Recurrence in the classical random walk is well known and described by the Pólya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on cycles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum state revival will occur.
Recurrence of quantum walks on lattices can be characterized by the generalized Pólya number. Its value reflects the difference between a classical and a quantum system. The dimension of the lattice is not a unique parameter in the quantum case; both the coin operator and the initial quantum state of the coin influence the recurrence in a nontrivial way. In addition, the definition of the Pólya number involves measurement of the system. Depending on how measurement is included in the definition, the recurrence properties vary. We show that in the limiting case of frequent, strong measurements, one can approach the classical dynamics. Comparing various cases, we have found numerical indication that our previous definition of the Pólya number provides an upper limit.
The quantum mechanical generalisation of random walks (called Quantum Walks) present us with a broader spectrum of possibilities compared to their classical counterparts. The aim of the presented study is to explore a new portion of this area by incorporating a new step in the process of the Quantum Walk unique to quantum mechanics: the measurement. Our focus lies in the characterising number of the recurrence behaviour of the walk (Polya-number). We observe the effect of the standard projective measurement, a yes-no measurement on the origin and the effect of different measurement schemes (periodic and random) on the definition and the numeric value of the Polya-number.
Recurrence of a random walk is described by the Pólya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full revival of its quantum state. Localization for two-dimensional quantum walks is known to exist in the sense of non-vanishing probability distribution in the asymptotic limit. We show, on the example of the 2D Grover walk, that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for a four-state quantum walk. Stationary states and revivals result from interference, which has no counterpart in classical random walks.
We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.
A continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix
U(t)=e−itA, where is
A is the adjacency matrix of the graph representing the network. Certain instantaneous phenomena occuring in quantum walks can be utilized in quantum computing algorithms. The two-vertex phenomenon fractional revival occurs between vertices
u and
v at time
t if the columns of
U(t) corresponding to
u and
v are only supported on the rows indexed by those same two vertices. This means that given any initial state supported only on vertices
u and
v, the state of the system at time
t will again be supported only on those same two vertices. Fractional revival can be used to generate entanglement between two quantum particles. We will discuss
K-fractional revival, the generalization of fractional revival to
K, a larger subset of the vertices in our graph. We will also explore the related concept approximate
K-fractional revival, which denotes the existence of a sequence of times for which
U(t) comes arbitrarily close to achieving
K-fractional revival. Next, we will examine some spectral characterizations of these phenomena, which will allow us to determine if a given graph exhibits either
K-fractional revival or approximate
K-fractional revival. Lastly, we will detail families of examples of approximate
K-fractional revival on path graphs.--Author's abstract