Computational Characteristics of Random Field Ising Model with Long-Range Interaction

TL;DR

This paper explores the computational complexity of the long-range interaction random field Ising model, showing NP-completeness in 2D and efficient approximation in 1D under certain decay conditions, relevant for quantum computation.

Contribution

It proves NP-completeness of 2D RFIM with long-range interactions and provides efficient approximation methods for 1D RFIM with fast-decaying interactions.

Findings

Solving 2D RFIM ground state is NP-complete for all decay exponents.

1D RFIM with fast decay can be efficiently approximated.

Results are applicable to current ion trap quantum systems.

Abstract

Ising model is a widely studied class of models in quantum computation. In this paper we investigate the computational characteristics of the random field Ising model (RFIM) with long-range interactions that decays as an inverse polynomial of distance, which can be achieved in current ion trap system. We prove that for an RFIM with long-range interaction embedded on a 2-dimensional plane, solving its ground state is NP-complete for all diminishing exponent, and prove that the 1-dimensional RFIM with long-range interaction can be efficiently approximated when the interaction decays fast enough.