  We develop a quantum-inspired numerical procedure for searching low-energy
states of a classical Hamiltonian composed of two-body fully-connected random
Ising interactions and a random local longitudinal magnetic field. In this
method, we introduce infinitesimal quantum interactions that do not commute
with the original Ising Hamiltonian, and repeatedly generate and truncate
direct product states, inspired by the Krylov subspace method, to obtain the
low-energy states of the original classical Ising Hamiltonian. The
computational cost is controlled by the form of infinitesimal quantum
interactions (e.g., one-body or two-body interactions) and the numbers of
infinitesimal interaction terms introduced, different initial states
considered, and low-energy states kept during the iteration. For a demonstrate
of the method, here we introduce as the infinitesimal quantum interactions pair
products of Pauli $X$ operators acting on different sites and on-site Pauli $X$
operators into the random Ising Hamiltonian, in which the numerical cost is
$O(N^3)$ per iteration with the system size $N$. We consider 120 instances of
the random coupling realizations for the random Ising Hamiltonian with $N$ up
to 600 and search the 120 lowest-energy states for each instance. We find that
the time-to-solution by the quantum-inspired method proposed here, with
parallelization in terms of the different initial states, for searching the
ground state of the random Ising Hamiltonian scales approximately as $N^5$ for
$N$ up to 600. We also examine the basic physical properties such as the
ensemble-averaged ground-state and first-excited energies and the
ensemble-averaged number of states in the low-energy region of the random Ising
Hamiltonian.
