  Dynamical evolution of systems with sparse Hamiltonians can always be
recognized as continuous time quantum walks (CTQWs) on graphs. In this paper,
we analyze the short time asymptotics of CTQWs. In recent studies, it was shown
that for the classical diffusion process the short time asymptotics of the
transition probabilities follows power laws whose exponents are given by the
usual combinatorial distances of the nodes. Inspired by this result, we perform
a similar analysis for CTQWs both in closed and open systems, including
time-dependent couplings. For time-reversal symmetric coherent quantum
evolutions, the short time asymptotics of the transition probabilities is
completely determined by the topology of the underlying graph analogously to
the classical case, but with a doubled power-law exponent. Moreover, this
result is robust against the introduction of on-site potential terms. However,
we show that time-reversal symmetry breaking terms and non-coherent effects can
significantly alter the short time asymptotics. The analytical formulas are
checked against numerics, and excellent agreement is found. Furthermore, we
discuss in detail the relevance of our results for quantum evolutions on
particular network topologies.
