  We consider the iteration of a unitary operator on a separable Hilbert space
and study the spreading rates of the associated discrete-time dynamical system
relative to a given orthonormal basis. We prove lower bounds for the transport
exponents, which measure the time-averaged spreading on a power-law scale, in
terms of dimensional properties of the spectral measure associated with the
unitary operator and the initial state. These results are the unitary analog of
results established in recent years for the dynamics of the Schr\"odinger
equation, which is a continuum-time dynamical system associated with a
self-adjoint operator. We discuss how these general results may be studied by
means of subordinacy theory in cases where the unitary operator is given by a
CMV matrix. An example of particular interest in which this scenario arises is
given by a time-homogeneous quantum walk on the integers. For the particular
case of the time-homogeneous Fibonacci quantum walk, we illustrate how these
components work together and produce explicit lower bounds for the transport
exponents associated with this model.
