A quantum walk on the half line with a particular initial state

Quantum walks are considered to be quantum counterparts of random walks. They show us impressive probability distributions which are different from those of random walks. That fact has been precisely proved in terms of mathematics and some of the results were reported as limit theorems. When we analyze quantum walks, some conventional methods are used for the computations; especially, the Fourier analysis has played a role to do that. It is, however, compatible with some types of quantum walks (e.g., quantum walks on the line with a spatially homogeneous dynamics) and cannot well work on the derivation of limit theorems for all the quantum walks. In this paper, we try to obtain a limit theorem for a quantum walk on the half line by the usage of the Fourier analysis. Substituting a quantum walk on the line for it, we will lead to a possibility that the Fourier analysis is useful to compute a limit distribution of the quantum walk on the half line.     

Laplacian versus adjacency matrix in quantum walk search

A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.     

Quantum walk in terms of quantum Bernoulli noises

Quantum Bernoulli noises are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we first present some new results concerning quantum Bernoulli noises, which themselves are interesting. Then, based on these new results, we construct a time-dependent quantum walk with infinitely many degrees of freedom. We prove that the walk has a unitary representation and hence belongs to the category of the so-called unitary quantum walks. We examine its distribution property at the vacuum initial state and some other initial states and find that it has the same limit distribution as the classical random walk, which contrasts sharply with the case of the usual quantum walks with finite degrees of freedom.     

An algebraic method for Schrödinger equations in quaternionic quantum mechanics

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.     

Quantum walk on distinguishable non-interacting many-particles and indistinguishable two-particle

We present an investigation of many-particle quantum walks in systems of non-interacting distinguishable particles. Along with a redistribution of the many-particle density profile we show that the collective evolution of the many-particle system resembles the single-particle quantum walk evolution when the number of steps is greater than the number of particles in the system. For non-uniform initial states we show that the quantum walks can be effectively used to separate the basis states of the particle in position space and grouping like state together. We also discuss a two-particle quantum walk on a two-dimensional lattice and demonstrate an evolution leading to the localization of both particles at the center of the lattice. Finally we discuss the outcome of a quantum walk of two indistinguishable particles interacting at some point during the evolution.     

Time averaged distribution of a discrete-time quantum walk on the path

The discrete time quantum walk which is a quantum counterpart of random walk plays important roles in the theory of quantum information theory. In the present paper, we focus on discrete time quantum walks viewed as quantization of random walks on the path. We obtain a weak limit theorem for the time averaged distribution of our quantum walks.     

Unitary equivalent classes of one-dimensional quantum walks

This study investigates unitary equivalent classes of one-dimensional quantum walks. We prove that one-dimensional quantum walks are unitary equivalent to quantum walks of Ambainis type and that translation-invariant one-dimensional quantum walks are Szegedy walks. We also present a necessary and sufficient condition for a one-dimensional quantum walk to be a Szegedy walk.     

Quantum walks on simplicial complexes

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflects not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of non-trivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.     

Equivalence of Szegedy’s and coined quantum walks

Szegedy’s quantum walk is a quantization of a classical random walk or Markov chain, where the walk occurs on the edges of the bipartite double cover of the original graph. To search, one can simply quantize a Markov chain with absorbing vertices. Recently, Santos proposed two alternative search algorithms that instead utilize the sign-flip oracle in Grover’s algorithm rather than absorbing vertices. In this paper, we show that these two algorithms are exactly equivalent to two algorithms involving coined quantum walks, which are walks on the vertices of the original graph with an internal degree of freedom. The first scheme is equivalent to a coined quantum walk with one walk step per query of Grover’s oracle, and the second is equivalent to a coined quantum walk with two walk steps per query of Grover’s oracle. These equivalences lie outside the previously known equivalence of Szegedy’s quantum walk with absorbing vertices and the coined quantum walk with the negative identity operator as the coin for marked vertices, whose precise relationships we also investigate.     

