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We study the dynamics given by the iteration of a (half-line) CMV matrix withsparse, high barriers. Using an approach of Tcheremchantsev, we are able toexplicitly compute the transport exponents for this model in terms of the givenparameters. In light of the connection between CMV matrices and quantum walkson the half-line due to Cantero-Gr\"unbaum-Moral-Vel\'azquez, our result alsoallows us to compute transport exponents corresponding to a quantum walk whichis sparsely populated with strong reflectors. To the best of our knowledge,this provides the first rigorous example of a quantum walk which exhibitsquantum intermittency, i.e., nonconstancy of the transport exponents. Whencombined with the CMV version of the Jitomirskaya-Last theory of subordinacyand the general discrete-time dynamical bounds from Damanik-Fillman-Vance, weare able to exactly compute the Hausdorff dimension of the associated spectralmeasure.

Quantum Intermittency for Sparse CMV Matrices with an Application to Quantum Walks on the Half-Line