Faster transport with a directed quantum walk
Stephan Hoyer, DAVID A. MEYER · Physical Review A · 2009
We give an 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 $\ensuremath{\Theta}(1)$ in constant time compared to $\ensuremath{\Theta}(1/n)$ classically, independent of the number of these small dimensions. The analysis proceeds by reducing this walk to a walk with a two-dimensional coin.