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 tr[O UρU†] for arbitrary input states ρ and observables O. Here we propose a parallel and non-adaptive CSEU protocol that uses 𝒪(min{dε−2,d3/2ε−1}) queries to U when the observables have constant effective size. This achieves Heisenberg scaling with respect to the additive error ε and is query-optimal, as we prove a matching lower bound Ω(min{dε−2,d3/2ε−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.
Related
- Optimal Diamond Distance Estimation for Unitary Channels
- Unitary equivalence of Schrödinger and Heisenberg pictures survives in nonlinear quantum mechanics
- Minimal unitary representations of time-invariant causal quantum channels
- Enabling chemically accurate quantum phase estimation in the early fault-tolerant regime via unitary weight concentration
- Generalized m-body mean-field projections for classical simulation of d-dimensional Heisenberg-picture dynamics
BibTeX
@misc{he2026optimal,
title = {{Optimal classical shadow estimation of unitary channels at Heisenberg limit}},
author = {Entong He and Zihao Li and Noam Scully and Sisi Zhou and Yuxiang Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.13638},
eprint = {2606.13638},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13638},
}
RIS
TY - UNPB
TI - Optimal classical shadow estimation of unitary channels at Heisenberg limit
AU - Entong He
AU - Zihao Li
AU - Noam Scully
AU - Sisi Zhou
AU - Yuxiang Yang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.13638
UR - https://arxiv.org/abs/2606.13638
AB - 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 tr[O UρU^†] for arbitrary input states ρ and observables O. Here we propose a parallel and non-adaptive CSEU protocol that uses 𝒪(min{dε^−2,d^3/2ε^−1}) queries to U when the observables have constant effective size. This achieves Heisenberg scaling with respect to the additive error ε and is query-optimal, as we prove a matching lower bound Ω(min{dε^−2,d^3/2ε^−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.
ER -