Low-temperature excitations within the Bethe approximation

TL;DR

This paper introduces a variational quantum cavity method within the Bethe approximation to estimate low-temperature excitations and upper bounds of energy costs in quantum systems, demonstrated on the transverse Ising model.

Contribution

It develops a novel variational approach using the cavity method to efficiently approximate low-energy excitations in quantum many-body systems.

Findings

The method provides upper bounds for excitation energies.

It accurately estimates overlaps between wave functions.

Comparisons show good agreement with exact solutions for small systems.

Abstract

We propose the variational quantum cavity method to construct a minimal energy subspace of wave vectors that are used to obtain some upper bounds for the energy cost of the low-temperature excitations. Given a trial wave function we use the cavity method of statistical physics to estimate the Hamiltonian expectation and to find the optimal variational parameters in the subspace of wave vectors orthogonal to the lower-energy wave functions. To this end, we write the overlap between two wave functions within the Bethe approximation which allows us to replace the global orthogonality constraint with some local constraints on the variational parameters. The method is applied to the transverse Ising model and different levels of approximations are compared with the exact numerical solutions for small systems.