Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature

TL;DR

This paper presents a new algorithm for efficiently simulating one-dimensional quantum systems with long-range interactions at any temperature, using a novel truncation scheme for matrix product operators.

Contribution

It introduces a quasi-polynomial runtime algorithm for quantum Gibbs states with long-range interactions, improving simulation accuracy and efficiency across all temperatures.

Findings

Achieves quasi-polynomial runtime for inverse temperatures up to poly(log n)

Provides a new truncation scheme with controlled error for matrix product operators

Enables more precise simulation of time evolution than Lieb-Robinson bounds

Abstract

We introduce a method that ensures efficient computation of one-dimensional quantum systems with long-range interactions across all temperatures. Our algorithm operates within a quasi-polynomial runtime for inverse temperatures up to $\beta={\rm poly}(\ln(n))$. At the core of our approach is the Density Matrix Renormalization Group algorithm, which typically does not guarantee efficiency. We have created a new truncation scheme for the matrix product operator of the quantum Gibbs states, which allows us to control the error analytically. Additionally, our method can be applied to simulate the time evolution of systems with long-range interactions, achieving significantly better precision than that offered by the Lieb-Robinson bound.