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

We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and O ( log ⁡ n ) non-Clifford gates. Specifically, for an n -qubit state | ψ ⟩ prepared with at most t non-Clifford gates, our algorithms use p o l y ( n , 2 t , 1 / ε ) time and copies of | ψ ⟩ to learn | ψ ⟩ to trace distance at most ε .The first algorithm for this task is more efficient, but requires entangled measurements across two copies of | ψ ⟩ . The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently 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.