Quantum Snake Walk on Graphs

Abstract

Quantum walks on graphs have been proven to be a useful tool in quantum algorithm construction for various problems. In this thesis we 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.

We first consider the quantum snake walk on the line. The analysis of the eigenvalues and the eigenvectors of the Hamiltonian governing the walk reveals that most states initially localized in a segment on the line always remain in that same segment. However, there are exponentially small (in the length of the snake) fraction of states which move on the line as wave packets with momentum inversely proportional to the length of the snake.

Next we show how an algorithm based on the quantum snake walk might 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. For that reason we consider a specific extension of the glued trees graph and analyze how the quantum snake walk behaves on it. In particular we show that the quantum snake walk on the infinite binary tree, restricted to certain superpositions, in many aspects is very similar to the quantum snake walk on the line. We also argue why the quantum snake walk, initialized in certain superpositions on one side of the glued trees graph, after certain amount of time is likely to be found on the other side of the graph. This seems to be crucial if we want our algorithm to work.