  A continuous-time quantum walk on a graph $G$ is given by the unitary matrix
$U(t) = \exp(-itA)$, where $A$ is the Hermitian adjacency matrix of $G$. We say
$G$ has pretty good state transfer between vertices $a$ and $b$ if for any
$\epsilon > 0$, there is a time $t$, where the $(a,b)$-entry of $U(t)$
satisfies $|U(t)_{a,b}| \ge 1-\epsilon$. This notion was introduced by Godsil
(2011). The state transfer is perfect if the above holds for $\epsilon = 0$. In
this work, we study a natural extension of this notion called universal state
transfer. Here, state transfer exists between every pair of vertices of the
graph. We prove the following results about graphs with this stronger property:
(1) Graphs with universal state transfer have distinct eigenvalues and flat
eigenbasis (where each eigenvector has entries which are equal in magnitude).
(2) The switching automorphism group of a graph with universal state transfer
is abelian and its order divides the size of the graph. Moreover, if the state
transfer is perfect, then the switching automorphism group is cyclic. (3) There
is a family of prime-length cycles with complex weights which has universal
pretty good state transfer. This provides a concrete example of an infinite
family of graphs with the universal property. (4) There exists a class of
graphs with real symmetric adjacency matrices which has universal pretty good
state transfer. In contrast, Kay (2011) proved that no graph with real-valued
adjacency matrix can have universal perfect state transfer. We also provide a
spectral characterization of universal perfect state transfer graphs that are
switching equivalent to circulants.
