efficient discrete approximations of quantum gates quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set it has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision e is only polynomial in log 1 e here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1 e a result which matches the lower bound from counting volume up to constant factor

chow s theorem and universal holonomic quantum computation a theorem from control theory relating the lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to holonomic quantum computation conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated with a non abelian gauge connection when applied to the optical holonomic computer these conditions determine that the holonomy group of the two qubit interaction model contains su 2 su 2 in particular a universal two qubit logic gate is attainable for this model

scalable programmable quantum gates and a new aspect of the additivity problem for the classical capacity of quantum channels we consider two apparently separated problems in the first part of the article we study the concept of a scalable approximate programmable quantum gate spqg these are special approximate programmable quantum gates with nice properties that could have implications on the theory of universal computation unfortunately as we prove such objects do not exist in the domain of usual quantum theory in the second part the problem of noisy dense coding and generalizations is addressed we observe that the additivity problem for the classical capacity obtained is of apparently greater generality than for the usual quantum channel completely positive maps i e the latter occurs as a special case of the former but as we shall argue with the help of the nonexistence result of the first part the former cannot be reduced to an instance of the latter we conclude by suggesting that the additivity problem for the classical capacity of quantum channels as posed until now may conceptually not be in its appropriate generality

analysis of noise properties of a class of exact methods of inverting the 2 d exponential radon transform spect application a general approach that the authors proposed elsewhere reveals the intrinsic relationship among methods for inversion of the 2 d exponential radon transform described by bellini et al 1979 by tretiak and metz 1980 by hawkins et al 1988 and by inouye et al 1989 moreover the authors approach provides an infinite class of linear methods for inverting the 2 d exponential radon transform in the work reported here the authors systematically investigated the noise characteristics of the methods in this class obtaining analytical forms for the autocovariance and the variance of the images reconstructed by use of various methods the noise properties of a new quasi optimal method were then compared theoretically to those of other methods of the class the authors analysis demonstrates that the quasi optimal method achieves smaller global variance in the reconstructed images than do the other methods of the class extensive numerical simulation studies confirm this prediction

normalization of rotationally symmetric shapes for pattern recognition abstract image normalization is useful for image recognition by the matching method recently a normalization algorithm was developed to normalize images under the distortion of translation rotation scaling and skew however the method fails when the image is rotationally symmetric a method is introduced called the modified fourier descriptor to normalize rotationally symmetric shapes this method is simple and fast and can be combined with the preceding normalization algorithm to make it more general and realizable

on the universality of almost every quantum logic gate lloyd phys rev lett 75 346 1995 showed that almost every quantum logic gate is universal in the sense that it can be used to approximate any unitary transformation the argument relied on a more general fact whose proof was not given in detail we give a complete proof of this more general fact

brief paper discrete time controllability for feedback quantum dynamics controllability properties for discrete time markovian quantum dynamics are investigated we find that while in general the controlled system is not finite time controllable feedback control allows for arbitrary asymptotic state to state transitions under further assumptions on the form of the measurement we show that finite time controllability can be achieved in a time that is twice the dimension of the system and we provide an iterative procedure to design the unitary control actions

the circular harmonic transform for spect reconstruction and boundary conditions on the fourier transform of the sinogram the circular harmonic transform cht solution of the exponential randon transform ert is applied to single photon emission computed tomography spect for uniform attenuation within a convex boundary an important special case also considered is the linear unattenuated radon transform lrt the solution is on the form of an orthogonal function expansion matched to projections that are in parallel ray geometry this property allows for efficient and accurate processing of the projections with fast fourier transform fft without interpolation or beam matching the algorithm is optimized by the use of boundary conditions on the 2 d fourier transform of the sinogram these boundary conditions imply that the signal energy of the sinogram is concentrated in well defined sectors in transform space the angle defining the sectors depends in a direct way on the radius of the field view these results are also obtained for fan beam geometry and the linear radon transform the fourier chebyshev transform of the sinogram to demonstrate that the boundary conditions are a more general property of the radon transform and a not a property unique to rectangular coordinates

