# Lazy Open Quantum Walks

## TL;DR

This paper introduces lazy open quantum walks, extending existing models to include the possibility of the walker remaining stationary, and proves a central limit theorem showing their distribution converges to a Gaussian.

## Contribution

It extends the mathematical framework of open quantum walks to include laziness and derives a central limit theorem for their behavior on lattices.

## Key findings

- Distribution of lazy OQWs converges to a Gaussian
- Properties of the Gaussian match theoretical predictions
- Extension of CPTP maps to lazy dynamics

## Abstract

Open quantum walks (OQWs) describe a quantum walker on an underlying graph whose dynamics is purely driven by dissipation and decoherence. Mathematically, they are formulated as completely positive trace preserving (CPTP) maps on the space of density matrices for the walker on the graph. Any microscopically derived OQW must include the possibility of remaining on the same site on the graph when the map is applied. We extend the CPTP map to describe a lazy OQW. We derive a central limit theorem for lazy OQWs on a $d$-dimensional lattice, where the distribution converges to a Gaussian. We show that the properties of this Gaussian computed using conventional methods agree with the general formulas derived from our central limit theorem.