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.
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.
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…
We introduce the model of generalized open quantum walks on networks using the Transition Operation Matrices formalism. We focus our analysis on the mean first passage time and the average return time in Apollonian networks. These results differ significantly from a classical walk on these networks. We show a comparison of the classical and quantum behaviour of walks on these networks.
We show that a one-dimensional discrete time quantum walk can be used to implement a generalized measurement in terms of positive operator value measure (POVM) on a single qubit. More precisely, we show that for a single qubit any set of rank 1 and rank 2 POVM elements can be generated by a properly engineered quantum walk. In such a scenario the measurement of particle at position x=i corresponds to a measurement of a POVM element E_i on a qubit. We explicitly construct quantum walks implementing unambiguous state discrimination and SIC-POVM.
We mainly study exceptional configuration for coined quantum walk search. For searching on a two-dimensional grid by AKR algorithm, we find some new classes of exceptional configurations that cannot be found by the AKR algorithm effectively and the known diagonal configuration can be regarded as its special case. Meanwhile, we give two modified quantum walk models that can improve the success probability in the exceptional configurations by numerical simulation. Furthermore, we introduce the concept of generalized exceptional configuration and consider search by quantum walk on a cycle with Gr…
We demonstrate an implementation of unambiguous state discrimination of two equally probable single-qubit states via a one-dimensional photonic quantum walk experimentally. Furthermore we experimentally realize a quantum walk algorithm for implementing a generalized measurement in terms of positive operator value measurement on a single qubit. The measurement of the single-photons' positions corresponds to a measurement of an element of the positive operator value measurement on the polarizations of the single-photons.
Quantum measurements play a fundamental role in quantum mechanics and quantum information processing, but it is not easy to implement generalized measurements, the most powerful measurements allowed by quantum mechanics. Here we propose a simple recipe for implementing generalized measurements on a qudit via quantum walks. With this recipe, any discrete quantum measurement can be implemented via a one-dimensional discrete quantum walk; the number of steps is only two times the number of measurement outcomes. As an illustration, we present a unified solution for implementing arbitrary symmetric…
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.
Quantum walks are more than tools for building quantum algorithms. They have been used effectively to model and simulate quantum dynamics in many complex physical processes. Particularly, a variant of discrete-time quantum walk known as split-step quantum walk is closely related to Dirac cellular automata and topological insulators whose realizations rely on position-dependent control of evolution operators. Owing to the ease of manipulating multiple degrees of freedom of photons, we provide an optical setup of split-step operators which in combination with position-dependent coin (PDC) operat…
Open quantum walks (OQWs) are a new type of quantum walks which are entirely driven by the dissipative interaction with external environments and are formulated as completely positive trace-preserving maps on graphs. A generalized quantum optical scheme for implementing OQWs that includes non-zero temperature of the environment is suggested. In the proposed quantum optical scheme, a two-level atom plays the role of the "walker", and the Fock states of the cavity mode correspond to the lattice sites for the "walker". Using the small unitary rotations approach the effective dynamics of the syste…
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.
Very recently, Lee et al. proposed the first secure quantum teleporation protocol, where quantum information shared by an arbitrary number of senders can be transferred to another arbitrary number of receivers. Here, by introducing quantum walks, a novel secure (n,m) quantum teleportation of shared quantum secret between n senders and $m$ receivers is presented. Firstly, two kinds of (n,2) teleportation schemes are proposed by n-walker quantum walks on the line, the first walker of which is driven by three coins, respectively, based on two kinds of coin operators: the homogeneous coins and the…
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…
We introduce a minimal set of physically motivated postulates that the Hamiltonian H of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e. the diagonal elements of H, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential energy lan…
This paper gives the quantum walks determined by graph zeta functions. The result enables us to obtain the characteristic polynomial of the transition matrix of the quantum walk, and it determines the behavior of the quantum walk. We treat finite graphs allowing multi-edges and multi-loops.
We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of order $(5^2,5)$ under this matrix and thus provide strongly regular counter-examples to the conjecture.