image normalization by complex moments the role of moments in image normalization and invariant pattern recognition is addressed the classical idea of the principal axes is analyzed and extended to a more general definition the relationship between moment based normalization moment invariants and circular harmonics is established invariance properties of moments as opposed to their recognition properties are identified using a new class of normalization procedures the application of moment based normalization in pattern recognition is demonstrated by experiment

a unified reconstruction framework for both parallel beam and variable focal length fan beam collimators by a cormack type inversion of exponential radon transform a variety of inversions of exponential radon transform has been derived based on the circular harmonic transform in fourier space by several research groups however these inversions cannot be directly applied to deal with the reconstruction for fan beam or variable focal length fan beam collimator geometries in single photon emission computed tomography spect in this paper the authors derived a cormack type inversion of the exponential radon transform by employing the circular harmonic transform directly in the projection space and the image space instead of the fourier space thus a unified reconstruction framework is established for parallel fan and variable focal length fan beam collimator geometries compared to many existing algorithms the presented one greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry and significantly reduces the computational burden of the special functions such as chebyshev or bessel functions by the well established fast fourier transform fft the authors algorithm is very efficient as demonstrated by several numerical simulations

noncommutative tori and universal sets of nonbinary quantum gates we address the problem of universality in simulation of evolution of quantum system and in theory of quantum computations related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific unitary transformations quantum gates from given set in an earlier paper application of clifford algebras to constructions of universal sets of binary quantum gates uk u 2n was shown for application of a similar approach to nonbinary quantum gates uk u ln in present work we used rational noncommutative torus t1 l2n a set of universal nonbinary two gates is presented here as one example

reconstruction from uniformly attenuated spect projection data using the dbh method an algorithm was developed for the 2 d reconstruction of truncated and nontruncated uniformly attenuated data acquired from single photon emission computed tomography spect the algorithm is able to reconstruct data from half scan 180deg and short scan 180deg fan angle acquisitions for parallel and fan beam geometries respectively as well as data from full scan 360deg acquisitions the algorithm is a derivative backprojection and hilbert transform dbh method which involves the backprojection of differentiated projection data followed by an inversion of the finite weighted hilbert transform the kernel of the inverse weighted hilbert transform is solved numerically using matrix inversion numerical simulations confirm that the dbh method provides accurate reconstructions from half scan and short scan data even when there is truncation however as the attenuation increases finer data sampling is required

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

noise characterization of block iterative reconstruction algorithms i theory researchers have shown increasing interest in block iterative image reconstruction algorithms due to the computational and modeling advantages they provide although their convergence properties have been well documented little is known about how they behave in the presence of noise in this work the authors fully characterize the ensemble statistical properties of the rescaled block iterative expectation maximization rbi em reconstruction algorithm and the rescaled block iterative simultaneous multiplicative algebraic reconstruction technique rbi smart also included in the analysis are the special cases of rbi em maximum likelihood em ml em and ordered subset em os em and the special case of rbi smart smart a theoretical formulation strategy similar to that previously outlined for ml em is followed for the rbi methods the theoretical formulations in this paper rely on one approximation namely that the noise in the reconstructed image is small compared to the mean image in a second paper the approximation will be justified through monte carlo simulations covering a range of noise levels iteration points and subset orderings the ensemble statistical parameters could then be used to evaluate objective measures of image quality

phase space dynamical model of an open system of interacting q bits the path integral form of the propagator for a stochastic schrodinger equation is used to derive stochastic differential equations on the quantum phase space of interacting q bits in a random environment the equations can be considered as an approximate classical model of the quantum system predictions of the model are compared with that of the exact quantum equations for various examples of the environment operators

on symmetry detection a straight line is an axis ofsymmetry of a planar figure if the figure is invariant to reflection with respect to that line the purpose of this correspondence is to describe an o n log n time algorithm for enumerating all the axes of symmetry of a planar figure which is made up of possibly intersecting segments circles points etc the solution involves a reduction of the problem to a combinatorial question on words our algorithm is optimal since we can establish an n log n time lower bound for this problem

