random time dependent quantum walks we consider the discrete time unitary dynamics given by a quantum walk on the lattice mathbb z d performed by a quantum particle with internal degree of freedom called coin state according to the following iterated rule a unitary update of the coin state takes place followed by a shift on the lattice conditioned on the coin state of the particle we study the large time behavior of the quantum mechanical probability distribution of the position observable in mathbb z d when the sequence of unitary updates is given by an i i d sequence of random matrices when averaged over the randomness this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute a moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation a generalization to unitary updates distributed according to a markov process is also provided

recurrence for discrete time unitary evolutions we consider quantum dynamical systems specified by a unitary operator u and an initial state vector phi in each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state we call the system recurrent if this eventually happens with probability one we show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of u with respect to phi we also show that in the recurrent case the expected first return time is an integer or infinite for which we give a topological interpretation a key role in our theory is played by the first arrival amplitudes which turn out to be the complex conjugated taylor coefficients of the schur function of the spectral measure on the one hand this provides a direct dynamical interpretation of these coefficients on the other hand it links our definition of first return times to a large body of mathematical literature

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

continuous time quantum walk on integer lattices and homogeneous trees this paper is concerned with the continuous time quantum walk on d and infinite homogeneous trees by using the generating function method we compute the limit of the average probability distribution for the general isotropic walk on and for nearest neighbor walks on d and infinite homogeneous trees in addition we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases

two dimensional quantum random walk we analyze several families of two dimensional quantum random walks the feasible region the region where probabilities do not decay exponentially with time grows linearly with time as is the case with one dimensional qrw the limiting shape of the feasible region is however quite different the limit region turns out to be an algebraic set which we characterize as the rational image of a compact algebraic variety we also compute the probability profile within the limit region which is essentially a negative power of the gaussian curvature of the same algebraic variety our methods are based on analysis of the space time generating function following the methods of pemantle and wilson j comb theory ser a 97 1 129 161 2002

a perron frobenius type of theorem for quantum operations we define a special class of quantum operations we call markovian and show that it has the same spectral properties as a corresponding markov chain we then consider a convex combination of a quantum operation and a markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the perron frobenius theorem if its markovian part does moreover under a compatibility condition of the two operations we show that its limiting distribution is the same as the corresponding markov chain we apply our general results to partially decoherent quantum random walks with decoherence strength 0 le p le 1 we obtain a quantum ergodic theorem for partially decoherent processes we show that for 0 p le 1 the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk

on factorization of m channel paraunitary filterbanks we systematically investigate the factorization of causal finite impulse response fir paraunitary filterbanks with given filter length based on the singular value decomposition of the coefficient matrices of the polyphase representation a fundamental order one factorization form is first proposed for general paraunitary systems then we develop a new structure for the design and implementation of paraunitary system based on the decomposition of hermitian unitary matrices within this framework the linear phase filterbank and pairwise mirror image symmetry filterbank are revisited their structures are special cases of the proposed general structures compared with the existing structures more efficient ones that only use approximately half the number of free parameters are derived the proposed structures are complete and minimal although the factorization theory with or without constraints is discussed in the framework of m channel filterbanks the results can be applied to wavelets and multiwavelet systems and could serve as a general theory for paraunitary systems

residence time control of systems subject to measurement noise abstract the residence time control problem is considered for linear systems that are subject to both input and measurement noise disturbances it is shown that the maximal residence time is bounded and an upper bound is derived necessary and sufficient conditions for the existence of controllers that achieve the upper bound are derived and design techniques for residence time controllers are considered connections with optimal output feedback control are explored the approach is based on an asymptotic version of the large deviations theory

pseudo hermitian continuous time quantum walks in this paper we present a model exhibiting a new type of continuous time quantum walk as a quantum mechanical transport process on networks which is described by a non hermitian hamiltonian possessing a real spectrum we call it pseudo hermitian continuous time quantum walk we introduce a method to obtain the probability distribution of walk on any vertex and then study a specific system we observe that the probability distribution on certain vertices increases compared to that of the hermitian case this formalism makes the transport process faster and can be useful for search algorithms

from quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

evanescence in coined quantum walks in this paper we complete the analysis begun by two of the authors in a previous work on the discrete quantum walk on the infinite line carteret et al 2003 j phys a math gen 36 8775 95 we obtain uniformly convergent asymptotics for the exponential decay regions at the leading edges of the main peaks in the schrodinger or wave mechanics picture this calculation required us to generalize the method of stationary phase and we describe this extension in some detail including self contained proofs of all the technical lemmas required we also rigorously establish the exact feynman equivalence between the path integral and wave mechanics representations for this system using some techniques from the theory of special functions taken together with the previous work we can now prove every theorem by both routes