We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give tables by computer on the identification of digraphs by their eigenvalues.
Recently, it was introduced a generalization of a nonstandard step operator named the elephant quantum walk (EQW). With proper statistical distribution for the steps, that generalized EQW (gEQW) can be tuned to exhibit a myriad of dynamical scaling behavior ranging from standard diffusion to %and superdiffusion to ballistic and hyperballistic spreading. In this work, we study the influence of the statistics of the step size and the delocalization of the initial states on the entanglement entropy of the coin. Our results show that the gEQW generates maximally entangled states for almost all ini…
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…
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 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…
Quantum walks are powerful tools for building quantum search algorithms or quantum sampling algorithms named the construction of quantum stationary state. However, the success probability of those algorithms are all far away from 1. Amplitude amplification is usually used to amplify success probability, but the soufflé problems follow. Only stop at the right step can we achieve a maximum success probability. Otherwise, as the number of steps increases, the success probability may decrease, which will cause troubles in practical application of the algorithm when the optimal number of steps is n…
Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this pap…
In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on one dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position, for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.
We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: (1) The graph obtained by augmenting the hypercube with an additive matching is instantaneous uniform mixing whenever the parity of the matching is even, but with a slower mixing time. This strictly includes Moore-Russell's result on the hyp…
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.
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.
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…
One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions.
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…
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 …
We show that a multi-step quantum walk can be realized for a single trapped ion with interpolation between quantum and random walk achieved by randomizing the generalized Hadamard coin flip phase. The signature of the quantum walk is manifested not only in the ion's position but also its phonon number, which makes an ion trap implementation of the quantum walk feasible.
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…
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…
In this paper we extend the study of three state lively quantum walks on cycles by considering the coin operator as a linear sum of permutation matrices, which is a generalization of the Grover matrix. First we provide a complete characterization of orthogonal matrices of order $3\times 3$ which are linear sum of permutation matrices. Consequently, we determine several groups of complex, real and rational orthogonal matrices. We establish that an orthogonal matrix of order $3\times 3$ is a linear sum of permutation matrices if and only if it is permutative. Finally we determine period of livel…
Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno-Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.
In this paper, following the recent paper on Walk/Zeta Correspondence by the first author and his coworkers, we compute the zeta function for the three- and four-state quantum walk and correlated random walk, and the multi-state random walk on the one-dimensional torus by using the Fourier analysis. We deal with also the four-state quantum walk and correlated random walk on the two-dimensional torus. In addition, we introduce a new class of models determined by the generalized Grover matrix bridging the gap between the Grover matrix and the positive-support of the Grover matrix. Finally, we gi…
$\mathcal{PT}$ symmetry, namely, a combined parity and time-reversal symmetry can make non-unitary quantum walks exhibit entirely real eigenenergy. However, it is known that the concept of $\mathcal{PT}$ symmetry can be generalized and an arbitrary anti-unitary symmetry has a possibility to substitute $\mathcal{PT}$ symmetry. The aim of the present work is to seek such non-unitary quantum walks with generalized $\mathcal{PT}$ symmetry by focusing on effects of spatially random disorder which breaks $\mathcal{PT}$ symmetry. We numerically find non-unitary quantum walks whose quasi-energy is ent…
Augmenting the unitary transformation which generates a quantum walk by a generalized phase gate G is a symmetry for both noisy and noiseless quantum walk on a line, in the sense that it leaves the position probability distribution invariant. However, this symmetry breaks down in the case of a quantum walk on an n-cycle, and hence can be regarded as a probe of the walk topology. Noise, modelled here as phase flip and generalized amplitude damping channels, tends to restore the symmetry because it classicalizes the walk. However, symmetry restoration happens even in the regime where the walker …
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 …
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 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 connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in the graph theory.
We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of $U$ without the spectral decomposition of $T$. Arguments in this direction reveal the eigenspaces of $U$ characterized by the generalized kernels of linear operators given by $T$.
