# Discrete Geometry from Quantum Walks

## TL;DR

This paper explores how discrete quantum walks can simulate fermion propagation in curved 2D space-time, extending concepts like metrics and curvature into the discrete setting and relating them to continuous geometry.

## Contribution

It introduces a framework for defining discrete covariant derivatives, spin-connections, metrics, and curvatures in quantum walks, bridging discrete models with continuous differential geometry.

## Key findings

- Two types of discrete Riemann curvatures are defined and related.
- One discrete curvature converges to the continuous Riemann curvature in the limit.
- A concrete example demonstrates the application of the framework.

## Abstract

A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in $2$D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and $2$-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.