One-dimensional Quantum Random Walk Research Articles

Entangling power of disordered quantum walks

We investigate how the introduction of different types of disorder affects the generation of entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional quantum random walks (QRW). Disorder is modeled by adding another random feature to QRW, i.e., the quantum coin that drives the system's evolution is randomly chosen at each position and/or at each time step, giving rise to either dynamic, fluctuating, or static disorder. The first one is position-independent, with every lattice site having the same coin at a given time, the second has time and position dependent randomness, while the third one is time-independent. We show for several levels of disorder that dynamic disorder is the most powerful entanglement generator, followed closely by fluctuating disorder. Static disorder is the less efficient entangler, being almost always less efficient than the ordered case. Also, dynamic and fluctuating disorder lead to maximally entangled states asymptotically in time for any initial condition while static disorder has no asymptotic limit and, similarly to the ordered case, has a long time behavior highly sensitive to the initial conditions.

A Novel Algorithm of Quantum Random Walk in Server Traffic Control and Task Scheduling

A quantum random walk optimization model and algorithm in network cluster server traffic control and task scheduling is proposed. In order to solve the problem of server load balancing, we research and discuss the distribution theory of energy field in quantum mechanics and apply it to data clustering. We introduce the method of random walk and illuminate what the quantum random walk is. Here, we mainly research the standard model of one-dimensional quantum random walk. For the data clustering problem of high dimensional space, we can decompose onem-dimensional quantum random walk intomone-dimensional quantum random walk. In the end of the paper, we compare the quantum random walk optimization method with GA (genetic algorithm), ACO (ant colony optimization), and SAA (simulated annealing algorithm). In the same time, we prove its validity and rationality by the experiment of analog and simulation.

Optical implementation of quantum random walks using weak cross-Kerr media

Weak cross-Kerr media provides additional degrees of freedom of qubits in quantum information processing. In this paper, by exploiting weak cross-Kerr nonlinearity, we propose an optical implementation scheme of one-dimensional quantum random walks. The random walks are described by the interaction of single photons with cross-Kerr media. The proposed scheme can also be used to implement one-dimensional quantum random walks on an infinite line.

One-dimensional quantum random walks with two entangled coins

We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRWs). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called ``localization,'' whereby the probability distribution of the walker's position is seen to exhibit a persistent major ``spike'' (or ``peak'') at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRWs of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location eventually becomes stationary and nonvanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator $U(k)$. An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases quadratically for large values of the position $x$. We locate the two minor spikes and demonstrate that their positions are determined by the phases of nondegenerated eigenvalues of $U(k)$. Finally, for fixed time $t$ sufficiently large, we examine the dependence on $t$ of the probability of finding a particle at a given location $x$.

Limiting distribution of decoherent quantum random walks

The behavior of one-dimensional quantum random walks is strikingly different from that of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we study the decoherence on both position and ``coin'' spaces of the particle. We propose an analytical approach to investigate these phenomena and obtain the generating functions which encode all the features of these walks. Specifically, from these generating functions, we find exact analytic expressions of several moments for the time and noise dependence of position. Moreover, the limiting position distributions of decoherent quantum random walks are shown to be Gaussian in an analytical manner. These results explicitly describe the relationship between the system and the level of decoherence.

Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons

Quantum random walks have attracted special interest because they could lead to new quantum algorithms. Photons can carry orbital angular momentum (OAM) thereby offering a practical realization of a high-dimensional quantum information carrier. By employing OAM of photons, we experimentally realized the one-dimensional discrete-time quantum random walk. Three steps of a one-dimensional quantum random walk were implemented in our protocol showing the obvious difference between quantum and classical random walks.

Quantum random walks in one dimension via generating functions

We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.

Optical implementation of one-dimensional quantum random walks using orbital angular momentum of a single photon

Photons can carry orbital angular momentum (OAM), which offers a practical realization of a high-dimensional quantum information carrier. In this paper, by employing OAM of a single photon, we propose an experimental scheme to implement one-dimensional two-state quantum random walks on an infinite line. Furthermore, we show that the scheme can be used to implement one-dimensional three-state quantum random walks on an infinite line.

A new type of limit theorems for the one-dimensional quantum random walk

In this paper we consider the one-dimensional quantum random walk X n ϕ at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U . We give a combinatorial expression for the characteristic function of X n ϕ . The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state ϕ . As a consequence, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X n ϕ / n converges weakly to a limit Z ϕ as n → ∞ , where Z ϕ has a density 1 / π ( 1 - x 2 ) 1 - 2 x 2 for x ∈ ( - 1 / 2 , 1 / 2 ) . Moreover we discuss some known simulation results based on our limit theorems.

Symmetry of Distribution for the One-Dimensional Hadamard Walk

In this paper we study a one-dimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Hadamard walk.

Limit theorems and absorption problems for quantum random walks in one dimension

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS...

Limit theorems and absorption problems for quantum radom walks in one dimension

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by $2 \times 2$ unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

One-dimensional quantum random walk for fermions and bosons.

With the help of quantum-scattering-theory methods and the approximation of the stationary phase, we propose a one-dimensional quantum-random-walk (QRW) model, which describes for both tunneling and scattering above the potential, the coherent diffusion of independent particles described by wave packets in a periodic one-dimensional lattice. The QRW model describes for each lattice cell the time evolution of modulating amplitudes of two opposite-moving wave packets as they are scattered by periodic potential barriers. Since the QRW model is a coherent process, interference contributions in the probabilities bring about strong departures from classical results. For many identical free particles we obtain the theoretical and graphical Bose and Fermi two-body QRW probability distribution. The result is generalized to N identical free particles and we obtain the N-body Bose and Fermi QRW probability distribution.