Quantum walks in the density operator picture

TL;DR

This paper introduces a novel framework for quantum walks using density operators, where the evolution is described by unitary channels acting via matrix multiplication, providing a new perspective on quantum dynamics in graphs.

Contribution

It presents a new density operator-based approach to quantum walks, defining unitary channels through reflections in tensor product spaces, and simplifies the evolution description.

Findings

Unitary channels are formulated using reflections in tensor product spaces.

Quantum walk dynamics are described by matrix multiplication on density operators.

The approach offers a new perspective on quantum system evolution in graphs.

Abstract

A new approach to quantum walks is presented. Considering a quantum system undergoing some unitary discrete-time evolution in a directed graph G, we think of the vertices of G as sites that are occupied by the quantum system, whose internal state is described by density operators. To formulate the unitary evolution, we define reflections in the tensor product of an internal Hilbert space and a spatial Hilbert space. We then construct unitary channels that govern the evolution of the system in the graph. The discrete dynamics of the system (called quantum walks) is obtained by iterating the unitary channel on the density operator of the quantum system. It turns out that in this framework, the action of the unitary channel on a density operator is described by the usual matrix multiplication.