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

TL;DR

This paper investigates continuous-time quantum walks on graphs, revealing invariants related to amplitude evolution and probability distributions, and provides formulas connecting these invariants to graph spectra.

Contribution

It introduces two invariants of quantum walks, including a new measure related to amplitude evolution and its maximum value linked to the graph's Laplace spectrum.

Findings

The quantity T is a time-invariant sum of squared amplitude derivatives.

The maximum of the minimized T, T_min^{max}, relates to the graph's maximum Laplace eigenvalue.

A universal relation for probability distributions in quantum walks on connected graphs.

Abstract

We examine the time dependent amplitude $ \phi_{j}\left( t\right)$ at each vertex $j$ of a continuous-time quantum walk on the cycle $C_{n}$. In many cases the Lissajous curve of the real vs. imaginary parts of each $ \phi_{j}\left( t\right)$ reveals interesting shapes of the space of time-accessible amplitudes. We find two invariants of continuous-time quantum walks. First, considering the rate at which each amplitude evolves in time we find the quantity $T = \displaystyle\sum_{j=0}^{n-1} \lvert\dfrac{d \phi_{j}\left( t\right)}{d t}\rvert^{2}$ is time invariant. The value of $T$ for any initial state can be minimized with respect to a global phase factor $e^{i \theta t}$ to some value $T_{min}$. An operator for $T_{min}$ is defined. For any simply connected graph $g$ the highest possible value of $T_{min}$ with respect to the initial state is found to be $T_{min}^{max}=\left( \frac{\lambda_{max}}{2}\right)^{2}$ where $\lambda_{max}$ is the maximum eigenvalue in the Laplace spectrum of $g$. A second invariant is found in the time-dependent probability distribution $P_{j}\left(t\right) = \lvert\phi_{j}\left(t\right)\rvert^{2}$ of any initial state satisfying $T_{min}^{max}$, with these conditions $\displaystyle\sum_{j=0}^{n-1}\left(P_{j}^{max} - P_{j}^{min}\right)^{2} = \dfrac{4}{n}$ for all simply connected graphs of $n$ vertices.