Quantum walks with infinite hitting times

Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choice of coin. Hitting times are not very well-defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.Comment: 28 pages, 3 figures in EPS forma

Quantum walks on quotient graphs

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup used to construct it. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A {\bf 74}, 042334 (2006)] that the hitting time can be infinite. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with fast hitting times correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.Comment: 18 pages, 7 figures in EPS forma

Symmetric Colorings of the Hypercube and Hyperoctahedron

A self-complementary graph G is a subgraph of the complete graph K_n that is isomorphic to its complement. A self-complementary graph can be thought of as an edge 2-coloring of K_n that admits a color-switching automorphism. An automorphism of K_n that is color-switching for some edge 2-coloring is called a complementing automorphism. Complementing automorphisms for K_n have been characterized in the past by such authors as Sachs and Ringel. We are interested in extending this notion of self-complementary to other highly symmetric families of graphs; namely, the hypercube Q_n and its dual graph, the hyperoctahedron O_n. To that end, we develop a characterization of the automorphism group of these graphs and use it to prove necessary and sufficient conditions for an automorphism to be complementing. Finally, we use these theorems to construct a computer search algorithm which finds all self-complementary graphs in Q_n and O_n up to isomorphism for n=2,3,4