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

the meeting problem in the quantum walk we study the motion of two non interacting quantum particles performing a random walk on a line and analyse the probability that the two particles are detected at a particular position after a certain number of steps meeting problem the results are compared to the corresponding classical problem and differences are pointed out analytic formulae for the meeting probability and its asymptotic behaviour are derived the decay of the meeting probability for distinguishable particles is faster than in the classical case but not quadratically entangled initial states and the bosonic or fermionic nature of the walkers are considered

ito formula for one dimensional continuous time quantum random walk abstract in the paper we first extend the discrete time quantum random walk dqrw to continuous time quantum random walk cqrw then we establish an ito formula for the one dimensional cqrw as an application we present tanaka s formula with the ito formula and characterize the relation between quantum local time l t a and dirac delta function finally we discuss quantum stochastic calculus for the cqrw

one dimensional three state quantum walk we study a generalized hadamard walk in one dimension with three inner states the particle governed by the three state quantum walk moves in superposition both to the left and to the right according to the inner state in addition to these two degrees of freedom it is allowed to stay at the same position we calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle in contrast with the hadamard walk with two inner states on a line the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states this implies that the particle is trapped near the origin after a long time with high probability

purely singular continuous spectrum for limit periodic cmv operators with applications to quantum walks abstract we show that a generic element of a space of limit periodic cmv operators has zero measure cantor spectrum we also prove a craig simon type theorem for the density of states measure associated with a stochastic family of cmv matrices and use our construction from the first part to prove that the craig simon result is optimal in general we discuss applications of these results to a quantum walk model where the coins are arranged according to a limit periodic sequence the key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a cmv matrix and its schur function

spectral properties of quantum walks on rooted binary trees we define coined quantum walks on the infinite rooted binary tree given by unitary operators u c on an associated infinite dimensional hilbert space depending on a unitary coin matrix c in u 3 and study their spectral properties for circulant unitary coin matrices c we derive an equation for the caratheodory function associated to the spectral measure of a cyclic vector for u c this allows us to show that for all circulant unitary coin matrices the spectrum of the quantum walk has no singular continuous component furthermore for coin matrices c which are orthogonal circulant matrices we show that the spectrum of the quantum walk is absolutely continuous except for four coin matrices for which the spectrum of u c is pure point

amenable isometry groups of hadamard spaces one of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology since the universal covering space of a complete space of nonpositive curvature is contractible this amounts to relations between the geometry of the space or its universal covering space and the algebraic structure of its fundamental group a paradigm is the well known result of avez av that the fundamental group of a compact riemannian manifold m of nonpositive sectional curvature has exponential growth if and only if m is not flat gromov observed that the proof of avez yields a somewhat stronger statement the fundamental group of m is amenable if and only if m is flat see g1 p 93 zimmer generalized this to the case where m is complete and of finite volume zi further generalizations were obtained by anderson an and burger and schroeder bs the objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are alexandrov spaces of nonpositive curvature for the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature it is convenient to study the isometric action of the fundamental group on the universal covering space in many situations it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature this is the case in most of our arguments below let x be a hadamard space that is a complete simply connected geodesic space of nonpositive curvature in the sense of alexandrov a flat in x is a

output residence time control the problem of residence time control is extended to systems with outputs necessary and sufficient conditions for output residence time controllability in linear systems with small amounts of additive noise are derived design techniques for state feedback controllers are developed and applied to a robotics control problem the approach is based on an extension of the asymptotic first passage time theory to output processes

