# Dynamical Triangulation Induced by Quantum Walk

## TL;DR

This paper introduces a quantum walk model on a dynamically evolving 2D triangulated surface, where the surface's geometry changes based on the walker's density, leading to emergent flatness over time.

## Contribution

It extends quantum walks to include dynamical triangulations induced by the walker, connecting quantum automata with evolving geometric structures.

## Key findings

- Number of triangles and curvature grow as t^α e^{-β t^2}
- Flatness emerges in the long-term evolution
- Global walker behavior remains stable under spacetime fluctuations

## Abstract

We present the single-particle sector of a quantum cellular automaton, namely a quantum walk, on a simple dynamical triangulated $2-$manifold. The triangulation is changed through Pachner moves, induced by the walker density itself, allowing the surface to transform into any topologically equivalent one. This model extends the quantum walk over triangular grid, introduced in a previous work, by one of the authors, whose space-time limit recovers the Dirac equation in (2+1)-dimensions. Numerical simulations show that the number of triangles and the local curvature grow as $t^\alpha e^{-\beta t^2}$, where $\alpha$ and $\beta$ parametrize the way geometry changes upon the local density of the walker, and that, in the long run, flatness emerges. Finally, we also prove that the global behavior of the walker, remains the same under spacetime random fluctuations.