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Institute of Physics eBooks

Quantum walking over triangles

Giuseppe Di Molfetta

Published Sep 1, 2024• 0 citations
Publisher

Abstract

In this chapter, we depart from the grid and we will introduce a new family of quantum walk (QW), which relies on triangulations. The motivation is to model discrete transport in all sorts of topologies as simplicial complexes.
DOI
10.1088/978-0-7503-5744-9ch6

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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 α e − β t 2 , where α and β 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.