spectral transition for random quantum walks on trees we define and analyze random quantum walks on homogeneous trees of degree q geq 3 such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in mathbb c q hopping on the neighboring sites of the tree in the presence of static disorder the one time step random unitary evolution operator of the particle depends on a unitary matrix c in u q which monitors the strength of the disorder we prove for any q that there exist open sets of matrices in u q for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum thereby showing the existence of a spectral transition driven by c in u q for q in 3 4 we establish properties of the spectral diagram which provide a description of the spectral transition

pseudo memory effects majorization and entropy in quantum random walks a quantum random walk on the integers exhibits pseudo memory effects in that its probability distribution after n steps is determined by reshuffling the first n distributions that arise in a classical random walk with the same initial distribution in a classical walk entropy increase can be regarded as a consequence of the majorization ordering of successive distributions the lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general nevertheless entropy can increase and computer experiments show that it does so on average varying the stages at which the quantum coin system is traced out leads to new quantum walks including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk

a quantum dynamical approach to matrix khrushchev s formulas khrushchev s formula is the cornerstone of the so called khrushchev theory a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle this formula can be understood as a factorization of the schur function for an orthogonal polynomial modification of a measure on the unit circle no such formula is known in the case of matrix valued measures this constitutes the main obstacle to generalize khrushchev theory to the matrix valued setting which we overcome in this paper it was recently discovered that orthogonal polynomials on the unit circle and their matrix valued versions play a significant role in the study of quantum walks the quantum mechanical analogue of random walks in particular schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system a topic in which khrushchev s formula has profound and surprising implications we will show that this connection between schur functions and quantum walks is behind a simple proof of khrushchev s formula via quantum diagrammatic techniques for cmv matrices this does not merely give a quantum meaning to a known mathematical result since the diagrammatic proof also works for matrix valued measures actually this path counting approach is so fruitful that it provides different matrix generalizations of khrushchev s formula some of them new even in the case of scalar measures furthermore the path counting approach allows us to identify the properties of cmv matrices which are responsible for khrushchev s formula on the one hand this helps to formalize and unify the diagrammatic proofs using simple operator theory tools on the other hand this is the origin of our main result which extends khrushchev s formula beyond the cmv case as a factorization rule for schur functions related to general unitary operators 2015 wiley periodicals inc

continuous time quantum walks on one dimensional regular networks in this paper we consider continuous time quantum walks ctqws on a one dimensional ring lattice of n nodes in which every node is connected to its 2m nearest neighbors m on either side in the framework of the bloch function ansatz we calculate the space time transition probabilities between two nodes of the lattice we find that the transport of ctqws between two different nodes is faster than that of the classical continuous time random walks ctrws the transport speed which is defined by the ratio of the shortest path length and propagating time increases with the connectivity parameter m for both ctqws and ctrws for fixed parameter m the transport of ctrws gets slower with the increase of the shortest distance while the transport speed of ctqws turns out to be a constant value in the long time limit depending on the network size n and connectivity parameter m the limiting probability distributions of ctqws show various patterns when the network size n is an even number the probability of being at the original node differs from that of being at the opposite node which also depends on the precise value of parameter m

search via quantum walk we propose a new method for designing quantum search algorithms for finding a marked element in the state space of a classical markov chain the algorithm is based on a quantum walk a la szegedy quantum speed up of markov chain based algorithms in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 32 41 that is defined in terms of the markov chain the main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator this operator is then used in an amplitude amplification scheme as a result we considerably expand the scope of the previous approaches of ambainis quantum walk algorithm for element distinctness in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 22 31 and szegedy 2004 our algorithm combines the benefits of these approaches in terms of being able to find marked elements incurring the smaller cost of the two and being applicable to a larger class of markov chains in addition it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk

