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Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State

Xun Tang, Haoxuan Chen, Yuehaw Khoo, Lexing Ying

TL;DR

Sketch Tomography leverages the classical shadow framework to recover a tensor-train representation of the density matrix for MPS-representable quantum states. By solving sketched linear systems for TT cores from shadow-obtained observable data, it produces a TT approximant tilde rho with provable Frobenius-norm convergence and a favorable sample complexity scaling as O(n^2). Across 1D and 2D quantum many-body models, tilde rho yields accurate global observable estimates and competitive local observables, often outperforming plain classical shadows and MLE-trained models for global predictions. The work demonstrates a scalable QST approach that combines shadow tomography with tensor-network structure, enabling efficient state characterization in larger systems and offering a path to extending to other tensor-network ansätze.

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.

Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State

TL;DR

Sketch Tomography leverages the classical shadow framework to recover a tensor-train representation of the density matrix for MPS-representable quantum states. By solving sketched linear systems for TT cores from shadow-obtained observable data, it produces a TT approximant tilde rho with provable Frobenius-norm convergence and a favorable sample complexity scaling as O(n^2). Across 1D and 2D quantum many-body models, tilde rho yields accurate global observable estimates and competitive local observables, often outperforming plain classical shadows and MLE-trained models for global predictions. The work demonstrates a scalable QST approach that combines shadow tomography with tensor-network structure, enabling efficient state characterization in larger systems and offering a path to extending to other tensor-network ansätze.

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.

Paper Structure

This paper contains 23 sections, 12 theorems, 137 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Procedure
  3. Performance guarantee
  4. Information-theoretical lower bound
  5. Numerical Experiments
  6. 1D Heisenberg
  7. 1D TFIM
  8. 2D Heisenberg
  9. Conclusion
  10. Literature review
  11. Notations
  12. Details for the sketch tomography procedure
  13. TT representation of rho
  14. Sketch tomography algorithm for quantum state tomography
  15. Proof of the upper bound
  16. ...and 8 more sections

Key Result

Proposition 1

(Informal version of the upper bound) Let $\rho$ be the density matrix of a matrix product state for $n$ qubits with $n$ sufficiently large. Let $\tilde{\rho}$ denote the output of the sketch tomography procedure, and suppose the observable estimations are done using the classical shadow protocol fr

Figures (3)

  • Figure 1: Predictions of two-point functions $\langle \vec{\sigma}_{1} \cdot \vec{\sigma}_{i} \rangle$ for the ground state of the 1D Heisenberg model with $n = 20$ lattice sites. One can see that sketch tomography reaches a similar accuracy to classical shadow. The prediction from the MLE model is accurate except at the boundary site $i = 20$.
  • Figure 4: Predictions of two-point functions $\langle \sigma_{1}^{Z} \sigma_{i}^{Z} \rangle$ for the ground state of the 1D TFIM model with $n = 40$ lattice sites. One can see that sketch tomography reaches the accuracy of classical shadow.
  • Figure 6: Predictions of two-point functions $\langle \vec{\sigma_{(1,1)}} \cdot \vec{\sigma_{(i,j)}}\rangle$ for the ground state of the 2D Heisenberg model with $n = 64$ lattice sites. One can see that our QST procedure successfully approximates the true 2-point correlation function.

Theorems & Definitions (24)

  • Proposition 1
  • Proposition 2: Informal version of the lower bound
  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Lemma 1
  • proof
  • Definition 3
  • ...and 14 more