Optimal Distributed Similarity Estimation of Quantum Channels

We study distributed similarity estimation of quantum channels (DSEC), a primitive for cross-platform verification where two remote quantum devices are compared by estimating the inner product of their Choi states. We show that the optimal channel query complexity of DSEC for two $d$-dimensional quantum channels is $\Theta(\max\{\sqrt{d}/\varepsilon, 1/\varepsilon^2\})$, where $\varepsilon$ is the additive error. We first prove an information-theoretic lower bound with this scaling, which holds even in the strongest setting, allowing adaptive strategies, multiple rounds of classical communication, and coherent access with arbitrary ancillas. We then give a matching upper bound in the weakest setting, namely non-adaptive and ancilla-free incoherent access, via a randomized measurement protocol achieving this bound. Finally, we show that our protocol achieves a quadratic improvement over classical shadow baselines. Our results provide theoretically optimal and practical methods for cross-platform verification, quantum device benchmarking, and distributed quantum learning.