LAUSR

We develop a quantum-inspired numerical procedure for searching low-energy states of a classical Hamiltonian including two-body full-connected random Ising interactions and a random local longitudinal magnetic field. In this method, we introduce infinitesimal quantum interactions that cannot commute with the original Ising Hamiltonian, and generate and truncate direct product states inspired by the Krylov subspace method to obtain the low-energy spectrum of the original classical Ising Hamiltonian. The numerical cost is controllable by the form of infinitesimal quantum interactions, the number $I$ of numerical iterations, the number $L$ of initial classical states, and the number $K$ of states kept. To demonstrate the method, here we introduce as the infinitesimal quantum interactions pair products of Pauli $X$ operators and on-site Pauli $X$ operators into the random Ising Hamiltonian, for which the numerical cost is $O(N^3)$ per a numerical iteration with the system size $N$. We consider 120 instances of the random coupling realizations for the random Ising Hamiltonian in each $N$ up to 30 and perform the calculations to search the 120 lowest-energy states of each Hamiltonian. We show that the procedure with $(L,K,I)=(N,N(N+1)/2+1,N)$ finds all the ground states successfully and about 99\% of the 120 lowest-energy states.