Using Hilbert transform and classical chains to simulate quantum walks

TL;DR

This paper introduces a classical simulation method using coupled chains of springs and Hilbert transforms to emulate quantum walks, providing insights into quantum dynamics and heat transport.

Contribution

It presents a novel classical simulation approach for quantum walks using coupled springs and Hilbert transforms, linking classical energy spreading to quantum wave functions.

Findings

Classical chains can simulate quantum wave functions.

Energy and heat spreading relate to quantum wave function modulus.

Provides a new perspective on ballistic heat transport.

Abstract

We propose a simulation strategy which uses a classical device of linearly coupled chain of springs to simulate quantum dynamics, in particular the quantum walks. Through this strategy, we obtain the quantum wave function from classical evolution. Specially, this goal is achieved with the classical momenta of the particles on the chain and their Hilbert transform, from which we construct the many-body momentum and Hilbert transformed momentum pair correlation functions yielding the real and imaginary parts of the wave function, respectively. With such wave function, we show that the classical chain's energy and heat spreading densities can be related to the wave function's modulus square. This relation indicates a concept of "phonon random walks", and thus it provides a new perspective to understand ballistic heat transport. The results here may give a definite answer to Feynman's idea of using a classical device to simulate quantum physics.