# Bifurcation Curves of Two-Dimensional Quantum Walks

## TL;DR

This paper analyzes the asymptotic behavior of two-dimensional quantum walks, showing that regions of polynomial decay are bounded by explicitly computable algebraic bifurcation curves, enhancing understanding of their long-term dynamics.

## Contribution

It introduces a method to explicitly compute bifurcation curves bounding polynomial decay regions in two-dimensional quantum walks.

## Key findings

- Regions of polynomial decay are bounded by algebraic curves.
- Explicit examples of bifurcation curves are provided for various quantum walks.
- The results improve understanding of the asymptotic structure of quantum walk dynamics.

## Abstract

The quantum walk differs fundamentally from the classical random walk in a number of ways, including its linear spreading and initial condition dependent asymmetries. Using stationary phase approximations, precise asymptotics have been derived for one-dimensional two-state quantum walks, one-dimensional three-state Grover walks, and two-dimensional four-state Grover walks. Other papers have investigated asymptotic behavior of a much larger set of two-dimensional quantum walks and it has been shown that in special cases the regions of polynomial decay can be parameterized. In this paper, we show that these regions of polynomial decay are bounded by algebraic curves which can be explicitly computed. We give examples of these bifurcation curves for a number of two-dimensional quantum walks.