One-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers

TL;DR

This paper demonstrates that simulating one-dimensional quantum systems at finite temperatures on classical computers is efficient, with polynomial complexity, by analyzing entanglement properties using quantum field theory and numerical validation.

Contribution

It introduces a method to efficiently simulate 1d quantum systems at finite temperatures using matrix product states, extending known results from ground states to finite-temperature states.

Findings

Simulation cost grows polynomially with inverse temperature

Thermofield double states can be efficiently represented as matrix products

Entanglement entropy obeys an area law at finite temperatures

Abstract

It is by now well-known that ground states of gapped one-dimensional (1d) quantum-many body systems with short-range interactions can be studied efficiently using classical computers and matrix product state techniques. A corresponding result for finite temperatures was missing. Using the replica trick in 1+1d quantum field theory, it is shown here that the cost for the classical simulation of 1d systems at finite temperatures grows in fact only polynomially with the inverse temperature and is system-size independent -- even for gapless systems. In particular, we show that the thermofield double state (TDS), a purification of the equilibrium density operator, can be obtained efficiently in matrix-product form. The argument is based on the scaling behavior of R\'enyi entanglement entropies in the TDS. At finite temperatures, they obey the area law. For gapless systems with central charge $c$, the entanglement is found to grow only logarithmically with inverse temperature, $S_\alpha\sim \frac{c}{6}(1+1/\alpha)\log \beta$. The field-theoretical results are tested and confirmed by quasi-exact numerical computations for integrable and non-integrable spin systems, and interacting bosons.