Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State

TL;DR

Sketch Tomography combines classical shadow techniques with matrix product state assumptions to efficiently and accurately reconstruct quantum states, outperforming existing methods in certain observable estimation tasks.

Contribution

It introduces a novel hybrid quantum state tomography method leveraging tensor train ansatz and classical shadows, with proven convergence and improved accuracy over traditional approaches.

Findings

Quadratic sample complexity in system size.

More accurate observable estimation than classical shadow protocol.

Superior to maximum likelihood estimation in certain tasks.

Abstract

We introduce Sketch Tomography, an efficient procedure for quantum state tomography based on the classical shadow protocol used for quantum observable estimations. The procedure applies to the case where the ground truth quantum state is a matrix product state (MPS). The density matrix of the ground truth state admits a tensor train ansatz as a result of the MPS assumption, and we estimate the tensor components of the ansatz through a series of observable estimations, thus outputting an approximation of the density matrix. The procedure is provably convergent with a sample complexity that scales quadratically in the system size. We conduct extensive numerical experiments to show that the procedure outputs an accurate approximation to the quantum state. For observable estimation tasks involving moderately large subsystems, we show that our procedure gives rise to a more accurate estimation than the classical shadow protocol. We also show that sketch tomography is more accurate in observable estimation than quantum states trained from the maximum likelihood estimation formulation.