The properties of the coinless quantum walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on $d$-dimensional lattices. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices and, using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.
A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely-studied coined walk. Coined walks require the direct product of the site basis with the coin space, coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argu…
Higher-order topological insulators not only exhibit exotic bulk-boundary correspondence principle, but also have an important application in quantum computing. However, they have never been achieved in quantum walk. In this paper, we construct a two-dimensional coinless discrete-time quantum walk to simulate second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after multi-step discrete-time quantum walks. Furthermore, we demonstrate the robustness of the topolo…
The spatial search problem on regular lattice structures in integer number of dimensions $d\geq2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here we investigate the spatial search problem on fractals of non-integer dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (an…
We present an implementation scheme for a quantum walk in the orbital angular momentum space of a laser beam. The scheme makes use of a ring interferometer, containing a quarter-wave plate and a q plate. This setup enables one to perform an arbitrary number of quantum walk steps. In addition, the classical nature of the implementation scheme makes it possible to observe the quantum walk evolution in real time. We use nonquantum entanglement of the laser beam's polarization with its orbital angular momentum to implement the quantum walk.
We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time.
We consider a new model of quantum walk on a one-dimensional momentum space that includes both discrete jumps and continuous drift. Its time evolution has two stages; a Markov diffusion followed by localized dynamics. As in the well known quantum kicked rotor, this model can be mapped into a localized one-dimensional Anderson model. For exceptional (rational) values of its scale parameter, the system exhibits resonant behavior and reduce to the usual discrete time quantum walk on the line.
Photons can carry spin angular momentum (SAM) and orbital angular momentum (OAM), which can be used to realize a qubit system and a high-dimension system respectively. This spin-orbital system is very suitable for implementing one-dimensional discrete-time quantum random walks. We propose a simple scheme of quantum walks on the spin-orbital angular momentum space of photons, where photons walk on the infinity OAM space controlled by their SAM. By employing the recent invention of an optical device, the so-called 'q-plate', our scheme is more simple and efficient than others because there is no…
A Quantum Walk (QW) simulating the flat $(1 + 2)$D Dirac Eq.\ on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group, but rather of its double cover, the QW can only be defined globally on an extended spacetime where the polar angle extends from $0$ to $4 π$. The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac Eq.\ are computed explicitly and simulated by the QW.
The first general analytic solutions for the one-dimensional walk in position and momentum space are derived. These solutions reveal, among other things, new symmetry features of quantum walk probability densities and further insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.
This letter treats the quantum random walk on the line determined by a 2 times 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P, Q, R and S given by U. The dependence of the mth moment on U and initial qubit state phi is clarified. A new type of limit theorems for the quantum walk is given. Furthermore necessary and sufficient conditions for symmetry of distribution for the quantum walk is presented. Our results show that the behavior of quantum random walk is striking different from that of the classical ramdom wa…
We analyze two families of three-state quantum walks which show the localization effect. We focus on the role of the initial coin state and its coherence in controlling the properties of the quantum walk. In particular, we show that the description of the walk simplifies considerably when the initial coin state is decomposed in the basis formed by the eigenvectors of the coin operator. This allows us to express the limit distributions in a much more convenient form. Consequently, striking features which are hidden in the standard basis description are easily identified. Moreover, the dependenc…
Classical optics can be used to efficiently implement certain quantum information processing tasks with a high degree of control, for example, one-dimensional quantum walks through the space of orbital angular momentum of light directed by its polarization. To explore the potential of quantum information processing with classical light, we here suggest a method to realize d-dimensional quantum walks with classical optics---an important step towards robust implementation of certain quantum algorithms. In this scheme, different degrees of freedom of light, such as frequency, orbital angular mome…
We study the quantum walk in momentum space using a coin arranged in quasi-periodic sequences following a Fibonacci prescription. We build for this system a classical map based on the trace of the evolution operator. The sub-ballistic behavior of this quantum walk is connected with the power-law decay of the time correlations of the trace map.
Quantum walks have by now been realized in a large variety of different physical settings. In some of these, particularly with trapped ions, the walk is implemented in phase space, where the corresponding position states are not orthogonal. We develop a general description of such a quantum walk and show how to map it into a standard one with orthogonal states, thereby making available all the tools developed for the latter. This enables a variety of experiments, which can be implemented with smaller step sizes and more steps. Tuning the non-orthogonality allows for an easy preparation of exte…
A discrete action principle for general quantum automata is proposed. This action principle is particularized to Discrete Time Quantum Walks (DTQWs) and then extended into an energy and momentum preserving, manifestly covariant formulation. Space-time coordinates are introduced as new variables of the action and their equations of motion enforce energy and momentum conservation. This guarantees that the proposed action can be used to build future, DTQW-based self-consistent models of spinors interacting with gauge fields. A discrete stress-energy tensor for the DTQW is also obtained by functio…
The atom-optics kicked rotor can be used to prepare specific momentum distributions on a discrete basis set. We implement a continuous-time quantum walk and a quantum search protocol in this momentum basis. In particular we propose ways to identify a specific marked state from the final momentum distribution after the walker's evolution. Our protocol is guided by current experimental possibilities making it accessible to experimentally implemented quantum walks with Bose-Einstein condensates.
