For irrational magnetic flux, the spectrum of the two-dimensional Hadamard magnetic quantum walk is a zero-measure Cantor set and the walk has no pure point spectrum, so the spectrum is purely singular continuous.
Universal quantum computation using the discrete time quantum walk
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abstract
A proof that continuous time quantum walks are universal for quantum computation, using unweighted graphs of low degree, has recently been presented by Childs [PRL 102 180501 (2009)]. We present a version based instead on the discrete time quantum walk. We show the discrete time quantum walk is able to implement the same universal gate set and thus both discrete and continuous time quantum walks are computational primitives. Additionally we give a set of components on which the discrete time quantum walk provides perfect state transfer.
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Singular continuous Cantor spectrum for magnetic quantum walks
For irrational magnetic flux, the spectrum of the two-dimensional Hadamard magnetic quantum walk is a zero-measure Cantor set and the walk has no pure point spectrum, so the spectrum is purely singular continuous.