In this paper, we present a learning algorithm aimed at learning states
obtained from computational basis states by Clifford circuits doped with a
finite number t of non-Clifford gates. To tackle this problem, we introduce a
novel algebraic framework for t-doped stabilizer states by utilizing tools from
stabilizer entropy. Leveraging this new structure, we develop an algorithm that
uses sampling from the distribution obtained by squaring expectation values of
Pauli operators that can be obtained by Bell sampling on the state and its
conjugate in the computational basis. The algorithm requires resources of
complexity O(\exp(t)\poly(n)) and exhibits an exponentially small probability
of failure.Comment: L.L. and S.O. contributed equally to this wor