index theory of one dimensional quantum walks and cellular automata if a one dimensional quantum lattice system is subject to one step of a reversible discrete time dynamics it is intuitive that as much quantum information as moves into any given block of cells from the left has to exit that block to the right for two types of such systems namely quantum walks and cellular automata we make this intuition precise by defining an index a quantity that measures the net flow of quantum information through the system the index supplies a complete characterization of two properties of the discrete dynamics first two systems s1 s2 can be pieced together in the sense that there is a system s which acts like s1 in one region and like s2 in some other region if and only if s1 and s2 have the same index second the index labels connected components of such systems equality of the index is necessary and sufficient for the existence of a continuous deformation of s1 into s2 in the case of quantum walks the index is integer valued whereas for cellular automata it takes values in the group of positive rationals in both cases the map s mapsto rm ind s is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems systems with trivial index are precisely those which can be realized by partitioned unitaries and the prototypes of systems with non trivial index are shifts

asymptotic evolution of quantum walks with random coin we study the asymptotic position distribution of general quantum walks on a lattice including walks with a random coin which is chosen from step to step by a general markov chain in the unitary i e nonrandom case we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other for example a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense with any lattice dimension and coin dimension we find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time namely as the distribution of the discrete time analog of the group velocity in the random case we let a markov chain control process pick in each step one of finitely many unitary walks in the sense described above in ballistic order we find a nonrandom drift which depends only on the mean of the control process and not on the initial state in diffusive scaling the limiting distribution is asymptotically gaussian with a covariance matrix diffusion matrix depending on momentum the diffusion matrix depends not only on the mean but also on the transition rates of the control process in the nonrandom limit i e when the coins chosen are all very close or the transition rates of the control process are small leading to long intervals of ballistic evolution the diffusion matrix diverges our method is based on spatial fourier transforms and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum

convergence of continuous time quantum walks on the line the position density of a particle performing a continuous time quantum walk on the integer lattice viewed on length scales inversely proportional to the time t converges as t tends to infinity to a probability distribution that depends on the initial state of the particle this convergence behavior has recently been demonstrated for the simplest continuous time random walk n konno phys rev e 72 026113 2005 in this brief report we use a different technique to establish the same convergence for a very large class of continuous time quantum walks and we identify the limit distribution in the general case

localization for a one dimensional split step quantum walk with bound states robust against perturbations for given two unitary and self adjoint operators on a hilbert space a spectral mapping theorem was proved in the work of higuchi et al e print arxiv 1506 06457 see also e segawa and a suzuki quantum stud math found 3 11 2016 in this paper as an application of the spectral mapping theorem we investigate the spectrum of a one dimensional split step quantum walk we give a criterion for when there are no eigenvalues around 1 in terms of a discriminant operator we also provide a criterion for when eigenvalues 1 exist in terms of birth eigenspaces moreover we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations

spectral characteristics of the unitary critical almost mathieu operator we discuss spectral characteristics of a one dimensional quantum walk whose coins are distributed quasi periodically the unitary update rule of this quantum walk shares many spectral characteristics with the critical almost mathieu operator however it possesses a feature not present in the almost mathieu operator namely singularity of the associated cocycles this feature is however present in the so called extended harper s model we show that this operator has empty absolutely continuous spectrum and that the lyapunov exponent vanishes on the spectrum hence this model exhibits cantor spectrum of zero lebesgue measure for all irrational frequencies and arbitrary phase which in physics is known as hofstadter s butterfly in fact we will show something stronger namely that all spectral parameters in the spectrum are of critical type in the language of avila s global theory of analytic quasiperiodic cocycles we further prove that it has empty point spectrum for each irrational frequency and away from a frequency dependent set of phases having lebesgue measure zero the key ingredients in our proofs are an adaptation of avila s global theory to the present setting self duality via the fourier transform and a johnson type theorem for singular dynamically defined cmv matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting

investigation of continuous time quantum walk on root lattice an and honeycomb lattice abstract the continuous time quantum walk ctqw on root lattice a n known as hexagonal lattice for n 2 and honeycomb one is investigated by using spectral distribution method to this aim some association schemes are constructed from abelian group z m n and two copies of finite hexagonal lattices such that their underlying graphs tend to root lattice a n and honeycomb one as the size of the underlying graphs grows to infinity the ctqw on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on terwilliger algebra where we get the required results for root lattice a n and honeycomb one from large enough underlying graphs moreover by using the stationary phase method the long time behavior of ctqw on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances the scaling behavior of the probability distribution is deduced also it is shown that the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit stratum of the graph of the point group of the lattice apart from physical results it is shown that the bose mesner algebras of our constructed association schemes called n variable p polynomial can be generated by n commuting generators where raising flat and lowering operators as elements of terwilliger algebra are associated with each generator a system of n variable orthogonal polynomials which are special cases of generalized gegenbauer polynomials is constructed where the probability amplitudes are given by integrals over these polynomials or their linear combinations finally the supersymmetric structure of finite honeycomb lattices is revealed

