Relation between quantum walks with tails and quantum walks with sinks on finite graphs

TL;DR

This paper establishes a theoretical connection between quantum walks with sinks and tails on finite graphs, revealing how the long-term behavior of one relates to the eigenspaces of the other.

Contribution

It introduces a novel framework linking quantum walks with sinks to those with tails via eigenspace analysis, advancing understanding of quantum walk dynamics on graphs.

Findings

Survival probability characterized by centered eigenspace

Centered eigenspace acts as attractor for sinks

Persistent eigenspace relates to boundary-free support

Abstract

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in the graph theory.