# Chirality from quantum walks without quantum coin

## TL;DR

This paper introduces a method to construct scalar quantum walks on various graphs with geometric interpretation and demonstrates how fundamental quantum walks like Weyl and Dirac can be derived from scalar walks in low-dimensional spaces.

## Contribution

It presents a novel strategy for building scalar quantum walks on graphs with Euclidean embedding and proves limitations on coin dimensions, connecting scalar walks to fundamental quantum models.

## Key findings

- Scalar quantum walks can be constructed on a wide class of graphs with geometric interpretation.
- No two-dimensional coin quantum walk can be obtained from an isotropic scalar walk.
- Weyl and Dirac quantum walks can be derived from scalar walks in spaces up to three dimensions.

## Abstract

Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom---called the coin and represented by a finite-dimensional Hilbert space---is associated to each node. Scalar quantum walks are QWs with a one-dimensional coin. We propose a general strategy allowing one to construct scalar QWs on a broad variety of graphs, which admit embedding in Eulidean spaces, thus having a direct geometric interpretation. After reviewing the technique that allows one to regroup cells of nodes into new nodes, transforming finite spatial blocks into internal degrees of freedom, we prove that no QW with a two-dimensional coin can be derived from an isotropic scalar QW in this way. Finally we show that the Weyl and Dirac QWs can be derived from scalar QWs in spaces of dimension up to three, via our construction.