Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State
This work addresses the problem of efficient quantum state tomography for MPS states, offering a practical improvement over existing methods for observable estimation in near-term quantum devices.
Sketch Tomography hybridizes classical shadow estimation with matrix product state (MPS) assumptions to perform efficient quantum state tomography. The method achieves provable convergence with sample complexity scaling quadratically in system size and outperforms classical shadow and maximum likelihood estimation in observable estimation tasks for moderately large subsystems.
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.