Enhancing quantum state reconstruction with structured classical shadows

While classical shadows can efficiently predict key quantum state properties, their suitability for certified quantum state tomography remains uncertain. In this paper, we address this challenge by introducing a projected classical shadow (PCS) that extends the standard classical shadow 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(4n) total state copies to achieve a bounded recovery error in the Frobenius norm between the reconstructed and true density matrices, reducing to O(2nr) for states of rank r < 2n—meeting information-theoretic optimal bounds in both cases. For matrix product operator states, we demonstrate that the PCS can recover the ground-truth state with O(n2) total state copies, improving upon the previously established Haar-random bound of O(n3). Numerical simulations validate our scaling results and demonstrate the practical accuracy of the proposed PCS method.