Here, There, Everywhere: Random walks are a powerful mathematical method that can be used to simulate certain processes in biology, chemistry, or even the stock market. They present a statistical method for mapping the possible routes that processes can take. Quantum walks are expected to be able to probe multiple paths simultaneously. Quantum walks have been demonstrated for one-dimensional, or straight-line, walks. Now, Schreiber et al. (p. 55 , published online 8 March) demonstrate an optical system that can simulate quantum walks over a two-dimensional system, thereby providing the capability of describing much more complex processes.

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.

The Pólya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. Štefaňák et al., Phys. Rev. Lett. 100, 020501 (2008)] which is based on a specific measurement scheme. The Pólya number of a quantum walk depends, in general, on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by Hadamard tensor-product coins, the Pólya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an estimation of the Pólya number for this class of quantum walks. Second, we examine the two-dimensional Grover walk, which exhibits localization and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with classical walks which are recurrent only for the dimensions $d=1,2$. Finally, we analyze the recurrence of the 2D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an estimation of the Pólya number in its dependence on the initial state.

Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices $N$ in $O\left(\sqrt{N}\right)$ steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large $N$.

The effects of projective measurements on the quantum mechanical evolution of a particle can be simulated with coherent light.