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.
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
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.
We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.
Quantum walks have wide applications in quantum information, such as universal quantum computation, so it is important to explore properties of quantum walks thoroughly. We propose a novel method to implement discrete-time quantum walks (DTQWs) using only a single qubit, in which both coin and walker are encoded in the two-dimensional state space of a single qubit, operations are realized using single-qubit gates only, and high-dimensional final states of DTQWs can be obtained naturally. With this "one-qubit" approach, DTQW experiments can be realized much more easily, compared with previous methods, in most quantum systems and all properties based on quantum states of DTQWs (such as quantum correlation and coherence) can be investigated. By this approach, we experimentally implement one-particle and two-particle DTQWs with seven steps using single photons. Furthermore, we systematically investigate quantum correlations and coherence (based on the full state of the coin and walker) of the DTQW systems with different initial states of the coin, which have not been obtained and studied in DTQW experiments. As an application, we also study the assisted distillation of quantum coherence using the full state of the two-particle DTQW from the experiment. The maximal increase in distillable coherence for high-dimensional mixed states is investigated for the first time by obtaining its upper and lower bounds. Our work opens a new door to implement DTQW experiments and to better explore properties of quantum walks.
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.
We demonstrate a quantum walk with time-dependent coin bias. With this technique we realize an experimental single-photon one-dimensional quantum walk with a linearly ramped time-dependent coin flip operation and thereby demonstrate two periodic revivals of the walker distribution. In our beam-displacer interferometer, the walk corresponds to movement between discretely separated transverse modes of the field serving as lattice sites, and the time-dependent coin flip is effected by implementing a different angle between the optical axis of half-wave plate and the light propagation at each step. Each of the quantum-walk steps required to realize a revival comprises two sequential orthogonal coin-flip operators, with one coin having constant bias and the other coin having a time-dependent ramped coin bias, followed by a conditional translation of the walker.