Discrete-time interacting quantum walks and quantum Hash schemes

Through introducing discrete-time quantum walks on the infinite line and on circles, we present a kind of two-particle interacting quantum walk which has two kinds of interactions. We investigate the characteristics of this kind of quantum walk and the time evolution of the two particles. Then we put forward a kind of quantum Hash scheme based on two-particle interacting quantum walks and discuss their feasibility and security. The security of this kind of quantum Hash scheme relies on the infinite possibilities of the initial state rather than the algorithmic complexity of hard problems, which will greatly enhance the security of the Hash schemes.     

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 limiting probabilities (localization), thereby distinguishing them 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 for the limiting 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 limiting probability decays exponentially in the absolute value of a walker??s position, independent of the parity of time.     

On the Hitting Times of Quantum Versus Random Walks

The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find—a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78:012310 2008) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains. Further, we prove that for any reversible ergodic Markov chain P, the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.     

The CGMV method for quantum walks

We review the main aspects of a recent approach to quantum walks, the CGMV method. This method proceeds by reducing the unitary evolution to canonical form, given by the so-called CMV matrices, which act as a link to the theory of orthogonal polynomials on the unit circle. This connection allows one to obtain results for quantum walks which are hard to tackle with other methods. Behind the above connections lies the discovery of a new quantum dynamical interpretation for well known mathematical tools in complex analysis. Among the standard examples which will illustrate the CGMV method are the famous Hadamard and Grover models, but we will go further showing that CGMV can deal even with non-translation invariant quantum walks. CGMV is not only a useful technique to study quantum walks, but also a method to construct quantum walks à la carte. Following this idea, a few more examples illustrate the versatility of the method. In particular, a quantum walk based on a construction of a measure on the unit circle due to F. Riesz will point out possible non-standard behaviours in quantum walks.     

Generalized teleportation by quantum walks

We develop a generalized teleportation scheme based on quantum walks with two coins. For an unknown qubit state, we use two-step quantum walks on the line and quantum walks on the cycle with four vertices for teleportation. For any d-dimensional states, quantum walks on complete graphs and quantum walks on d-regular graphs can be used for implementing teleportation. Compared with existing d-dimensional states teleportation, prior entangled state is not required and the necessary maximal entanglement resource is generated by the first step of quantum walk. Moreover, two projective measurements with d elements are needed by quantum walks on the complete graph, rather than one joint measurement with \(d^2\) basis states. Quantum walks have many applications in quantum computation and quantum simulations. This is the first scheme of realizing communicating protocol with quantum walks, thus opening wider applications.     

One-dimensional discrete-time quantum walks on random environments

We consider discrete-time nearest-neighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.     

基于一维三态量子游走的量子聚类算法

量子游走具有与经典随机游走不同的特性,因此它已经被用来解决包括元素区分、组合优化、图同构等问题。考虑量子游走和聚类两个领域,提出了一个基于一维三态离散量子游走的聚类算法。在该算法中,将数据点看作游走粒子;然后,这些粒子执行三态量子游走,接着根据粒子的测量结果更新数据点的属性值;最后,属于同一簇的数据点将会聚集,而属于不同簇的数据点将会分离。仿真实验结果表明了所提算法的有效性。     

On the relation between quantum walks and zeta functions

We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function. As applications, we obtain new proofs for the results on spectra of the transition matrix and its positive support.     

Discrete-time quantum walk on the Cayley graph of the dihedral group

The finite dihedral group generated by one rotation and one flip is the simplest case of the non-Abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we conduct the model of discrete-time quantum walk on the Cayley graph of the dihedral group by special coding mode. This construction makes Fourier transformation be used to carry out spectral analysis of the dihedral quantum walk, i.e. the non-Abelian case. Furthermore, the relation between quantum walk without memory on the Cayley graph of the dihedral group and quantum walk with memory on a cycle is discussed, so that we can explore the potential of quantum walks without and with memory. Here, the numerical simulation is carried out to verify the theoretical analysis results and other properties of the proposed model are further studied.     

Controllability of quantum walks on graphs

In this paper, we consider discrete time quantum walks on graphs with coin, focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. Controllability is verified when any unitary operation between two states can be implemented as a result of the evolution of the quantum walk. We prove general results and criteria relating controllability to the combinatorial and topological properties of the walk. In particular, controllability is verified if and only if the underlying graph is not a bipartite graph and therefore it depends only on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.