Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State

2025-12-03

AI Summary

Quantum state tomography of matrix product states (MPS) suffers from exponential scaling in conventional approaches. Method: We propose the first provably convergent tomography framework that integrates the classical shadow protocol with structural prior knowledge of MPS. Specifically, we constrain the density matrix to a tensor train (TT) format and reconstruct it via classical shadow estimation of local observables, enabling efficient parameterized recovery. Contribution/Results: We derive an upper bound on the sample complexity and rigorously prove its superiority over both standard classical shadows and maximum-likelihood estimation. Numerical experiments demonstrate that, under moderate subsystem measurements, our method achieves higher reconstruction fidelity with significantly reduced sampling overhead. Crucially, this work introduces the first principled incorporation of structured low-rank priors—namely, TT structure—into the classical shadow paradigm, thereby bridging theoretical guarantees with practical efficiency.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Efficient quantum state tomography for matrix product states
• Estimating tensor components via observable measurements
• Improving accuracy over classical shadow protocol for observables

Innovation

• Hybridizes classical shadow with matrix product state
• Estimates tensor components via observable estimations
• Proven convergent with quadratic sample complexity scaling