Strict Hierarchy for Quantum Channel Certification to Unitary

2026-04-29

AI Summary

This work addresses the problem of testing whether an unknown $d$-dimensional quantum channel is equal to a target unitary channel or differs from it by at least $\varepsilon$ in diamond norm. The authors design optimal quantum algorithms for this unitarity certification task under three distinct access models—incoherent, coherent, and source-code—and rigorously establish a strict hierarchy in their query complexities. Specifically, they achieve query complexities of $\Theta(d/\varepsilon^2)$, $\Theta(d/\varepsilon)$, and $\Theta(\sqrt{d}/\varepsilon)$, respectively, each tightly matching the known lower bounds for the corresponding model. These results fully characterize how varying degrees of access to the quantum channel fundamentally influence the efficiency of certification.

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) $Θ(d/\varepsilon^2)$ queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023). (ii) $Θ(d/\varepsilon)$ queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008). (iii) $Θ(\sqrt{d}/\varepsilon)$ 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.

Problem

• quantum channel certification
• unitary channel
• diamond norm
• access models
• query complexity

Innovation

• quantum channel certification
• unitary testing
• query complexity
• access models
• diamond norm