Nearly query-optimal classical shadow estimation of unitary channels

TL;DR

This work develops a two-phase classical shadow estimation framework for learning properties of unknown unitary channels (CSEU), achieving near-optimal query scaling with the system dimension $d$ via collective measurements and providing a memory-free variant for practical use. It establishes a concrete information-theoretic lower bound and analyzes average-case performance, revealing a separation between worst-case and average-case query costs. The authors further connect CSEU to Hamiltonian learning via polynomial interpolation, delivering an ancilla-free protocol with tilde-$\mathcal{O}$ scaling in queries and evolution time for arbitrary Hamiltonians, and extend the shadow approach to nonlinear quantities such as OTOCs. Overall, the results offer scalable, practically implementable strategies for learning quantum dynamics and predicting a broad class of channel properties, with clear implications for quantum sensing, simulation, and verification.

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 $\mathcal{U}$ 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 $\mathcal{U}$ 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 non-linear properties, such as out-of-time-ordered correlators.

Figures (3)

• Figure 1: Illustration of the CSEU task. The task contains two separate phases. In the learning phase, we apply the unknown unitary channel $\mathcal{U}$ multiple times in some quantum experiments and obtain some classical data. In the prediction phase, we are given a collection of quantum states $\rho_1,\dots, \rho_M$ and observables $O_1,\dots, O_M$. By performing classical postprocessing on the data collected from the learning phase, the goal is to accurately predict linear properties $\mathop{\mathrm{Tr}}\nolimits\mathopen{}\mathclose{\left( O_l \,\mathcal{U}(\rho_l)\right)$ for all $l=1,2,\dots, M$.
• Figure 2: Learning phase of our protocol for CSEU: procedure and intuition. (a) Schematic view of our learning algorithm $\mathcal{A}_{\text{learn}}$. In each round, we first prepare $s$ copies of a pure state $\hat{\psi}$ chosen from a 4-design ensemble, and rotate them with the unitary channel $\mathcal{U}$, then measure the resulting states using the symmetric collective measurement $\mathcal{M}_s$. (b) Geometric intuition of our protocol illustrated on the single-qubit Bloch sphere.
• Figure 3: The query complexity of our protocol [see Eq. (\ref{['eq:generalUB']})] with respect to the dimension $d$. Here $s$ denotes the number of systems collectively measured in our learning phase. The optimal complexity $\mathcal{O}\mathopen{}\mathclose{\left(d\right)$ is attained when $s=\Theta(d)$.

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• Proposition 1
• Theorem 1
• Definition 1: CSEU task
• Theorem 2
• Definition 2
• Lemma 3: Kothari14arunachalam2017survey
• Lemma 4
• Definition 3