Quantum walks with infinite hitting times

Abstract

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.

Decoherence in quantum walks – a review

“…Analogous results for coined quantum walks on the 'glued trees' graph were noted in Tregenna et al (2003). Further work by Krovi and Brun on the question of whether and under what conditions quantum walks show dramatically different properties (speed up or slow down) compared with classical random walks suggests that it is highly dependent on the symmetry of the graph (Krovi and Brun 2006b). For the Grover coin, this is exemplified by the quantum walk search algorithm of Shenvi et al (2003) (which is described in Section 4.1), where any disturbance of the symmetry causes the walker to converge on the marked state.…”

Quantum walks on quotient graphs

“…(46) is not always well defined, because the matrix I−N may not be invertible. In [23], it is shown that when this matrix is not invertible, then the quantum walk has an infinite hitting time for certain initial states. An infinite hitting time means that the probability that the particle reaches the final vertex at any time step (i.e., ∞ t=0 p(t)) is less than unity.…”

“…In such a walk, after the application of the unitary evolution operator, a measurement is performed to see if the particle is in the final vertex or not. In [23] it was shown that graphs with sufficient symmetry can have infinite hitting times for certain initial states, a phenomenon with no classical analogue.…”

“…We determine the structure of the quantum walk on the quotient graph. We then apply the analysis of hitting times developed in [22] and [23] to quotient graphs, and investigate the possibility of both infinite hitting times and reduced hitting times on quotient graphs. This paper is organized as follows.…”

“…There has also been considerable work on other regular graphs. The N -cycle is treated in [18,19], and the hypercube in [6,20,21,22,23]. Quantum walks on general undirected graphs are defined in [24,25], and on directed graphs in [26].…”