Efficient Learning of Clifford-Scrambled Product States

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\(n²\) 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\(n²\) queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.