  Quantum imaginary-time evolution (QITE) is a promising tool to prepare
thermal or ground states of Hamiltonians, as convergence is guaranteed when the
evolved state overlaps with the ground state. However, its implementation using
a parameterized quantum circuit is impractical as the number of parameters $m$
increases, since each step in the evolution takes $\Theta(m^2)$ state
preparations to calculate the quantum Fisher information matrix (QFIM). In this
work, we accelerate QITE by rapid estimation of the QFIM, while conserving the
convergence guarantees to the extent possible. To this end, we prove that if a
parameterized state is rotated by a 2-design and measured in the computational
basis, then the QFIM can be inferred from partial derivative cross correlations
of the probability outcomes. One sample estimate costs only $\Theta(m)$ state
preparations, leading to rapid QFIM estimation when a few samples suffice. The
second family of estimators take greater liberties and replace QFIMs with
averaged classical Fisher information matrices (CFIMs). In an extreme special
case optimized for rapid (over accurate) descent, just one CFIM sample is
drawn. We justify the second estimator family by proving rapid descent. Guided
by these results, we propose the random-measurement imaginary-time evolution
(RMITE) algorithm, which we showcase and test in several molecular systems,
with the goal of preparing ground states.