qudit basis universal quantum computation using χ 2 interactions we prove that universal quantum computation can be realized using only linear optics and 2 three wave mixing interactions in any n 1 dimensional qudit basis of the n pump photon subspace first we exhibit a strictly universal gate set for the qubit basis in the one pump photon subspace next we demonstrate qutrit basis universality by proving that 2 hamiltonians and photon number operators generate the full u 3 lie algebra in the two pump photon subspace and showing how the qutrit controlled z gate can be implemented with only linear optics and 2 interactions we then use proof by induction to obtain our general qudit result our induction proof relies on coherent photon injection or subtraction a technique enabled by 2 interaction between the encoding modes and ancillary modes finally we show that coherent photon injection is more than a conceptual tool in that it offers a route to preparing high photon number fock states from single photon fock states

quantum gates on hybrid qudits we introduce quantum hybrid gates that act on qudits of different dimensions in particular we develop two representative two qudit hybrid gates sum and swap and many qudit hybrid toffoli and fredkin gates we apply the hybrid sum gate to generating entanglement and find that operator entanglement of the sum gate is equal to the entanglement generated by it for certain initial states we also show that the hybrid sum gate acts as an automorphism on the pauli group for two qudits of different dimensions under certain conditions finally we describe a physical realization of these hybrid gates for spin systems

automatic symmetry determination and normalization for rotationally symmetric 2d shapes and 3d solid objects abstract determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision application however the usual normalization methods are inapplicable when the shape is n fold rotationally symmetric n 3 in this paper an efficient and reliable new method using modified fourier transform is introduced to normalize such a 2d rotationally symmetric shape furthermore we extend this new method to the normalization of a n fold axial rotationally symmetric 3d object there are two obvious advantages when using this new method 1 it prevents the high computation load and the difficulty in normalizing the shape s orientation which occurred in the old methods using full matching or correlation functions and 2 the weakness introduced by the old methods using data reduction from 2d data to 1d data is remedied experiments will show the validity of our method

analytic controllability of time dependent quantum control systems the question of controllability is investigated for a quantum control system in which the hamiltonian operator components carry explicit time dependence which is not under the control of an external agent we consider the general situation in which the state moves in an infinite dimensional hilbert space a drift term is present and the operators driving the state evolution may be unbounded however considerations are restricted by the assumption that there exists an analytic domain dense in the state space on which solutions of the controlled schrodinger equation may be expressed globally in exponential form the issue of controllability then naturally focuses on the ability to steer the quantum state on a finite dimensional submanifold of the unit sphere in hilbert space and thus on analytic controllability a relatively straightforward strategy allows the extension of lie algebraic conditions for strong analytic controllability derived earlier for the simpler time independent system in which the d

a new reconstruction approach for reflection mode diffraction tomography reflection mode diffraction tomography rm dt is an inversion scheme used to reconstruct the acoustical refractive index distribution of a scattering object in this work we reveal the existence of statistically complementary information inherent in the backscattered data and propose reconstruction algorithms that exploit this information for achieving a bias free reduction of image variance in rm dt images such a reduction of image variance can potentially enhance the detectability of subtle image features when the signal to noise ratio of the measured scattered data is low in rm dt the proposed reconstruction algorithms are mathematically identical but they propagate noise and numerical errors differently we investigate theoretically and validate numerically the noise properties of images reconstructed using one of the reconstruction algorithms for several different multifrequency sources and uncorrelated data noise

quantum computer with mixed states and four valued logic in this paper we discuss a model of quantum computer in which a state is an operator of density matrix and gates are general quantum operations not necessarily unitary a mixed state operator of density matrix of n twolevel quantum systems is considered as an element of 4 n dimensional operator hilbert space liouville space it allows us to use a quantum computer model with four valued logic the gates of this model are general superoperators which act on n ququat state ququat is a quantum state in a four dimensional operator hilbert space unitary two valued logic gates and quantum operations for an n qubit open system are considered as four valued logic gates acting on n ququats we discuss properties of quantum four valued logic gates in the paper we study universality for quantum four valued logic gates

universality in quantum computation we show that in quantum computation almost every gate that operates on two or more bits is a universal gate we discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates

