Strict Hierarchy for Quantum Channel Certification to Unitary

TL;DR

This paper establishes optimal quantum algorithms for certifying whether an unknown quantum channel is a specific unitary, revealing a hierarchy of complexities across different access models.

Contribution

It provides the first optimal algorithms with matching lower bounds for quantum channel certification to unitaries in three access models, demonstrating a strict complexity hierarchy.

Findings

Optimal query complexities are established for each access model.

The results match previously known lower bounds, confirming optimality.

A strict hierarchy of complexities across access models is demonstrated.

Abstract

We consider the problem of quantum channel certification to unitary, where one is given access to an unknown $d$-dimensional channel $\mathcal{E}$, and wants to test whether $\mathcal{E}$ is equal to a target unitary channel or is $\varepsilon$-far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that: (i) $\Theta(d/\varepsilon^2)$ queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023). (ii) $\Theta(d/\varepsilon)$ queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008). (iii) $\Theta(\sqrt{d}/\varepsilon)$ queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026). This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.