Classical Shadow Tomography with Locally Scrambled Quantum Dynamics

TL;DR

This work extends classical shadow tomography to locally scrambled quantum dynamics, enabling efficient state and observable prediction on near-term devices by exploiting an entanglement-feature (EF) formalism. The key idea is that the reconstruction map $\mathcal{M}^{-1}$ depends only on the EF $W_{\mathcal{E}_{\sigma}}^{(2)}$, allowing a linear estimator $\rho=\mathcal{M}^{-1}[\sigma]$ computed entirely in classical post-processing. A central result is a concrete procedure to obtain reconstruction coefficients $r_A$ by solving $\sum_{A,C} r_A f_{A,B,C} W_{\mathcal{E}_{\sigma},C}^{(2)}=\delta_{B,\Omega_N}$, together with a bound on sample complexity $M \ge {\lVert O\rVert}_{\mathcal{E}_{\sigma}}^2/(\epsilon^2 \delta)$ for predicting observables via the shadow norm ${\lVert O\rVert}_{\mathcal{E}_{\sigma}}^2$. Numerically, the authors demonstrate unbiased fidelity estimation for GHZ states with shallow circuits, scaling analyses showing a finite-depth optimum, and approaches for fixed-circuit or Hamiltonian-dynamics implementations, including approximate locally scrambled ensembles with frame-potential diagnostics. These results broaden the applicability of classical shadow tomography to realistic, hardware-limited quantum devices and flexible data-acquisition strategies.

Abstract

We generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.

Figures (20)

• Figure 1: Illustration of classical shadow tomography protocol. This work focuses on the case when the unitary channel is of finite depth and respects locality.
• Figure 2: Two-qudit unitary channel in (a) the long-time (Page state) limit and (b) the short-time (product state) limit. (c) Reconstruction coefficients $r_A$ and (d) the shadow norm ${\lVert O\rVert}_{\mathcal{E}_{\sigma}}^2$ v.s. the single-qudit purity $w$, for $d=2$. $w$ varying from $1$ to $4/5$ effectively models the circuit depth (or evolution time) growing from 0 to $\infty$.
• Figure 3: Classical shadow tomography with (a) finite-depth random unitary/Clifford circuits (of $L$ layers), (b) a fixed unitary twirled by single qubit random Clifford gates, and (c) discrete-time Hamiltonian dynamics (of $T$ steps).
• Figure 4: (a) Fidelity estimation of GHZ state with RUC of different circuit depth $L$ using entanglement-feature-based reconstruction $\mathcal{M}^{-1}_\text{EF}$ (denoted by EF) over different number $N$ of qubits. (b) Fidelity estimation of GHZ state using shallow RUC (3-layer, with $\mathcal{M}^{-1}_\text{EF}$, denoted by EF), random on-site (local Haar) gates (0-layer, with $\mathcal{M}^{-1}_\text{LH}$, denoted by LH) and global Haar unitary ($\infty$-layer, with $\mathcal{M}^{-1}_\text{GH}$, denoted by GH). The inset shows the variance $\operatorname{Var} F$ of the predicted fidelity as a function of system size $N$. In both subfigures, the sample size is 5000. Error bar indicates 3-standard-deviation estimated by the bootstrap method. Points are split horizontally to avoid the overlap of markers.
• Figure 5: (a) Fidelity estimation of the reconstructed GHZ state with RUC of finite depth $L$. (b) Estimation of observable $P_0=|\langle \Psi|00\cdots 0\rangle|^2$ (the projection operator to the ${|00\cdots 0 \rangle}$ state) on the reconstructed GHZ state with RUC of finite depth $L$. In both cases, the reconstruction uses the global Haar reconstruction map. The sample size is 5000. Error bar indicates 3-standard-deviation estimated by the bootstrap method.