decoherent quantum walks driven by a generic coin operation abstract we consider the effect of different unitary noise mechanisms on the evolution of a quantum walk qw on a linear chain with a generic coin operation i bit flip channel noise restricted to the coin subspace of the qw and ii topological noise caused by randomly broken links in the linear chain similarities and differences in the respective decoherent dynamics of the walker as a function of the probability per unit time of a decoherent event taking place are discussed

quantum decomposition of random walk on cayley graph of finite group in the paper a quantum decomposition qd for short of random walk on cayley graph of finite group is introduced which contains two cases one is qd of quantum random walk operator qrwo for short another is qd of quantum random walk state qrws for short using these findings i finally obtain some applications for quantum random walk qrw for short which are of interest in the study of qrw highlighting the role played by qrwo and qrws

quantum walks on cayley graphs we address the problem of the construction of quantum walks on cayley graphs our main motivation is the relationship between quantum algorithms and quantum walks in particular we discuss the choice of the dimension of the local hilbert space and consider various classes of graphs on which the structure of quantum walks may differ we completely characterize quantum walks on free groups and present partial results on more general cases some examples are given including a family of quantum walks on the hypercube involving a clifford algebra

limit theorem for continuous time quantum walk on the line concerning a discrete time quantum walk x t d with a symmetric distribution on the line whose evolution is described by the hadamard transformation it was proved by the author that the following weak limit theorem holds x t d t ensuremath rightarrow dx ensuremath pi 1 ensuremath x 2 sqrt 1 ensuremath 2 x 2 as t ensuremath rightarrow ensuremath infty the present paper shows that a similar type of weak limit theorem is satisfied for a continuous time quantum walk x t c on the line as follows x t c t ensuremath rightarrow dx ensuremath pi sqrt 1 ensuremath x 2 as t ensuremath rightarrow ensuremath infty these results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete and continuous time classical random walks y t sqrt t ensuremath rightarrow e ensuremath x 2 2 dx sqrt 2 ensuremath pi as t ensuremath rightarrow ensuremath infty the work deals also with the issue of the relationship between discrete and continuous time quantum walks this topic subject of a long debate in the previous literature is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases

on the c chevalley s theorem the group of orthogonal matrices k so n acts by conjugation on the vector space p of all n x ii symmetric matrices and the spectral theorem implies that every orbit intersects the subspace a c p of the diagonal matrices in fact every orbit k x is orthogonal to a under the invariant inner product x y tr xy and the intersection is the orbit of the permutation group on n letters w s which permutes the eigenvalues of x if we let

open quantum random walks a new model of quantum random walks is introduced on lattices as well as on finite graphs these quantum random walks take into account the behavior of open quantum systems they are the exact quantum analogues of classical markov chains we explore the quantum trajectory point of view on these quantum random walks that is we show that measuring the position of the particle after each time step gives rise to a classical markov chain on the lattice times the state space of the particle this quantum trajectory is a simulation of the master equation of the quantum random walk the physical pertinence of such quantum random walks and the way they can be concretely realized is discussed differences and connections with the already well known quantum random walks such as the hadamard random walk are established

mixing and decoherence in continuous time quantum walks on long range interacting cycles we study the effect of small decoherence in continuous time quantum walks on long range interacting cycles which are constructed by connecting all the two nodes of distance m on the cycle graph in our investigation each node is continuously monitored by an individual point contact which induces the decoherence process we obtain the analytical probability distribution and the mixing time upper bound our results show that for small rates of decoherence the mixing time upper bound is independent of distance parameter m and is proportional to inverse of decoherence rate

