Quantuminspired search method for lowenergy states of classical Ising Hamiltonians
Abstract
We develop a quantuminspired numerical procedure for searching lowenergy states of a classical Hamiltonian composed of twobody fullyconnected 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 lowenergy states of the original classical Ising Hamiltonian. The computational cost is controlled by the form of infinitesimal quantum interactions (e.g., onebody or twobody interactions) and the numbers of infinitesimal interaction terms introduced, different initial states considered, and lowenergy 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 onsite 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 lowestenergy states for each instance. We find that the timetosolution by the quantuminspired 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 ensembleaveraged groundstate and firstexcited energies and the ensembleaveraged number of states in the lowenergy region of the random Ising Hamiltonian.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2010.00180
 Bibcode:
 2020arXiv201000180U
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Quantum Physics
 EPrint:
 13 pages, 11 figures