Scalable and Flexible Classical Shadow Tomography with Tensor Networks

Ahmed A. Akhtar1, Hong-Ye Hu1,2, and Yi-Zhuang You1

1Department of Physics, University of California San Diego, La Jolla, CA 92093, USA
2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA

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Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.

Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. The measurement protocol is defined in terms of a unitary ensemble that is applied to the state of interest before measurement, and different choices of unitary ensemble produce efficient protocols for different types of operators. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with local, finite-depth random Clifford circuits. Using this framework, we show that the shallow-circuit measurement protocol provides immediate, exponential advantages over random, single-qubit measurements for predicting quasi-local operators and performing fidelity estimation.

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