The Fast Mode-Fourier-Transform and the bands of the Bethe chain

TL;DR

This paper introduces a fast Fourier transform protocol for many-body systems, applied to the Bethe chain, revealing interaction-related band structures and enabling efficient simulations of translationally invariant quantum systems.

Contribution

A novel logarithmic-scaling Fourier transformation protocol for many-body systems, demonstrated on the Bethe chain, with broad applicability for quantum simulation and analysis.

Findings

Band diagram shows flat patterns linked to interactions

Protocol achieves logarithmic scaling in two-site gates

Applicable to diverse quantum many-body scenarios

Abstract

A many-body Fourier transformation with logarithmic scaling in the number of necessary two-site gates is implemented. The protocol is applied to study the Bethe chain as a prototype of a translationally invariant system. The resulting band diagram features a flat pattern highly correlated to interaction mechanisms. The introduced protocol can be applied to a wide spectrum of scenarios and can offer new possibilities of simulation and analysis.