Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

We give an algorithm that efficiently learns a quantum state prepared by Clifford gates and $O(\log(n))$ non-Clifford gates. Specifically, for an $n$-qubit state $\lvert \psi \rangle$ prepared with at most $t$ non-Clifford gates, we show that $\mathsf{poly}(n,2^t,1/\epsilon)$ time and copies of $\lvert \psi \rangle$ suffice to learn $\lvert \psi \rangle$ to trace distance at most $\epsilon$. This result follows as a special case of an algorithm for learning states with large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.