$(H,\rho)$-induced dynamics and the quantum game of life

TL;DR

This paper introduces a novel quantum dynamics framework combining Hamiltonian evolution with rule-based updates, extending the classical game of life into a quantum setting with diverse possible behaviors.

Contribution

It develops a general $(H, ho)$-induced dynamics model and applies it to extend and analyze the quantum version of the classical game of life.

Findings

The dynamics can be non-periodic and vary with initial conditions.

The model generalizes classical life to quantum systems.

Different rule sets lead to diverse behaviors.

Abstract

We propose an extended version of quantum dynamics for a certain system S, whose evolution is ruled by a Hamiltonian $H$, its initial conditions, and a suitable set $\rho$ of {\em rules}, acting repeatedly on S. The resulting dynamics is not necessarily periodic or quasi-periodic, as one could imagine for conservative systems with a finite number of degrees of freedom. In fact, it may have quite different behaviors depending on the explicit forms of $H$, $\rho$ as well as on the initial conditions. After a general discussion on this $(H,\rho)$-{\em induced dynamics}, we apply our general ideas to extend the classical game of life, and we analyze several aspects of this extension.