learning, simulation, and algorithmic tasks where identifying unknown processes efficiently is essential. Overall, the work connects nonasymptotic hypothesis testing, channel divergences, and practical discrimination strategies to yield practically relevant bounds and tight results for fundamental quantum-information-processing tasks.
Abstract
Quantum channel discrimination has been studied from an information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of unknown channel accesses. In this paper, we study the query complexity of quantum channel discrimination, wherein the goal is to determine the minimum number of channel uses needed to reach a desired error probability. To this end, we show that the query complexity of binary channel discrimination depends logarithmically on the inverse error probability and inversely on the negative logarithm of the (geometric and Holevo) channel fidelity. As a special case of these findings, we precisely characterize the query complexity of discriminating two classical channels and two classical-quantum channels. Furthermore, by obtaining an optimal characterization of the sample complexity of quantum hypothesis testing, including prior probabilities, we provide a more precise characterization of query complexity when the error probability does not exceed a fixed threshold. We also provide lower and upper bounds on the query complexity of binary asymmetric channel discrimination and multiple quantum channel discrimination. For the former, the query complexity depends on the geometric Rényi and Petz Rényi channel divergences, while for the latter, it depends on the negative logarithm of the (geometric and Uhlmann) channel fidelity. For multiple channel discrimination, the upper bound scales as the logarithm of the number of channels.