# Continuous limits of linear and nonlinear quantum walks

## TL;DR

This paper rigorously demonstrates that nonlinear quantum walks on a lattice converge to solutions of nonlinear Dirac equations as the lattice spacing approaches zero, bridging discrete quantum models with continuous relativistic quantum equations.

## Contribution

The paper provides a rigorous proof of the continuous limit of nonlinear quantum walks, connecting them to nonlinear Dirac equations using Shannon interpolation.

## Key findings

- Nonlinear quantum walks converge to nonlinear Dirac solutions as lattice spacing decreases.
- Uniform convergence in Sobolev space $H^s$ is established.
- The approach bridges discrete quantum models with continuous relativistic quantum equations.

## Abstract

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice $\delta \mathbb Z$ uniformly converges (in Sobolev space $H^s$) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as $\delta\to 0$. Here, to compare the walker defined on $\delta\mathbb Z$ and the solution to the NLD defined on $\mathbb R$, we use Shannon interpolation.