On the query complexity of unitary channel certification

Abstract

Certifying the correct functioning of a unitary channel is a critical step toward reliable quantum information processing. In this work, we investigate the query complexity of the unitary channel certification task: testing whether a given d-dimensional unitary channel is identical to or ε-far in diamond distance from a target unitary operation. We show that incoherent algorithms—those without quantum memory—require Ω(d/ε2) queries, matching the known upper bound. In addition, for general quantum algorithms, we prove a lower bound of Ω(d/ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\sqrt{d}/\varepsilon )$$\end{document} and present a matching quantum algorithm based on quantum singular value transformation, establishing a tight query complexity of Θ(d/ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (\sqrt{d}/\varepsilon )$$\end{document}. On the other hand, notably, we prove that for almost all unitary channels drawn from a natural average-case ensemble, certification can be accomplished with only O(1/ε2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{O}}(1/{\varepsilon }^{2})$$\end{document} queries. This demonstrates an exponential query complexity gap between worst- and average-case scenarios in certification, implying that certification is significantly easier for most unitary channels encountered in practice. Together, our results offer both theoretical insights and practical tools for verifying quantum processes.