Faster transport with a directed quantum walk

TL;DR

This paper demonstrates a quantum walk on a directed graph that achieves faster transport than classical methods, utilizing self-loops and Fourier transform coins to enhance efficiency.

Contribution

It introduces the first example of accelerated quantum transport on a directed graph with self-loops, reducing the walk to a simpler two-dimensional coin model.

Findings

Quantum walk achieves constant-time transport on a directed line.

Speedup is independent of the number of self-loops.

Reduction to a two-dimensional coin simplifies analysis.

Abstract

We give the first example of faster transport with a quantum walk on an inherently directed graph, on the directed line with a variable number of self-loops at each vertex. These self-loops can be thought of as adding a number of small dimensions. This is a discrete time quantum walk using the Fourier transform coin, where the walk proceeds a distance $\Theta(1)$ in constant time compared to $\Theta(1/n)$ classically, independent of the number of these small dimensions. The analysis proceeds by reducing this walk to a walk with a two dimensional coin.