Coin Parameter Research Articles

Analytical solutions for quantum walks on 1D chain with different shift operators

In this paper, we study the discrete-time quantum walks on 1D Chain with the moving and swapping shift operators. We derive analytical solutions for the eigenvalues and eigenstates of the evolution operator Uˆ using the Chebyshev polynomial technique, and calculate the long-time averaged probabilities for the two different shift operators respectively. It is found that the probability distributions for the moving and swapping shift operators display completely different characteristics. For the moving shift operator, the probability distribution exhibits high symmetry where the probabilities at mirror positions are equal. The probabilities are inversely proportional to the system size N and approach to zero as N→∞. On the contrary, for the swapping shift operator, the probability distribution is not symmetric, the probability distribution approaches to a power-law stationary distribution as N→∞ under certain coin parameter condition. We show that such power-law stationary distribution is determined by the eigenstates of the eigenvalues ±1 and calculate the intrinsic probability for different starting positions. Our findings suggest that the eigenstates corresponding to eigenvalues ±1 play an important role for the swapping shift operator.

Continuous deformations of the Grover walk preserving localization

The three-state Grover walk on a line exhibits the localization effect characterized by a non-vanishing probability of the particle to stay at the origin. We present two continuous deformations of the Grover walk which preserve its localization nature. The resulting quantum walks differ in the rate at which they spread through the lattice. The velocities of the left and right-traveling probability peaks are given by the maximum of the group velocity. We find the explicit form of peak velocities in dependence on the coin parameter. Our results show that localization of the quantum walk is not a singular property of an isolated coin operator but can be found for entire families of coins.

Asymptotic distributions of quantum walks on the line with two entangled coins

We advance the previous studies of quantum walks on the line with two coins. Such four-state quantum walks driven by a three-direction shift operator may have nonzero stationary distributions (localization), thus distinguishing themselves from the quantum walks on the line in the basic scenario (i.e., driven by a single coin). In this work, asymptotic position distributions of the quantum walks are examined. We derive a weak limit for the quantum walks and explicit formulas of stationary probability distribution, whose dependencies on the coin parameter and the initial state of quantum walks are presented. In particular, it is shown that the weak limit for the present quantum walks can be of the form in the basic scenario of quantum walks on the line, for certain initial states of the walk and certain values of the coin parameter. In the case where localization occurs, we show that the stationary probability exponentially decays with the absolute value of a walker's position, independent of the parity of time.

Disordered-quantum-walk-induced localization of a Bose-Einstein condensate

We present an approach to induce localization of a Bose-Einstein condensate in a one-dimensional lattice under the influence of unitary quantum walk evolution using disordered quantum coin operation. We introduce a discrete-time quantum walk model in which the interference effect is modified to diffuse or strongly localize the probability distribution of the particle by assigning a different set of coin parameters picked randomly for each step of the walk, respectively. Spatial localization of the particle/state is explained by comparing the variance of the probability distribution of the quantum walk in position space using disordered coin operation to that of the walk using an identical coin operation for each step. Due to the high degree of control over quantum coin operation and most of the system parameters, ultracold atoms in an optical lattice offer opportunities to implement a disordered quantum walk that is unitary and induces localization. Here we present a scheme to use a Bose-Einstein condensate that can be evolved to the superposition of its internal states in an optical lattice and control the dynamics of atoms to observe localization. This approach can be adopted to any other physical system in which controlled disordered quantum walk can be implemented.

Discrete-time quantum walks on one-dimensional lattices

In this paper, we study discrete-time quantum walks on one-dimensional lattices. We find that the coherent dynamics depends on the initial states and coin parameters. For infinite size of lattices, we derive an explicit expression for the return probability, which shows scaling behavior P(0, t) ~ t -1 and does not depends on the initial states of the walk. In the long-time limit, the probability distribution shows various patterns, depending on the initial states, coin parameters and the lattice size. The time-averaged probability mixes to the limiting probability distribution in linear time, i.e., the mixing time M ε is a linear function of N (size of the lattices) for large values of thresholds ϵ. Finally, we introduce another kind of quantum walk on infinite or even-numbered size of lattices, and show that by the method of mathematical induction, the walk is equivalent to the traditional quantum walk with symmetrical initial state and coin parameter.

Quantum phase transition using quantum walks in an optical lattice

We present an approach using quantum walks (QWs) to redistribute ultracold atoms in an optical lattice. Different density profiles of atoms can be obtained by exploiting the controllable properties of QWs, such as the variance and the probability distribution in position space using quantum coin parameters and engineered noise. The QW evolves the density profile of atoms in a superposition of position space, resulting in a quadratic speedup of the process of quantum phase transition. We also discuss implementation in presently available setups of ultracold atoms in optical lattices.

Optimizing the discrete time quantum walk using a SU(2) coin

We present a generalized version of the discrete time quantum walk, using the SU(2) operation as the quantum coin. By varying the coin parameters, the quantum walk can be optimized for maximum variance subject to the functional form ${\ensuremath{\sigma}}^{2}\ensuremath{\approx}{N}^{2}$ and the probability distribution in the position space can be biased. We also discuss the variation in measurement entropy with the variation of the parameters in the SU(2) coin. Exploiting this we show how the quantum walk can be optimized for improving the mixing time in an $n$-cycle and for a quantum walk search.