Simulating Dirac Hamiltonian in Curved Space-time by Split-step Quantum Walk

TL;DR

This paper develops a quantum walk-based simulation of Dirac particles in curved space-time, incorporating gauge potentials, and demonstrates implementation in simple qubit systems, advancing quantum simulation of fundamental physics.

Contribution

It introduces a modified split-step quantum walk that simulates Dirac dynamics in curved space-time with gauge fields, including multi-particle interactions.

Findings

Effective Hamiltonian mimics Dirac particles in curved space-time.

Simulation scheme successfully implemented in qubit systems.

Extension to higher dimensions and gauge groups discussed.

Abstract

Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the non-classical nature of dynamics in its discrete form. In this work, starting from a modified version of one-spatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved $(1+1)$ space-time dimension coupled to $U(1)$ gauge potential---which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent $(2+1)$ space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include $U(N)$ gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operations.