Continuous-time quantum walks over connected graphs, amplitudes and invariants

Phillip Dukes

Published 2015-06-09Version 1

We examine the time dependent amplitude $ \phi_{i}\left( t\right)$ at each vertex of a continuous-time quantum walk on a variety of connected graphs. The Lissajous curve of the real vs. imaginary parts of each $ \phi_{i}\left( t\right)$ often reveals an interesting shape of the "space of accessible amplitudes." We find that, depending on the graph and initial state of the walker, the time evolution of the quantum walk can result in a static probability distribution over the vertices or have the walker never visit some connected vertices. We also find two invariants of continuous-time quantum walks. First, considering the rate at which each amplitude changes in time we find the scalar quantity $T = \sum_{i=0}^{n-1} \|\dfrac{d \phi_{i}\left( t\right) }{d t}\|^{2}$ is invariant. Second, the vector area of a continuous patch of contingent amplitude space at each vertex is time dependent; however the total vector area is invariant.