Discrete-time Quantum Walk Model Research Articles

The uniform measure for quantum walk on hypercube: A quantum Bernoulli noises approach

In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators ∂k*+∂k behave actually as the shift operators, where ∂k and ∂k* are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above-mentioned operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.

Perfect state transfer in Grover walks between states associated to vertices of a graph

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time.

Formation of a Morphogen Gradient - Acceleration by Quantum Walks

The establishment of graded concentration distributions of certain signalling molecules, called morphogens, is crucial for the successful embryonic development of multicellular organisms. The morphogen system that has been studied in the greatest detail is the Bicoid (Bcd) gradient established during the early embryonic development of fruit flies and provides the embryo cells with the necessary positional information for the proper implementation of the body plan of the organism. Both continuous as well as discrete-state classical random-walk models fail to provide a completely satisfactory explanation of the dynamics of Bcd gradient formation within the developmentally relevant timescales. By presuming the existence of an arbitrary chirality degree of freedom that characterizes each Bcd molecule, we here therefore develop a discrete-time quantum walk model of the process. Our approach is based on the recent suggestion that high temperatures and random interactions need not necessarily destroy many-body quantum coherence. In this way we are led to a system of non-classical master equations bearing an interference term that can be numerically solved on a computer. Our analysis predicts a large value of diffusion rate for the Bcd system in the very early nuclear division cycles that needs to be verified experimentally. Additionally, with the chirality and position degrees of freedom at our disposal entanglement shows up as a natural consequence in the system. Taking inspiration from a very recent demonstration of the existence of entanglement phenomenon in a hot, strongly-interacting atomic system, the discussion takes the point of view that the emergence of geometry in the embryo system is consequence of quantum entanglement that needs further understanding and analysis.

Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises.

As a discrete-time quantum walk model on the one-dimensional integer lattice , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

Discrete-Time Quantum Walk with Memory on the Cayley Graph of the Dihedral Group

By adding one-step memory to enrich the model of discrete-time quantum walk on the Caylay graph of the dihedral group, the model of quantum walk with memory on the Cayley graph of the dihedral group is constructed. The Fourier Transform is used to analyze the walk, and a formula for probability distribution and time-average probability distribution is given. According to the one-to-one correspondence between quantum walk with memory on a regular graph and quantum walk without memory on the corresponding line digraph, the diagrammatic counterpart of quantum walk with one-step memory on the Cayley graph of the dihedral group is presented. Moreover, basic probabilistic properties of the proposed model are further studied using numerical simulation method. Furthermore, the similarities and differences are discussed in comparison to the memoryless model.

Eigenbasis of the evolution operator of 2-tessellable quantum walks

Staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well-known discrete-time quantum walk models. In this work, we address the class of 2-tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator. By interpreting the evolution operator as a quantum Markov chain on an underlying multigraph, we define the concept of quantum detailed balance, which helps to obtain the eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum Markov chain. To obtain the remaining eigenvectors, we have to use the quantum detailed balance conditions. If the quantum Markov chain has a quantum detailed balance, there is an eigenvector for each fundamental cycle of the underlying multigraph. If the quantum Markov chain does not have a quantum detailed balance, we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors. We exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice (the line graph of the hexagonal lattice), which has symmetry properties that help in the calculation process.

Mimicking the Hadamard discrete-time quantum walk with a time-independent Hamiltonian

We implement the discrete-time quantum walk model using the continuous-time evolution of the Hamiltonian that includes both the shift and the coin generators. Based on the Trotter-Suzuki first-order approximation, we consider an optimization problem in which the Hellinger distance between the walker probability distributions resulted from the evolution of such Hamiltonian and the quantum walk dynamics is minimized. The phase space implementation of the quantum walk is considered where the walker state is encoded on the coherent state of a resonator and the coin on the two-level state of a qubit. In this approach, no mechanism for switching between the coin and the shift operators is included. We show the Hellinger distance is bounded for large number of time steps. The distance is small when we deviate from the standard quantum walk model, namely, when the walker is allowed to move in between the sites. In simulating the standard quantum walk model, the distance is large but bounded by $26\%$ for a relevant number of time steps. Even so, the system evolution shows the essential characteristics of the standard quantum walk dynamics, namely, the ballistic evolution of the probability distribution and the linear growth of the corresponding standard deviation. Moreover, the entanglement generated in this approach is approximately the same as the entanglement generated in the standard quantum walk dynamics. Finally, the system dynamics under the influence of the decoherence also shows the similar quantum-to-classical transition as in the standard quantum walk dynamics.

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.

Boundary-induced coherence in the staggered quantum walk on different topologies

The staggered quantum walk is a type of discrete-time quantum walk model without a coin which can be generated on a graph using particular partitions of the graph nodes. We design Hamiltonians for potential realization of the staggered dynamics on a two-dimensional lattice composed of superconducting microwave resonators connected with tunable couplings. The naive generalization of the one-dimensional staggered dynamics generates two uncoupled one-dimensional quantum walks thus more complex partitions need to be employed. However, by analyzing the coherence of the dynamics, we show that the quantumness of the evolution corresponding to two independent one-dimensional quantum walks can be elevated to the level of a single two-dimensional quantum walk, only by modifying the boundary conditions. In fact, by changing the lattice boundary conditions (or topology), we explore the walk on different surfaces such as torus, Klein bottle, real projective plane and sphere. The coherence and the entropy reach different levels depending on the topology of the surface. We observe that the entropy captures similar information as coherence, thus we use it to explore the effects of boundaries on the dynamics of the continuous-time quantum walk and the classical random walk.

Discrete-time quantum walk with feed-forward quantum coin

Constructing a discrete model like a cellular automaton is a powerful method for understanding various dynamical systems. However, the relationship between the discrete model and its continuous analogue is, in general, nontrivial. As a quantum-mechanical cellular automaton, a discrete-time quantum walk is defined to include various quantum dynamical behavior. Here we generalize a discrete-time quantum walk on a line into the feed-forward quantum coin model, which depends on the coin state of the previous step. We show that our proposed model has an anomalous slow diffusion characterized by the porous-medium equation, while the conventional discrete-time quantum walk model shows ballistic transport.

Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models

The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position-dependent transition amplitudes and any topology (Andrade et al 2009 Phys. Rev. A 80 052301). Although the proof explicitly describes the mapping obtention, its high technicality may hinder relevant physical aspects involved in the equivalence. Discussing concrete examples—the most general constructions for the line, square and honeycomb lattices—here we unveil the similarities and differences of these two versions of quantum walks. We moreover show how to derive the dynamics of one from the other by means of proper projections. We perform calculations for different probability amplitudes such as Hadamard, Grover, discrete Fourier transform and the uncommon in the area (but interesting) discrete Hartley transform, comparing the evolutions. Our study illustrates the models’ interplay, an important issue for implementations and applications of such systems.

Limit theorems for quantum walks associated with Hadamard matrices

We study a one-parameter family of discrete-time quantum walk models on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ is proved. Symmetrization on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on $\mathbb{Z}$. A necessary condition for the existence of a phenomenon known as localization is given.

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.

Limit distributions of two-dimensional quantum walks

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, ${X}_{t}/t$ and ${Y}_{t}/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.