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arXiv:1506.03086

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

Phillip Dukes

Published Jun 9, 2015• 0 citations
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Abstract

We examine the time dependent amplitude $ ϕ_{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 $ ϕ_{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 $T = \displaystyle\sum_{j=0}^{n-1} \lvert\dfrac{d ϕ_{j}\left( t\right)}{d t}\rvert^{2}$ is time invariant. The value of $T$ for any initial state can be minimized with respect to a global phase factor $e^{i θt}$ 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 $T_{min}^{max}=\left( \frac{λ_{max}}{2}\right)^{2}$ where $λ_{max}$ is the maximum eigenvalue in the Laplace spectrum of $g$. A second invariant is found in the time-dependent probability distribution $P_{j}\left(t\right) = \lvertϕ_{j}\left(t\right)\rvert^{2}$ 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.

DOI
10.48550/arxiv.1506.03086

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Continuous-time quantum walk (CTQW) on a given graph is investigated using the techniques of the spectral analysis and inverse Laplace transform of the Stieltjes function (Stieltjes transform of the spectral distribution) associated with the graph. It is shown that the probability amplitude of observing the CTQW at a given site at time t is related to the inverse Laplace transformation of the Stieltjes function, namely, one can calculate the probability amplitudes only by taking the inverse laplace transform of the function iGμ(is), where Gμ(x) is the Stieltjes function of the graph. The preference of this procedure is that there is no need to know the spectrum of the graph.
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