On the query complexity of unitary channel certification

TL;DR

This paper establishes tight bounds on the number of queries needed to certify quantum unitary channels, revealing that certification is generally easier for most channels than in worst-case scenarios, with implications for quantum verification.

Contribution

The work provides tight bounds on query complexity for unitary channel certification, including worst-case and average-case analyses, and introduces a quantum algorithm based on singular value transformation.

Findings

Incoherent algorithms require (d/^2) queries, matching upper bounds.

Quantum algorithms can certify with (^{1/2}/) queries, matching lower bounds.

Average-case certification requires only O(1/^2) queries, showing exponential gap with worst-case.

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 $\varepsilon$-far in diamond distance from a target unitary operation. We show that incoherent algorithms-those without quantum memory-require $\Omega(d/\varepsilon^2)$ queries, matching the known upper bound. In addition, for general quantum algorithms, we prove a lower bound of $\Omega(\sqrt{d}/\varepsilon)$ and present a matching quantum algorithm based on quantum singular value transformation, establishing a tight query complexity of $\Theta(\sqrt{d}/\varepsilon)$. 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 $O(1/\varepsilon^2)$ 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.