For two unknown 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 in quantum computing and quantum information, for example verification and characterization of the outputs of a quantum computer, and design and analysis of quantum algorithms. In this Letter, we propose a quantum algorithm that solves this problem in $text{poly}(log (N), r)$ time, where $r$ is the lower rank of $rho$ and $sigma$. This algorithm exhibits an exponential improvement over the best-known algorithm (based on quantum state tomography) in $text{poly}(N, r)$ time.
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