Quantum Algorithm for Fidelity Estimation
Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Sheng Fang, Junyi Liu, Mingsheng Ying · IEEE Transactions on Information Theory · 2022
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$.