Title: Topological classification of time-asymmetry in unitary quantum processes,
Abstract: Effective gauge fields have allowed the emulation of matter under strong
magnetic fields leading to the realization of Harper-Hofstadter, Haldane
models, and led to demonstrations of one-way waveguides and topologically
protected edge states. Central to these discoveries is the chirality induced by
time-symmetry breaking. Due to the discovery of quantum search algorithms based
on walks on graphs, recent work has discovered new implications the effect of
time-reversal symmetry breaking has on the transport of quantum states and has
brought with it a host of new experimental implementations. We provide a full
classification of the unitary operators defining quantum processes which break
time-reversal symmetry in their induced transition properties between basis
elements in a preferred site-basis. Our results are furthermore proven in terms
of the geometry of the corresponding Hamiltonian support graph and hence
provide a topological classification. A quantum process of this type is
necessarily time-symmetric for any choice of time-independent Hamiltonian if
and only if the underlying support graph is bipartite. Moreover, for
non-bipartite support, there exists a time-independent Hamiltonian with
necessarily complex edge weights that induces time-asymmetric transition
probabilities between edge(s). We further prove that certain bipartite graphs
give rise to transition probability suppression, but not broken time-reversal
symmetry. These results fill an important missing gap in understanding the role
this omnipresent effect has in quantum physics. Furthermore, through our
development of a general framework, along the way to our results we completely
characterize gauge potentials on combinatorial graphs.