Massless Dirac Equation from Fibonacci Discrete-Time Quantum Walk

TL;DR

This paper demonstrates that certain Fibonacci discrete-time quantum walks can simulate the massless Dirac equation, revealing ballistic transport and continuous limits similar to relativistic quantum mechanics.

Contribution

It analytically shows that specific Fibonacci quantum walks can replicate the massless Dirac equation's dynamics, expanding quantum simulation capabilities.

Findings

Fibonacci quantum walks exhibit six-step periodic dynamics.

Models show ballistic transport properties.

Continuous limits match the massless Dirac equation.

Abstract

Discrete-time quantum walks can be regarded as quantum dynamical simulators since they can simulate spatially discretized Schr\"{o}dinger, massive Dirac, and Klein-Gordon equations. Here, two different types of Fibonacci discrete-time quantum walks are studied analytically. The first is the Fibonacci coin sequence with a generalized Hadamard coin and demonstrates six-step periodic dynamics. The other model is assumed to have three- or six-step periodic dynamics with the Fibonacci sequence. We analytically show that these models have ballistic transportation properties and continuous limits identical to those of the massless Dirac equation with coin basis change.