  We study the dynamics given by the iteration of a (half-line) CMV matrix with
sparse, high barriers. Using an approach of Tcheremchantsev, we are able to
explicitly compute the transport exponents for this model in terms of the given
parameters. In light of the connection between CMV matrices and quantum walks
on the half-line due to Cantero-Gr\"unbaum-Moral-Vel\'azquez, our result also
allows us to compute transport exponents corresponding to a quantum walk which
is sparsely populated with strong reflectors. To the best of our knowledge,
this provides the first rigorous example of a quantum walk which exhibits
quantum intermittency, i.e., nonconstancy of the transport exponents. When
combined with the CMV version of the Jitomirskaya-Last theory of subordinacy
and the general discrete-time dynamical bounds from Damanik-Fillman-Vance, we
are able to exactly compute the Hausdorff dimension of the associated spectral
measure.
