Efficient Learning of Clifford-Scrambled Product States

Abstract:Clifford circuits acting on product magic states provide a compact ansatz exhibiting extensive magic, volume-law entanglement, and even classically hard sampling under standard complexity assumptions. Here, we efficiently recover its hidden product subsystems from two-copy Bell sampling. Quadratic relations between Bell samples determine the irreducible blocks after removing Pauli stabilizers, and binary linear algebra constructs a Clifford disentangler. For logarithmic-size blocks, approximate learning of the full state is efficient whenever the state remains inverse-polynomially separated from acquiring additional Pauli stabilizers. We efficiently learn $n$-qubit states prepared by random $T$-doped Clifford circuits with $T$-gate density below one via $O(n^2)$ Bell samples, and disentangle hidden product blocks in Clifford-augmented matrix product states. For unitary learning, we exactly learn $T$-depth-one circuits using $O(n^2)$ queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.

arXiv:2609.27128 [quant-ph]

(or arXiv:2609.27128v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.27128 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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