Polynomial-Time Algorithm for Simulation of Weakly Interacting Quantum Spin Systems

Abstract

We describe an algorithm that computes the ground state energy and correlation functions for 2-local Hamiltonians in which interactions between qubits are weak compared to single-qubit terms. The running time of the algorithm is polynomial in n and δ⁻¹, where n is the number of qubits, and δ is the required precision. Specifically, we consider Hamiltonians of the form ${H=H_0+ \epsilon V}$ , where H ₀ describes non-interacting qubits, V is a perturbation that involves arbitrary two-qubit interactions on a graph of bounded degree, and ${\epsilon}$ is a small parameter. The algorithm works if ${|\epsilon|}$ is below a certain threshold value ${\epsilon_0}$ that depends only upon the spectral gap of H ₀, the maximal degree of the graph, and the maximal norm of the two-qubit interactions. The main technical ingredient of the algorithm is a generalized Kirkwood-Thomas ansatz for the ground state. The parameters of the ansatz are computed using perturbative expansions in powers of ${\epsilon}$ . Our algorithm is closely related to the coupled cluster method used in quantum chemistry.