the t 2 law for quantum random walks on the line starting in the classical state based on the theory of unitary matrices our treatment of the theory of quantum random walks simplifies and clarifies certain prior derivations based on fourier transform methods given a quantum random walk on the line determined by a 2 2 unitary matrix u we show how the first two moments of the position probability distribution are determined by the eigenvalues of u by varying the coin operator a we show that the leading term of the standard deviation of the position probability distribution is ct where t denotes time and 0 c 1 however it turns out that the maximum value of c namely c 1 is achievable when and only when the coin operator a is diagonal and the initial state is unbiased starting in the classical state 0 1 our approach confirms that the maximum value of the leading term of the standard deviation of the position probability distribution is which by way of known examples is verified to be achievable

residence time control the problem of residence time controllability in dynamical systems with stochastic perturbations is formulated the solution is given for linear systems with small additive white noise perturbation it is shown that the existence of the desired residence time controller depends on the relationship between the column spaces of the control and noise matrices if the former includes the latter any residence time is possible if this inclusion does not occur the achievable residence time is bounded lower and upper estimates of this bound are given for each of these cases controller design techniques are suggested and illustrative examples are considered the development is based on an asymptotic version of the large deviations theory

matrix valued szegő polynomials and quantum random walks we consider quantum random walks qrw on the integers a subject that has been considered in the last few years in the framework of quantum computation we show how the theory of cmv matrices gives a natural tool to study these processes and to give results that are analogous to those that karlin and mcgregor developed to study classical birth and death processes using orthogonal polynomials on the real line in perfect analogy with the classical case the study of qrws on the set of nonnegative integers can be handled using scalar valued laurent polynomials and a scalar valued measure on the circle in the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions we show how our tools yield results in the well known case of the hadamard walk but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods more precisely we consider qrws on the set of nonnegative integers the analysis of these cases leads to phenomena that are absent in the case of qrws on the integers even if one restricts oneself to a constant coin this is illustrated here by studying recurrence properties of the walk but the same method can be used for other purposes the presentation here aims at being self contained but we refrain from trying to give an introduction to quantum random walks a subject well surveyed in the literature we quote for two excellent reviews see 1 9 see also the recent notes 20 2009 wiley periodicals inc

quantum walks with memory on cycles abstract we study the model of quantum walks on cycles enriched by the addition of 1 step memory we provide a formula for the probability distribution and the time averaged limiting probability distribution of the introduced quantum walk using the obtained results we discuss the properties of the introduced model and the difference in comparison to the memoryless model

quantum recurrence of a subspace and operator valued schur functions a notion of monitored recurrence for discrete time quantum processes was recently introduced in grunbaum et al commun math phys 2 320 543 569 2013 taking the initial state as an absorbing one we extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension the generating function approach leads to a connection with the well known theory of operator valued schur functions this is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in grunbaum et al commun math phys 2 320 543 569 2013 the spectral decomposition of the unitary step operator driving the evolution yields a spectral measure which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time this notion of subspace recurrence also links the concept of expected return time to an aharonov anandan phase that in contrast to the case of state recurrence can be non integer even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning so that the averaged expected return time is always a rational number moreover state recurrence can occasionally give higher return probabilities than subspace recurrence a fact that reveals once more the counter intuitive behavior of quantum systems all these phenomena are illustrated with explicit examples including as a natural application the analysis of site recurrence for coined walks

asymptotic behavior of quantum walks on the line abstract this paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line the formulae depend heavily on the normalized position of the walk when the position is in the support of the weak limit distribution obtained by konno 2005 5 one observes in addition to the limit distribution itself an oscillating phenomenon in the leading term of the asymptotic formula when the position lies outside of the support one can establish an asymptotic formula of large deviation type the rate function which expresses the exponential decay rate is explicitly given around the boundary of the support of the limit distribution called the wall the asymptotic formula is described in terms of the airy function

asymptotics of multivariate sequences part iii quadratic points abstract we consider a number of combinatorial problems in which rational generating functions may be obtained whose denominators have factors with certain singularities specifically there exist points near which one of the factors is asymptotic to a nondegenerate quadratic we compute the asymptotics of the coefficients of such a generating function the computation requires some topological deformations as well as fourier laplace transforms of generalized functions we apply the results of the theory to specific combinatorial problems such as aztec diamond tilings cube groves and multi set permutations

