The nonlocalized case of the spatial density probability of the two-dimensional Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions. This significantly reduces the resources necessary for its feasible experimental realization. We present a formal proof of this correspondence and analyze the behavior of the coin-position entanglement as well as the x-y spatial entanglement in our scheme with respect to the Grover one. Our scheme allows us to entangle the two orthogonal directions of the walk more efficiently.
The nonlocalized case of the spatial density probability of the two-dimensional Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions. This significantly reduces the resources necessary for its feasible experimental realization. We present a formal proof of this correspondence and analyze the behavior of the coin-position entanglement as well as the x-y spatial entanglement in our scheme with respect to the Grover one. Our scheme allows us to entangle the two orthogonal directions of the walk more efficiently.
We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of x-y spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.
One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed on the coin degree of freedom after the evolution, allowing a further comparison with the standard two-dimensional Grover walk.
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
We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.
Exploiting multidimensional quantum walks as feasible platforms for quantum computation and quantum simulation attracts constantly growing attention from a broad experimental physics community. Here, we propose a two-dimensional quantum walk scheme with a single-qubit coin that presents, in the considered regimes, a strong localizationlike effect on the walker. The result could provide new possible directions for the implementation of quantum algorithms or from the point of view of quantum simulation. We characterize the localizationlike effect in terms of the parameters of a step-dependent qubit operation that acts on the coin space after any standard coin operation, showing that a proper choice can guarantee a nonnegligible probability of finding the walker in the origin even for large times. We finally discuss the robustness to imperfections, a qualitative relation with coherences behavior, and possible experimental realizations of this model with the current state-of-the-art settings.
Entanglement with single-particle states is advantageous in quantum technology because of the states' ability to encode and process information more securely than their multiparticle analogs. Three-way- and nonlocal two-way-entangled single-particle states are desirable in this context. Herein we generate genuine three-way entanglement from an initially separable state involving three degrees of freedom of a quantum particle, which evolves via a two-dimensional alternate quantum walk employing a resource-saving single-qubit coin. We achieve maximum possible values for the three-way entanglement quantified by the $\ensuremath{\pi}$-tangle between the three degrees of freedom. We also generate optimal nonlocal two-way entanglement, quantified by the negativity between the nonlocal position degrees of freedom of the particle. This prepared architecture using quantum walks can be experimentally realized with a photon.