Quantum Algorithm for Fidelity Estimation

For two unknown mixed quantum states $\rho$ and $\sigma$ in an $N$-dimensional Hilbert space, computing their fidelity $F(\rho,\sigma)$ is a basic problem with many important applications information, for example verification characterization of the outputs computer, design analysis algorithms. In this paper, we propose algorithm that solves $\operatorname{poly}(\log (N), r, 1/\varepsilon)$ time, where $r$ lower rank $\sigma$, $\varepsilon$ desired precision, provided purifications are prepared by oracles. This exhibits exponential speedup over best known (based on state tomography) which has time complexity polynomial $N$.