  In current studies of mean-field quantum spin systems, much attention is
placed on the calculation of the ground-state energy and the excitation gap,
especially the latter which plays an important role in quantum annealing. In
pure systems, the finite gap can be obtained by various existing methods such
as the Holstein-Primakoff transform, while the tunneling splitting at
first-order phase transitions has also been studied in detail using instantons
in many previous works. In disordered systems, however, it remains challenging
to compute the gap of large-size systems with specific realization of disorder.
Hitherto, only quantum Monte Carlo techniques are practical for such studies.
Recently, Knysh [Nature Comm. \textbf{7}, 12370 (2016)] proposed a method where
the exponentially large dimensionality of such systems is condensed onto a
random potential of much lower dimension, enabling efficient study of such
systems. Here we propose a slightly different approach, building upon the
method of static approximation of the partition function widely used for
analyzing mean-field models. Quantum effects giving rise to the excitation gap
and non-extensive corrections to the free energy are accounted for by
incorporating dynamical paths into the path integral. The time-dependence of
the trace of the time-ordered exponential of the effective Hamiltonian is
calculated by solving a differential equation perturbatively, yielding a
finite-size series expansion of the path integral. Formulae for the first
excited-state energy are proposed to aid in computing the gap. We illustrate
our approach using the infinite-range ferromagnetic Ising model and the
Hopfield model, both in the presence of a transverse field.

Disordered Systems and Neural Networks