Pólya number of the continuous-time quantum walks

Abstract

We propose a definition for the Pólya number of continuous-time quantum walks to characterize their recurrence properties. The definition involves a series of measurements on the system, each carried out on a different member from an ensemble in order to minimize the disturbance caused by it. We examine various graphs, including the ring, the line, the higher-dimensional integer lattices, and a number of other graphs, and we calculate their Pólya number. For the timing of the measurements, a Poisson process as well as regular timing are discussed. We find that the speed of decay for the probability at the origin is the key for recurrence.

Quantized recurrence time in unital iterated open quantum dynamics

“…In this latter case, it should be possible to cast the requirement of unitality, as well as the dimension of the relevant Hilbert space, in a simple formula for the Lindblad operators of the master equation. As an aside, there is a continuous-time generalization of the ensemble approach, with measurements that are either randomly distributed or regularly timed 34 . Our results give a concrete quantitative measure of the size of the part of the Hilbert space accessible from |Ψ .…”

Transport properties of continuous-time quantum walks on Sierpinski fractals

“…Ref. [34] suggests a possible quantum definition for the Pólya number, which is directly related to the return proba-bility to the initial node (|ψ(0) = |1 ):…”

“…∞} is an infinite time series which can be chosen regularly or be determined by some random process. It can be shown that its value depends on the convergence speed of π 1,1 (t) to zero: if π 1,1 (t) converges faster than t −1 then the CTQW is transient, otherwise it is recurrent [34]. For a finite network of N sites the probability that we find the walker at the origin can be written as a finite sum of cosine functions,…”

Recurrence and Pólya Number of General One-Dimensional Random Walks

“…As a consequence, Pólya showed that for one and two dimensional infinite lattices the walks are recurrent, while for three dimension or higher dimensions the walks are transient and a unique Pólya number is calculated for them [8]. Recently, M.Štefaňák et al extend the concept of Pólya number to characterize the recurrence properties of quantum walks [9][10][11]. They point out that the recurrence behavior of quantum walks is not solely determined by the dimensionality of the structure, but also depend on the topology of the walk, choice of coin operators, and the initial coin state, etc [9][10][11].…”

“…Recently, M.Štefaňák et al extend the concept of Pólya number to characterize the recurrence properties of quantum walks [9][10][11]. They point out that the recurrence behavior of quantum walks is not solely determined by the dimensionality of the structure, but also depend on the topology of the walk, choice of coin operators, and the initial coin state, etc [9][10][11]. This suggests the Pólya number of random walks or quantum walks may depends on a variety of ingredients including the structural dimensionality and model parameters.…”