Localization in two-dimensional alternate quantum walks with periodic coin operations

Abstract

The field of quantum computation and quantum simulation has been recently driven to a new rising edge by the experimental realization of quantum walks in various setups, highlighting that different physical systems can be adapted for the implementation of these models. In particular, optical systems have shown their full potential, allowing the experimental demonstration of two-dimensional quantum walks for the first time [1, 2], even if several further progresses will sur ely be obtained also in the other physical scenarios that have been already exploited for the one-dimensional case [3, 4]. From the theoretical point of view, the interest in twodimensional quantum walks has been boosted by the fact that, differently from the one-dimensional version, highe rdimensional schemes (i.e., walkers moving on structures with dimension larger than one) can be exploited for the efficient implementation of quantum search algorithms [5]. In particular, the Grover walk has been intensively studied due to its localization feature [6]. It has been proved that the non localized case of the Grover walk can be simulated by a walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions i n which the walker can move [7]. In the quest for more feasible quantum walk models, an important step forward is presented in this paper: a strong localization-like effect can be indeed obtained in the modified version of the alternate quantum walk presented here, reducing the gap with the standard two-dimensional version. This result could pave the way for adapting this scheme to the realization of quantum algorithms, providing a clear advantage in terms of experimental resources. After a short introduct ion on the quantum walk studied here, we characterize it in terms of its relevant parameters and show that a proper choice can guarantee a non-negligible probability of finding the walke r at the origin even for large times. We then hint at a qualitative relation between the ability of the system to localize and th e behavior of coherences established in the state of the parti cles performing a two-particle equivalent scheme of the walk, which in turns provides information on the way correlations are set up between them. We discuss the robustness of this model to imperfections and its possible experimental realization with the current state-of-the-art settings in linear o ptics and cold-atom devices. Let us consider a quantum system with two degrees of freedom, and thus described by a vector in the composite Hilbert space H = HW ⊗ HC . The coin space HC is a two-dimensional Hilbert space spanned by {|0i,|1i} and the walker space HW is an infinite-dimensional Hilbert space spanned by {|x,yi}, with x and y assuming all possible integer values. We take as a basis of this space H the set {|x,y,ci}, with |x,y,ci= |x,yiW ⊗ |ciC ; x and y could denote, for instance, the position of a particle ( walker) along the x and y directions, respectively, while |ciC is an internal twolevel degree of freedom. From now on, we consider|0,0iW as the initial state of our walker, and |+yiC = (|0iC+i|1iC)/ √ 2 as the coin one. The evolution of the system is given by a sequence of conditional shift, coin operations, and phase gates. The effec t of the two different conditional shift operations