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 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
$N$
.