  Suzuki-Trotter decomposition is a well-known technique used to calculate the
partition function of quantum spin systems, in which the imaginary-time
dependence of the partition function occurs inevitably. Since it is very
difficult to explicitly treat the imaginary-time dependence of the partition
function, we usually neglect the imaginary-time dynamical effect, which is
called the static approximation. Although the static approximation is the first
approach, it is not even clear when the static approximation is justified for
mean-field quantum spin systems, that is, mean-field quantum spin systems have
not been solved exactly so far. In this study, we solve exactly the partition
function for a particular class of mean-field quantum spin systems including
randomness without the static approximation. The partition function can be
regarded as a result of time evolution in the imaginary-time Schr\"odinger
equation, and solving the exact solution of the partition function is
equivalent to solving the optimal control problem in the imaginary-time
Schr\"odinger equation. As the result, the solution of the optimal control
problem coincides exactly with the static approximate solution of the partition
function and, therefore, the static approximation is exact for the particular
class of mean-field quantum spin systems including randomness in general.
Furthermore, we prove that the analysis of the previous study in quantum
annealing is exact where the non-stoquastic interaction and the inhomogeneous
transverse field accelerate the computational time exponentially for mean-field
quantum spin systems.