We investigate the possibility of implementing a sequence of quantum walks whose probability distributions give an overall positive winning probability, while it is negative for the single walks (Parrondo's paradox). In particular, we have in mind an experimental realisation with a Bose-Einstein condensate in which the walker's space is momentum space. Experimental problems in the precise implementation of the coin operations for our discrete-time quantum walks are analysed in detail. We study time-dependent phase fluctuations of the coins as well as perturbations arising from the finite momen…
Over the last decade there have been many advances in studies of quantum walks (QWs) including a momentum-space QW recently realized in our spinor Bose-Einstein condensate system. This QW possessed behaviors that generally agreed with theoretical predictions; however, it also showed momentum distributions that were not adequately explained by the theory. We present a theoretical model which proves that the coherent dynamics of the spinor condensate is sufficient to explain the experimental data without invoking the presence of a thermal cloud of atoms as in the original theory. Our numerical f…
We report on a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. Each step of the walk consists of a coin toss (a microwave pulse) followed by a unitary shift operator (a resonant ratchet pulse). We carry out a comprehensive experimental study on the effects of various parameters, including the strength of the shift operation, coin parameters, noise, and initialization of the system on the behavior of the walk. The walk dynamics can be well controlled in our experiment; potential app…
We present a discrete-time, one-dimensional quantum walk based on the entanglement between the momentum of ultracold rubidium atoms (the walk space) and two internal atomic states (the "coin" degree of freedom). Our scheme is highly flexible and can provide a platform for a wide range of applications such as quantum search algorithms, the observation of topological phases, and the realization of walks with higher dimensionality. Along with the investigation of the quantum-to-classical transition, we demonstrate the distinctive features of a quantum walk and contrast them to those of its classi…
Each step in a quantum random walk is typically understood to have two basic components; a `coin-toss' which produces a random superposition of two states, and a displacement which moves each component of the superposition by different amounts. Here we suggest the realization of a walk in momentum space with a spinor Bose-Einstein condensate subject to a quantum ratchet realized with a pulsed, off-resonant optical lattice. By an appropriate choice of the lattice detuning, we show how the atomic momentum can be entangled with the internal spin states of the atoms. For the coin-toss, we propose …
Many phenomena in solid-state physics can be understood in terms of their topological properties. Recently, controlled protocols of quantum walks are proving to be effective simulators of such phenomena. Here we report the realization of a photonic quantum walk showing both the trivial and the non-trivial topologies associated with chiral symmetry in one-dimensional periodic systems, as in the Su-Schrieffer-Heeger model of polyacetylene. We find that the probability distribution moments of the walker position after many steps behave differently in the two topological phases and can be used as …
Quantum walks are powerful tools for quantum applications and for designing topological systems. Although they are simulated in a variety of platforms, genuine two-dimensional realizations are still challenging. Here we present an innovative approach to the photonic simulation of a quantum walk in two dimensions, where walker positions are encoded in the transverse wavevector components of a single light beam. The desired dynamics is obtained by means of a sequence of liquid-crystal devices, which apply polarization-dependent transverse "kicks" to the photons in the beam. We engineer our quant…
A massless Dirac particle is considered, moving along the x-axis while Pauli-coupled by its anomalous magnetic moment to a piecewise constant magnetic field along the same axis, with stochastically varying sign. The motion is approximated as a quantum walk with unitary noise, for which the evolution can be found exactly. Initially ballistic, the motion approaches a classical diffusion on a time-scale determined by the speed of light, the size of the magnetic moment, the strength of the field and the time interval between changes in its direction. It is suggested that a process of this type cou…
We investigate continuous-time quantum walks of two indistinguishable anyons in one-dimensional lattices with both on-site and nearest-neighbor interactions based on the fractional Jordan-Wigner transformation. It is shown that the two-body correlations in position space are symmetric about the initial sites of two quantum walkers in the Bose limit ($χ=0$ ) and Fermi limit ( $χ=1$), while in momentum space this happens only in the Bose limit. An interesting asymmetry arises in the correlation for most cases with the statistical parameter $χ$ varying in between. It turns out that the origin of …
We show how the Weyl quantum walk derived from principles in Ref. [1], enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras--the usual Poincaré and the k-Poincaré algebras.
We have realized a quantum walk in momentum space with a rubidium spinor Bose-Einstein condensate by applying a periodic kicking potential as a walk operator and a resonant microwave pulse as a coin toss operator. The generated quantum walks appear to be stable for up to ten steps and then quickly transit to classical walks due to spontaneous emissions induced by laser beams of the walk operator. We investigate these quantum to classical walk transitions by introducing well controlled spontaneous emissions with an external light source during quantum walks. Our findings demonstrate a scheme to…
It has been discovered that open quantum walks diffusively distribute in space, since they were introduced in 2012. Indeed, some limit distributions have been demonstrated and most of them are described by Gaussian distributions. We operate an open quantum walk on $\mathbb{Z}=\left\{0, \pm 1, \pm 2,\ldots\right\}$ with parameterized operations in this paper, and study its 1st and 2nd moments so that we find its standard deviation. The standard deviation tells us whether the open quantum walker shows diffusive or ballistic behavior, which results in a phase transition of the walker.
We study the generation of hybrid entanglement in a one-dimensional quantum walk. In particular, we explore the preparation of maximally entangled states between position and spin degrees of freedom. We address it as an optimization problem, where the cost function is the Schmidt norm. We then benchmark the algorithm and compare the generation of entanglement between the Hadamard quantum walk, the random quantum walk and the optimal quantum walk. Finally, we discuss an experimental scheme with a photonic quantum walk in the orbital angular momentum of light. The experimental measurement of ent…
The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin 1/2, in which each wave number k of walker's wave function is mapped to a point \vec{q}(k) in the three-dimensional momentum space and \vec{q}(k) makes a planar orbit as k changes its value in [-π, π). The integration over k providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstate…
We study a one-parameter family of discrete-time quantum walk models on the line and in the xy-plane associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudo-velocity on the line and in the xy-plane is proved. Symmetrization on the line and in the xy-plane 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 the line . A necessary condition for the existence of a phenomenon known as localization is given.