bulk edge correspondence of one dimensional quantum walks we outline a theory of symmetry protected topological phases of one dimensional quantum walks we assume spectral gaps around the symmetry distinguished points 1 and 1 in which only discrete eigenvalues are allowed the phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure however our theory requires no translation invariance whatsoever and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations even those that cannot be continuously contracted to the identity more precisely we define two indices for every walk characterizing the behavior far to the right and far to the left respectively their sum is a lower bound on the number of eigenstates at 1 and 1 for a translation invariant system the indices add up to zero so one of them already characterizes the phase by joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel so the theory predicts bound states at 1 or 1 this is a rigorous statement of bulk edge correspondence the results also apply to the hamiltonian case with a single gap at zero

on g invariant norms abstract a result of r mathias and horn cf linear algebra appl 142 1990 63 on the representation of the unitarily invariant norm is extended in the context of eaton triples and of real semisimple lie algebras the representation is related to a function criteria for being a norm is given in terms of and the longest element of the underlying finite reflection group in particular for the real simple lie algebras is a norm on its cartan subspace for a given nonzero if and only if 0 where 0 is the longest element of the weyl group this is the case if the weyl group contains id some related results are obtained

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

recurrence and polya number of quantum walks we analyze the recurrence probability polya number for d dimensional unbiased quantum walks a sufficient condition for a quantum walk to be recurrent is derived as a by product we find a simple criterion for localization of quantum walks in contrast with classical walks where the polya number is characteristic for the given dimension the recurrence probability of a quantum walk depends in general on the topology of the walk choice of the coin and the initial state this allows us to change the character of the quantum walk from recurrent to transient by altering the initial state

three routes to the exact asymptotics for the one dimensional quantum walk we demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one coin quantum walk on the infinite line via the jacobi polynomials that arise in the path integral representation we calculate the asymptotics using a method that is significantly easier to use than the darboux method it also provides a single integral representation for the wavefunction that works over the full range of positions n including throughout the transitional range where the behaviour changes from oscillatory to exponential previous analyses of this system have run into difficulties in the transitional range because the approximations on which they were based break down here the fact that there are two different kinds of approach to this problem path integral versus schrodinger wave mechanics is ultimately a manifestation of the equivalence between the path integral formulation of quantum mechanics and the original formulation developed in the 1920s we also discuss how and why our approach is related to the two methods that have already been used to analyse these systems

weak limits for quantum random walks we formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions with x n denoting position at time n we show that x n n converges weakly as stackrel ensuremath rightarrow n ensuremath infty to a certain distribution which is absolutely continuous and of bounded support the proof is rigorous and makes use of fourier transform methods this approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods

affine images of riemannian manifolds we describe all affine maps from a riemannian manifold to a metric space and all possible image spaces

a path integral formula with applications to quantum random walks in z d we consider general quantum random walks in a d dimensional half space we first obtain a path integral formula for general quantum random walks in a d dimensional space our path integral formula is valid for general quantum random walks on cayley graphs as well then the path integral formula is applied to obtain the scaling limit of the exit distribution the expectation of exit time and the asymptotic behaviour of the exit probabilities for general quantum random walks in a half space under some conditions on amplitude functions the conditions are shown to be satisfied by both the hadamard and grover quantum random walks in two dimensional half spaces for the two dimensional case we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero i e the natural magnitude of the hitting position is of order o 1 if the lattice spacing is set to be 1 n we also show that the rate of convergence of the total hitting probability has lower bound n 2 and upper bound n 2 for any 0 for a quantum random walk with a fixed starting point we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk in both one and two dimensions given the event of a hit the conditional expectation of hitting times is finite in contrast to being infinite for the classical case in the one dimensional case we also obtain an exact order of the probability distribution of the hitting time at 0

