# Path-sum solution of the Weyl Quantum Walk in 3+1 dimensions

## TL;DR

This paper presents an analytical path-sum solution for the Weyl quantum walk in 3+1 dimensions, providing explicit propagator expressions that reveal quantum interference effects on a Cayley graph.

## Contribution

It introduces a novel analytical solution for the Weyl quantum walk in higher dimensions, leveraging path encoding and semigroup structures to derive the propagator.

## Key findings

- Explicit analytical propagator for the 3+1D Weyl quantum walk
- Demonstration of quantum interference effects in the walk
- Framework for analyzing quantum walks on Cayley graphs

## Abstract

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group $\mathbb Z^3$, that in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of $\mathbb Z^3$ that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.