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  • Open Access

Nearly Query-Optimal Classical Shadow Estimation of Unitary Channels

Zihao Li1,2,3,*, Changhao Yi1,2,3, You Zhou4,5,†, and Huangjun Zhu1,2,3,5,‡

  • 1State Key Laboratory of Surface Physics, Department of Physics, and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China
  • 2Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China
  • 3Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
  • 4Key Laboratory for Information Science of Electromagnetic Waves (Ministry of Education), Fudan University, Shanghai 200433, China
  • 5Hefei National Laboratory, Hefei 230088, China

  • *Contact author: zihaoli20@fudan.edu.cn
  • †Contact author: you_zhou@fudan.edu.cn
  • ‡Contact author: zhuhuangjun@fudan.edu.cn
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PRX Quantum 6, 030366 – Published 30 September, 2025

DOI: https://doi.org/10.1103/wccm-zys6

Abstract

Classical shadow estimation (CSE) is a powerful tool for learning the properties of quantum states and quantum processes. Here we consider the CSE task for quantum unitary channels. By querying an unknown unitary channel U multiple times in quantum experiments, the goal is to learn a classical description from which one can accurately predict many different linear properties of the channel, i.e., the expectation values of arbitrary observables measured on the output of U upon arbitrary input states. Based on collective measurements on multiple systems, we propose a query-efficient protocol for this task, whose query complexity has a quadratic advantage over the previous best approach for this problem, and almost saturates the information-theoretic lower bound. To further enhance practicality, we also present a variant protocol using only single-copy measurements, which still offers much better query performance than previous protocols that do not use quantum memory, and can serve as a key subroutine for learning an arbitrary unknown Hamiltonian from dynamics. In addition to linear properties of unitary channels, our protocol can also be applied to simultaneously predict many nonlinear properties, such as out-of-time-ordered correlators.

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Popular Summary

In quantum mechanics, the evolution of a closed system is governed by a unitary channel acting on quantum states. Learning such channels is a fundamental problem in physics and has applications across various fields, including characterizing complex quantum dynamics, learning quantum Hamiltonians, probing quantum chaos, and variational quantum algorithms. Traditional tomographic approaches can fully characterize an unknown unitary channel but require a prohibitively large number of experiments, rendering them resource consuming and inefficient.

Our research provides a more efficient way to learn key behaviors of unitary channels under the framework of classical shadow estimation. We propose a protocol that lets scientists predict many different properties of an unknown unitary after applying it significantly fewer times. Our protocol achieves a quadratic improvement in query complexity, meaning that the required number of accesses to the unitary grows much slower as the system gets large, compared to the best previous methods. We prove that this improvement is essentially nearly optimal, that is, the access number cannot be further reduced. For practical implementation, we further introduce a variant protocol that operates without using quantum memories. As a key application, we show that this protocol can serve as a key subroutine for learning an arbitrary Hamiltonian from short-time dynamics, outperforming existing approaches in terms of both the query complexity and total evolution time.

Our work offers both theoretical insights and state-of-the-art tools for learning unitary channels, advancing the understanding of complex quantum systems, and paving the way for future research.

Article Text

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PRX Quantum 6, 030366– Published 30 September, 2025
  • Received 9 December 2024
  • Revised 5 May 2025
  • Accepted 1 August 2025
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