Variational quantum Monte Carlo simulations with tensor-network states

TL;DR

This paper introduces a variational quantum Monte Carlo method utilizing tensor-network states, specifically matrix product states, to efficiently simulate large quantum systems with reduced computational complexity.

Contribution

It presents a novel approach combining tensor-network states with stochastic optimization for quantum Monte Carlo, enabling simulations of larger systems at criticality with lower computational costs.

Findings

Successfully simulated up to 256 spins at criticality

Achieved lower computational scaling (ND^3) compared to traditional methods

Demonstrated effectiveness on the transverse Ising chain with periodic boundary conditions

Abstract

We show that the formalism of tensor-network states, such as the matrix product states (MPS), can be used as a basis for variational quantum Monte Carlo simulations. Using a stochastic optimization method, we demonstrate the potential of this approach by explicit MPS calculations for the transverse Ising chain with up to N=256 spins at criticality, using periodic boundary conditions and D*D matrices with D up to 48. The computational cost of our scheme formally scales as ND^3, whereas standard MPS approaches and the related density matrix renromalization group method scale as ND^5 and ND^6, respectively, for periodic systems.