Localization of two-dimensional quantum walks

Abstract

The Grover walk, which is related to the Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is striking different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover's operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.

Localization of multi-state quantum walk in one dimension

“…To compare the probability distribution between the Hadamard walk and four-state quantum walk, Fig.1 shows the snapshot of probability distribution at t = 50 starting from an initial wave function: In contrast with the Hadamard walk we clearly find a single spike at the origin in four-state quantum walk. A similar spike has also been observed in twodimensional quantum walk by simulation [7] and the height of time-averaged probability at the origin is exactly calculated [8]. However, the probability distribution has not been obtained explicitly as a function of the location.…”

“…This basic condition is also satisfied in the two-dimensional Grover walk. As a result, the walk shows localization [8]. From this criterion we conclude that the Hadamard walk does not exhibit localization.…”

“…We know little about the spiky probability distribution in the later case. The existence of the spikes are already reported in the two-dimensional Grover walk [7,8], one-dimensional quantum walk as many coins [9], but, no probability distributions are calculated explicitly. In this paper we consider quantum walk with four internal states, which is the same as the quantum walk with many coins essentially, and compute analyti-cally the probability distribution.…”

Decoherence in quantum walks – a review

“…It was first used in quantum walks in Watrous (2001), and is the key ingredient in the quantum walk searching algorithm in Shenvi et al (2003) -see Section 4.1. Inui et al (2004) studied the localisation properties related to searching on a two-dimensional lattice. Szegedy (2004a;2004b) introduced a generalisation of the Grover coin that quantises an arbitrary Markov chain: essentially this allows for edge weights on the graph, and works for graphs of variable degree as well as regular graphs.…”

The QWalk simulator of quantum walks

“…In most situations the diagonal lattice provides better visualization than the natural lattice. This kind of non-diagonal lattice was used by Innui et al [15] with a slightly different shift operator. Our evolution equation, however, has the advantage of preserving the final probability distribution-except for the above mentioned rotation-with the same coin operator.…”