In this study we show a way of achieving the reverse evolution of n-dimensional quantum walks by introducing interventions on the coin degree of freedom during the forward progression of the coin-walker system. Only a single intervention is required to reverse a quantum walker on a line to its initial positon and the number of interventions increases with the dimensionality of the walk. We present an analytical treatment to prove these results. This reversion scheme can be used to generate periodic bounded quantum walks and to control the locations where particle can be found with highest probability. From the point of view of quantum computations and simulations, this scheme could be useful in resetting quantum operations and implementing certain quantum gates.
In this study we show a way of achieving the reverse evolution of n-dimensional quantum walks by introducing interventions on the coin degree of freedom during the forward progression of the coin-walker system. Only a single intervention is required to reverse a quantum walker on a line to its initial positon and the number of interventions increases with the dimensionality of the walk. We present an analytical treatment to prove these results. This reversion scheme can be used to generate periodic bounded quantum walks and to control the locations where particle can be found with highest probability. From the point of view of quantum computations and simulations, this scheme could be useful in resetting quantum operations and implementing certain quantum gates.
Abstract Full state revivals in a quantum walk can be viewed as returning of the walker to the initial quantum state in a periodic fashion during the propagation of the walk. In this paper we show that for any given number of spatial dimensions, a coin operator can be constructed to generate a quantum walk having full revivals with any desired period. From the point of view of quantum computation and simulations, these coin operators can be useful in implementing quantum walks, which oscillate between any two states with a finite periodicity.
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.
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
The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this paper, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a d-dimensional lattice governed by a c-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admit such a protocol. Moreover, we modify the quantity known as the P\'olya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.
We report on the possibility of controlling quantum random walks (QWs) with a step-dependent coin (SDC). The coin is characterized by a (single) rotation angle. Considering different rotation angles, one can find diverse probability distributions for this walk including: complete localization, Gaussian and asymmetric likes. In addition, we explore the entropy of walk in two contexts; for probability density distributions over position space and walker's internal degrees of freedom space (coin space). We show that entropy of position space can decrease for a SDC with the step-number, quite in contrast to a walk with step-independent coin (SIC). For entropy of coin space, a damped oscillation is found for walk with SIC while for a SDC case, the behavior of entropy depends on rotation angle. In general, we demonstrate that quantum walks with simple initiatives may exhibit a quite complex and varying behavior if SDCs are applied. This provides the possibility of controlling QW with a SDC.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator, constant in both space and time, and a localized initial state using numerical modeling. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study on general coin operators also includes the popular coins – Hadamard, Grover, and Fourier.