Quantumness of discrete Hamiltonian cellular automata

TL;DR

This paper explores how discrete Hamiltonian cellular automata can emulate quantum mechanics through linear, unitary-like evolution, establishing a connection between automata and continuous quantum models with a fundamental scale.

Contribution

It demonstrates that discrete CA dynamics must be linear to be consistent, enabling a mapping to quantum models and analyzing their observables and stationary solutions.

Findings

CA dynamics are linear, akin to Schrödinger evolution

A invertible map exists between CA and quantum models

Solutions to modified dispersion relations are found for stationary states

Abstract

We summarize a recent study of discrete (integer-valued) Hamiltonian cellular automata (CA) showing that their dynamics can only be consistently defined, if it is linear in the same sense as unitary evolution described by the Schr\"odinger equation. This allows to construct an invertible map between such CA and continuous quantum mechanical models, which incorporate a fundamental scale. Presently, we emphasize general aspects of these findings, the construction of admissible CA observables, and the existence of solutions of the modified dispersion relation for stationary states.