Quantum Algorithm for Fidelity Estimation - Wang, Qisheng; Zhang, Zhicheng; Chen, Kean; Guan, Ji; Fang, Wang; Liu, Junyi; Ying, Mingsheng (IEEE transactions on information theory, 2023)
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 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 paper, we propose a quantum algorithm that solves this problem in {\mathrm{ poly}}(\log (N), r, 1/\varepsilon) time, where r is the lower rank of \rho and \sigma , and \varepsilon is the desired precision, provided that the purifications of \rho and \sigma are prepared by quantum oracles. This algorithm exhibits an exponential speedup over the best known algorithm (based on quantum state tomography) which has time complexity polynomial in N .