Quantum Repeated Interactions and the Chaos Game

TL;DR

This paper models the chaos game within quantum systems using repeated interactions, demonstrating how fermionic and bosonic dynamics can physically realize chaos game algorithms through quantum state evolution.

Contribution

It introduces a quantum open system framework that physically implements the chaos game using fermionic and bosonic repeated interactions.

Findings

Fermionic systems maintain Gaussian states during evolution.

Particle number averages follow the chaos game equation.

Bosonic systems' eigenvalues evolve according to a chaos game-like rule.

Abstract

Inspired by the algorithm of Barnsley's chaos game, we construct an open quantum system model based on the repeated interaction process. We shown that the quantum dynamics of the appropriate fermionic/bosonic system (in interaction with an environment) provides a physical model of the chaos game. When considering fermionic operators, we follow the system's evolution by focusing on its reduced density matrix. The system is shown to be in a Gaussian state (at all time $t$) and the average number of particles is shown to obey the chaos game equation. Considering bosonic operators, with a system initially prepared in coherent states, the evolution of the system can be tracked by investigating the dynamics of the eigenvalues of the annihilation operator. This quantity is governed by a chaos game-like equation from which different scenarios emerge.