Enhancing Quantum State Reconstruction with Structured Classical Shadows

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.