Grover's search meets Ising models: a quantum algorithm for finding low-energy states

TL;DR

This paper introduces a quantum algorithm that leverages Grover's search and Ising model evolution operators to efficiently find low-energy states in disordered Ising models, offering a quadratic speedup over classical methods.

Contribution

It presents a novel approach integrating Ising model evolution as the oracle in Grover's algorithm for quantum simulation of disordered systems.

Findings

Achieves quadratic speedup over classical algorithms.

Effectively identifies low and high energy states with high probability.

Provides a method to determine optimal evolution time for phase flips.

Abstract

We propose a methodology for implementing Grover's algorithm in the digital quantum simulation of disordered Ising models. The core concept revolves around using the evolution operator for the Ising model as the quantum oracle within Grover's search. This operator induces phase shifts for the eigenstates of the Ising Hamiltonian, with the most pronounced shifts occurring for the lowest and highest energy states. Determining these states for a disordered Ising Hamiltonian using classical methods presents an exponentially complex challenge with respect to the number of spins (or qubits) involved. Within our proposed approach, we determine the optimal evolution time by ensuring a phase flip for the target states. This method yields a quadratic speedup compared to classical computation methods and enables the identification of the lowest and highest energy states (or neighboring states) with a high probability $\lesssim 1$.