quantum to classical transition for random walks we look at two possible routes to classical behavior for the discrete quantum random walk on the integers decoherence in the quantum coin which drives the walk or the use of higher dimensional or multiple coins to dilute the effects of interference we use the position variance as an indicator of classical behavior and find analytical expressions for this in the long time limit we see that the multicoin walk retains the quantum quadratic growth of the variance except in the limit of a new coin for every step while the walk with decoherence exhibits classical linear growth of the variance even for weak decoherence

decoherence in the quantum walk on the line abstract we investigate the quantum walk on the line when decoherences are introduced either through simultaneous measurements of the chirality and particle position or as a result of broken links both mechanisms drive the system to a classical diffusive behavior in the case of measurements we show that the diffusion coefficient is proportional to the variance of the initially localized quantum random walker just before the first measurement when links between neighboring sites are randomly broken with probability p per unit time the evolution becomes decoherent after a characteristic time that scales as 1 p the fact that the quadratic increase of the variance is eventually lost even for very small frequencies of disrupting events suggests that the implementation of a quantum walk on a real physical system may be severely limited by thermal noise and lattice imperfections

disordered quantum walks in one lattice dimension we study a spin 12 particle moving on a one dimensional lattice subject to disorder induced by a random space dependent quantum coin the discrete time evolution is given by a family of random unitary quantum walk operators where the shift operation is assumed to be deterministic each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices we derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization put differently the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance our findings imply that this effect takes place if the coin is chosen at random from the haar measure or some measure continuous with respect to it but also for a class of discrete probability measures which support consists of two coins one of them being the hadamard coin

optimal output feedback by minimizing k f 2 consider lqp with output feedback f initial condition x 0 and integral quadratic cost value given by x 0 k f x 0 choice of f to minimize parallel k f parallel 2 instead of the levine athans cost trace k f is considered an algorithm is presented

quantization and asymptotic behaviour of quantum random walk on integers abstract quantization and asymptotic behaviour of a variant of discrete random walk on integers are investigated this variant the e v k walk has the novel feature that it uses many identical quantum coins keeping at the same time characteristic quantum features like the quadratically faster than the classical spreading rate and unexpected distribution cutoffs a weak limit of the position probability distribution pd is obtained and universal properties of this arch sine asymptotic distribution function are examined questions of driving the walk are investigated by means of a quantum optical interaction model that reveals robustness of quantum features of walker s asymptotic pd against stimulated and spontaneous quantum noise on the coin system

limit theorems for open quantum random walks we consider the limit distributions of open quantum random walks on one dimensional lattice space we introduce a dual process to the original quantum walk process which is quite similar to the relation of schrodinger heisenberg representation in quantum mechanics by this we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them in particular it is possible to get rid of the initial state when we consider the evolution of the walk it appears only in the last step of the computation

time evolution of continuous time quantum walks on dynamical percolation graphs we study the time evolution of continuous time quantum walks on randomly changing graphs at certain moments edges of the graph appear or disappear with a given probability as in percolation we treat this problem in a strong noise limit we focus on the case when the time interval between the subsequent changes of the graph tends to zero we derive explicit formulae for the general evolution in this limit we find that the percolation in this limit causes an effective time rescaling independently of the graph and the initial state of the walk the time is rescaled by the probability of keeping an edge both the individual trajectories for a single system and average properties with a superoperator formalism are discussed we give an analytical proof for our theorem and we also present results from the numerical simulations of the phenomena for different graphs we also analyze the effect of finite step size on the evolution

orthogonal polynomials on the unit circle with fibonacci verblunsky coefficients ii applications we consider cmv matrices with verblunsky coefficients determined in an appropriate way by the fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics in our first application we estimate the spreading rates of quantum walks on the line with time independent coins following the fibonacci sequence the estimates we obtain are explicit in terms of the parameters of the system in our second application we establish a connection between the classical nearest neighbor ising model on the one dimensional lattice in the complex magnetic field regime and cmv operators in particular given a sequence of nearest neighbor interaction couplings we construct a sequence of verblunsky coefficients such that the support of the lee yang zeros of the partition function for the ising model in the thermodynamic limit coincides with the essential spectrum of the cmv matrix with the constructed verblunsky coefficients under certain technical conditions we also show that the zeros distribution measure coincides with the density of states measure for the cmv matrix