Boundary-induced coherence in the staggered quantum walk on different topologies

TL;DR

This paper investigates how changing boundary conditions and topology in a staggered quantum walk on a 2D lattice can induce coherence and alter quantum dynamics, with potential physical realization in superconducting resonators.

Contribution

It demonstrates that boundary modifications can unify independent 1D quantum walks into a 2D walk and explores topological effects on quantum coherence and entropy.

Findings

Boundary conditions can elevate two 1D walks to a 2D quantum walk.

Topology influences coherence and entropy levels in quantum walks.

Different surfaces like torus and Klein bottle affect quantum dynamics distinctly.

Abstract

The staggered quantum walk is a type of discrete-time quantum walk model without a coin which can be generated on a graph using particular partitions of the graph nodes. We design Hamiltonians for potential realization of the staggered dynamics on a two-dimensional lattice composed of superconducting microwave resonators connected with tunable couplings. The naive generalization of the one-dimensional staggered dynamics generates two uncoupled one-dimensional quantum walks thus more complex partitions need to be employed. However, by analyzing the coherence of the dynamics, we show that the quantumness of the evolution corresponding to two independent one-dimensional quantum walks can be elevated to the level of a single two-dimensional quantum walk, only by modifying the boundary conditions. In fact, by changing the lattice boundary conditions (or topology), we explore the walk on different surfaces such as torus, Klein bottle, real projective plane and sphere. The coherence and the entropy reach different levels depending on the topology of the surface. We observe that the entropy captures similar information as coherence, thus we use it to explore the effects of boundaries on the dynamics of the continuous-time quantum walk and the classical random walk.