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Enhancing quantum state reconstruction with structured classical shadows
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DOI：10.1038/s41534-025-01101-1.png)
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Abstract
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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.
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