  We formalize the treatment of directed (or chiral) quantum walks using
Hermitian adjacency matrices, bridging two developing fields of research in
quantum information and spectral graph theory. We display results and
simulations which highlight the conceptual differences between having
directions encoded in the Hamiltonians or not. This leads to a construction of
a new type of quantum phenomenon: zero transfer between pairs of sites in a
connected coupled network, which is only possible in the directed model we
study. Our main result is a description of several families of directed cycles
that admit zero transfer.
