Quantum Snake Walk on Graphs

TL;DR

This paper introduces the quantum snake walk, a new continuous-time quantum walk on graphs using fixed-length path states, analyzing its behavior on the line and exploring potential applications for solving complex graph problems.

Contribution

It presents the quantum snake walk, analyzes its localization and wave packet properties, and discusses its potential for quantum algorithms on graph connectivity problems.

Findings

Most states remain localized during evolution

Certain states form wave packets with inverse proportional momentum

Potential application in solving extended glued trees problem

Abstract

I introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. First I analyze the quantum snake walk on the line, and I show that, even though most states stay localized throughout the evolution, there are specific states which most likely move on the line as wave packets with momentum inversely proportional to the length of the snake. Next I discuss how an algorithm based on the quantum snake walk might potentially be able to solve an extended version of the glued trees problem which asks to find a path connecting both roots of the glued trees graph. No efficient quantum algorithm solving this problem is known yet.