Optimal classical shadow estimation of unitary channels at Heisenberg limit

Abstract:Full tomography of an unknown quantum evolution is resource-intensive and often unnecessary when the goal is only to predict selected properties. This motivates the study of classical shadow estimation of unitary channels (CSEU), a task in which one queries an unknown $d$-dimensional unitary $U$ and stores classical data that can later be used to predict expectation values $\mathrm{tr}[O U\rho U^\dagger]$ for arbitrary input states $\rho$ and observables $O$. Here we propose a parallel and non-adaptive CSEU protocol that uses $\mathcal{O}(\min\{d\varepsilon^{-2},d^{3/2}\varepsilon^{-1}\})$ queries to $U$ when the observables have constant effective size. This achieves Heisenberg scaling with respect to the additive error $\varepsilon$ and is query-optimal, as we prove a matching lower bound $\Omega(\min\{d\varepsilon^{-2},d^{3/2}\varepsilon^{-1}\})$ for the query complexity, which remains valid even when stronger access to the unknown unitary is available. Our CSEU protocol is a versatile primitive with applications to several quantum learning tasks. In particular, it yields a near-optimal algorithm for learning an arbitrary Hamiltonian from its real-time evolution in the high-precision regime and recovers the best-known bounds for several tasks in a unified way.

arXiv:2606.13638 [quant-ph]

(or arXiv:2606.13638v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.13638 Focus to learn more arXiv-issued DOI via DataCite

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