Pauli Measurements Are Near-Optimal for Pure State Tomography

Abstract: We give an algorithm for pure state tomography with near-optimal copy complexity using single-qubit measurements. Specifically, given $\widetilde{O}(2^n/ε)$ copies of an unknown pure $n$-qubit state $\lvertψ\rangle$, the algorithm performs only \textit{nonadaptive Pauli measurements}, runs in time $\mathrm{poly}(2^n,1/ε)$, and outputs $\lvert \widehatψ \rangle$ that has fidelity $1-ε$ with $\lvert ψ\rangle$ with high probability. This improves upon the previous best copy complexity bound of $\widetilde{O}(3^n/ε)$.