# Eigenvalue gaps and continuous walks

A complex matrix is a *gain matrix* if it is Hermitian, its diagonal entries are zero, and any non-zero entry has norm one. We view a gain matrix as a weighted adjacency matrix of a graph, and one actual example is the adjacency matrix of a graph. For a second example, if $S$ is the skew adjacency matrix of an oriented graph, then $iS$ is a gain matrix. Given a gain matrix $H$, we have a continuous quantum walk given by the group of matrices $U(t)=\exp(itH)$ (for $t\in\mathbb{R}$).

We prove that if $\sigma$ is the minimum distance between two eigenvalues of an $n\times n$ gain matrix, then $\sigma^2 \le \frac{12}{n+1}$.

From this we derive two things. First: the proportion of graphs on $n$ vertices that admit perfect state transfer goes to zero as $n\to\infty$. Second: an oriented graph that admits perfect state transfer between any two vertices has at most 11 vertices. (One example is known where this happens, on three vertices. We did hope to find more.)