Finite correlation length implies efficient preparation of quantum thermal states

TL;DR

This paper presents a method to efficiently prepare quantum thermal states with finite correlation length on a quantum computer, leveraging properties like exponential decay of correlations and Markovianity, especially in one dimension and above certain temperatures.

Contribution

It introduces a logarithmic depth circuit approach for thermal state preparation under specific correlation and Markov properties, applicable to a broad class of models.

Findings

Efficient thermal state preparation is possible with finite correlation length.

Exponential decay of correlations enables local observable expectation estimation.

Method breaks down for states with topological order.

Abstract

Preparing quantum thermal states on a quantum computer is in general a difficult task. We provide a procedure to prepare a thermal state on a quantum computer with a logarithmic depth circuit of local quantum channels assuming that the thermal state correlations satisfy the following two properties: (i) the correlations between two regions are exponentially decaying in the distance between the regions, and (ii) the thermal state is an approximate Markov state for shielded regions. We require both properties to hold for the thermal state of the Hamiltonian on any induced subgraph of the original lattice. Assumption (ii) is satisfied for all commuting Gibbs states, while assumption (i) is satisfied for every model above a critical temperature. Both assumptions are satisfied in one spatial dimension. Moreover, both assumptions are expected to hold above the thermal phase transition for models without any topological order at finite temperature. As a building block, we show that exponential decay of correlation (for thermal states of Hamiltonians on all induced subgraph) is sufficient to efficiently estimate the expectation value of a local observable. Our proof uses quantum belief propagation, a recent strengthening of strong sub-additivity, and naturally breaks down for states with topological order.