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Enhancing Quantum State Reconstruction with Structured Classical Shadows

Zhen Qin, Joseph M. Lukens, Brian T. Kirby, Zhihui Zhu

TL;DR

This work addresses the scalability gap in quantum state tomography by marrying classical shadow estimators with a projection step onto physically meaningful subspaces, forming the projected classical shadow (PCS) method. PCS provides guaranteed recovery performance under Haar-random measurements and achieves optimal or near-optimal sample complexity for general and low-rank states, while significantly improving scaling for matrix product operator (MPO) states via TT-SVD-based projections. Theoretical guarantees are complemented by numerical experiments showing that PCS—especially LR-PCS and MPO-PCS—outperforms standard CS in reconstructing full quantum states across varying system sizes and entanglement structures. This approach broadens the applicability of classical-shadow techniques to full-state tomography and offers a versatile framework for incorporating prior structural knowledge into quantum state estimation, with potential extensions to more advanced tensor-network representations.

Abstract

Quantum state tomography (QST) remains the prevailing method for benchmarking and verifying quantum devices; however, its application to large quantum systems is rendered impractical due to the exponential growth in both the required number of total state copies and classical computational resources. Recently, the classical shadow (CS) method has been introduced as a more computationally efficient alternative, capable of accurately predicting key quantum state properties. Despite its advantages, a critical question remains as to whether the CS method can be extended to perform QST with guaranteed performance. In this paper, we address this challenge by introducing a projected classical shadow (PCS) method with guaranteed performance for QST based on Haar-random projective measurements. PCS extends the standard CS method by incorporating a projection step onto the target subspace. For a general quantum state consisting of $n$ qubits, our method requires a minimum of $O(4^n)$ total state copies to achieve a bounded recovery error in the Frobenius norm between the reconstructed and true density matrices, reducing to $O(2^n r)$ for states of rank $r<2^n$ -- meeting information-theoretic optimal bounds in both cases. For matrix product operator states, we demonstrate that the PCS method can recover the ground-truth state with $O(n^2)$ total state copies, improving upon the previously established Haar-random bound of $O(n^3)$. Simulation results further validate the effectiveness of the proposed PCS method.

Enhancing Quantum State Reconstruction with Structured Classical Shadows

TL;DR

This work addresses the scalability gap in quantum state tomography by marrying classical shadow estimators with a projection step onto physically meaningful subspaces, forming the projected classical shadow (PCS) method. PCS provides guaranteed recovery performance under Haar-random measurements and achieves optimal or near-optimal sample complexity for general and low-rank states, while significantly improving scaling for matrix product operator (MPO) states via TT-SVD-based projections. Theoretical guarantees are complemented by numerical experiments showing that PCS—especially LR-PCS and MPO-PCS—outperforms standard CS in reconstructing full quantum states across varying system sizes and entanglement structures. This approach broadens the applicability of classical-shadow techniques to full-state tomography and offers a versatile framework for incorporating prior structural knowledge into quantum state estimation, with potential extensions to more advanced tensor-network representations.

Abstract

Quantum state tomography (QST) remains the prevailing method for benchmarking and verifying quantum devices; however, its application to large quantum systems is rendered impractical due to the exponential growth in both the required number of total state copies and classical computational resources. Recently, the classical shadow (CS) method has been introduced as a more computationally efficient alternative, capable of accurately predicting key quantum state properties. Despite its advantages, a critical question remains as to whether the CS method can be extended to perform QST with guaranteed performance. In this paper, we address this challenge by introducing a projected classical shadow (PCS) method with guaranteed performance for QST based on Haar-random projective measurements. PCS extends the standard CS method by incorporating a projection step onto the target subspace. For a general quantum state consisting of qubits, our method requires a minimum of total state copies to achieve a bounded recovery error in the Frobenius norm between the reconstructed and true density matrices, reducing to for states of rank -- meeting information-theoretic optimal bounds in both cases. For matrix product operator states, we demonstrate that the PCS method can recover the ground-truth state with total state copies, improving upon the previously established Haar-random bound of . Simulation results further validate the effectiveness of the proposed PCS method.
Paper Structure (10 sections, 8 theorems, 50 equations, 5 figures, 2 tables)

This paper contains 10 sections, 8 theorems, 50 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classical Shadows
  3. Projected Classical Shadow (PCS) for QST
  4. Simulation Results
  5. Conclusion
  6. Proof of Equation (\ref{['proof of expectation of MSE main result']})
  7. Proof of Theorem \ref{['project CS:general']}
  8. Proof of Theorem \ref{['project CS conclusion']}
  9. Proof of Theorem \ref{['project CS MPO conclusion']}
  10. Auxiliary Materials

Key Result

Theorem 1

For a given $\boldsymbol{\rho}^\star \in \mathbb{X}$, let $\boldsymbol{\rho}_{\textup{PCS}}$ be the projected CS in Eq. projected classical shadow any set. Then with probability at least $1- e^{- \Omega(\log N_{1/2}(\overline \mathbb{X}))}$,

Figures (5)

  • Figure 1: Illustration of proposed PCS method. Given an initial CS estimate $\boldsymbol{\rho}_{\text{CS}}$ lying in the space of Hermitian and unit-trace matrices (not necessarily PSD), we compute the closest state $\boldsymbol{\rho}_{\text{PCS}}$ in the physical space of interest---either the space of all possible states (left) or a subspace possessing a desired structure (right).
  • Figure 2: Mean squared error as a function of state copies $M$ for CS and LR-PCS methods on $n=4$ qubits, averaged over trials on ten randomly chosen ground truth states for each rank $r\in\{1,4,16\}$. The figures span $M=250$ to $M=10000$.
  • Figure 3: Mean squared error as a function of state copies $M$ for CS and MPO-PCS methods on seven-qubit MPO states, where each point is an average over trials on ten randomly chosen ground truth states for each bond dimension $D\in\{1,4\}$.
  • Figure 4: Mean square error as a function of the number of state copies $M$ for (a) thermal state ($T = 0.2$), (b) thermal state ($T = 2$), and (c) GHZ state. Comparison between different methods for (a) thermal state ($T = 0.2$), (b) thermal state ($T = 2$), and (c) GHZ state. All figures have $M=100$ as the starting point.
  • Figure 5: Mean squared error as a function of the total qubit number with $M = 3000$ for (a) thermal state ($T = 0.2$), (b) thermal state ($T = 2$), and (c) GHZ state.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • proof
  • proof
  • proof
  • Lemma 1
  • Lemma 2
  • ...and 2 more