Optimal Distributed Similarity Estimation for Unitary Channels
Congcong Zheng, Kun Wang, Xutao Yu, Ping Xu, Zaichen Zhang
TL;DR
This work scrutinizes cross-platform benchmarking of quantum devices by addressing distributed similarity estimation for unitary channels (DSEU). It proves that estimating $\mathrm{Tr}[U^\dagger V]^2/d^2$ with LOCC requires $\Theta(\sqrt{d})$ queries, and presents two randomized-measurement protocols (incoherent with SPAM shared randomness and coherent with state-prep shared randomness) achieving this bound. The authors also establish matching lower bounds under both access models, demonstrating that coherence does not reduce the fundamental cost, while shared randomness provides a quadratic advantage over independent classical shadow. A comparative analysis shows the proposed methods outperform independent classical shadow by a square-root factor in query complexity, underscoring the practical value for distributed quantum learning and device benchmarking. Overall, the paper delivers theoretically optimal tools for assessing similarity of unitary channels across quantum devices and clarifies the role of shared randomness in distributed quantum learning.
Abstract
We study distributed similarity estimation for unitary channels (DSEU), the task of estimating the similarity between unitary channels implemented on different quantum devices. We completely address DSEU by showing that, for $n$-qubit unitary channels, the query complexity of DSEU is $Θ(\sqrt{d})$, where $d=2^n$, for both incoherent and coherent accesses. First, we propose two estimation algorithms for DSEU with these accesses utilizing the randomized measurement toolbox. The query complexities of these algorithms are both $O(\sqrt{d})$. Although incoherent access is generally weaker than coherent access, our incoherent algorithm matches this complexity by leveraging additional shared randomness between devices, highlighting the power of shared randomness in distributed quantum learning. We further establish matching lower bounds, proving that $Θ(\sqrt{d})$ queries are both necessary and sufficient for DSEU. Finally, we compare our algorithms with independent classical shadow and show that ours have a square-root advantage. Our results provide practical and theoretically optimal tools for quantum devices benchmarking and for distributed quantum learning.