a unified analysis of exact methods of inverting the 2 d exponential radon transform with implications for noise control in spect exact methods of inverting the two dimensional 2 d exponential radon transform have been proposed by bellini et al 1979 and by inouye et al 1989 both of whom worked in the spatial frequency domain to estimate the 2 d fourier transform of the unattenuated sinogram by hawkins et al 1988 who worked with circularly harmonic bessel transforms and by tretiak and metz 1980 who followed filtering of appropriately modified projections by exponentially weighted backprojection with perfect sampling all four of these methods are exact in the absence of projection data noise but empirical studies have shown that they propagate noise differently and no underlying theoretical relationship among the methods has been evident here an analysis of the 2 d fourier transform of the modified sinogram reveals that all previously proposed linear methods can be interpreted as special cases of a broad class of methods and that each method in the class can be implemented in principle by any one of four distinct techniques moreover the analysis suggests a new member of the class that is predicted to have noise properties better than those of previously proposed members

detection of generalized principal axes is rotationally symmetric shapes abstract automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper the inapplicability of a well known equation which is used to compute the direction of the principal axis of a given shape using moment functions to a class of so called degenerate shapes is first pointed out rotationally symmetric shapes often encountered in real applications are shown to belong to this class this problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions analytic equations for computing the direction of high order principal axes are derived they include the well known equation for computing the direction of the second order principal axis as a special case some experimental results are included finally to show the effectiveness of the derived analytic equations

consistency conditions and linear reconstruction methods in diffraction tomography because an image can be reconstructed from knowledge of its radon transform rt the task of reconstructing an image is tantamount to that of estimating its rt based upon the fourier diffraction projection fdp theorem from the statistical perspective of unbiased reduction of image variance the author previously proposed an infinite family of estimation methods for obtaining the rt from the scattered data in diffraction tomography dt in this work using the fdp theorem the authors define the diffraction radon transform drt which can be treated as the data function in dt subsequently using strategies similar to those that analyze the consistency conditions on the exponential radon transform in two dimensional 2 d single photon emission computed tomography with uniform attenuation the author studied the consistency condition on the drt and the author shows that there is a hierarchy of estimation methods that actually project the noisy data function onto its consistency space in different ways in terms of a weighted inner product of the consistency and inconsistency parts of a noisy data function the author further demonstrates that a subset of the family of estimation methods can be interpreted as orthogonal projections onto the consistency space of the drt in particular the statistically suboptimal estimation method in the family corresponds to an orthogonal projection associated with an ordinary inner product of the consistency and inconsistency parts of a noisy data function

affine permutation invariance of 2 d shapes shapes provide a rich set of clues on the identity and topological properties of an object in many imaging environments however the same object appears to have different shapes due to distortions such as translation rotation reflection scaling or skewing further the order by which the object s feature points are scanned changes i e the order of the pixels may be permuted relating two dimensional shapes of the same object distorted by different affine and permutation transformations is a challenge we introduce a shape invariant that we refer to as the intrinsic shape of an object and describe an algorithm blaiser to recover it the intrinsic shape is invariant to affine permutation distortions it is a uniquely defined representative of the equivalence class of all affine permutation distortions of the same object blaiser computes the intrinsic shape from any arbitrarily affine permutation distorted image of the object without prior knowledge regarding the distortions or the undistorted shape of the object the critical step of blaiser is the determination of the shape orientation and we provide a detailed discussion on this topic the operations of blaiser are based on low order moments of the input shape and thus robust to error and noise examples illustrate the performance of the algorithm

decidable and undecidable problems about quantum automata we study the following decision problem is the language recognized by a quantum finite automaton empty or nonempty we prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds this result is in contrast with the corresponding situation for probabilistic finite automata for which it is known that strict and nonstrict thresholds both lead to undecidable problems

on the detection of the axes of symmetry of symmetric and almost symmetric planar images a simple algorithm is presented for finding all the axes of symmetry of symmetric and almost symmetric planar images having nonuniform gray level intensity images the technique which can also be used in conjunction with planar curves is based on the identification of the centroids of the given image and other related sets of points followed by a maximization of a specially defined coefficient of symmetry owing to the nature of the required procedures which are strictly based on the computation of mean values the method is not particularly affected by the presence of noise and is especially suited for application in conjunction with digitized figures

criteria for reachability of quantum states we address the question of which quantum states can be inter converted under the action of a time dependent hamiltonian in particular we consider the problem as applied to mixed states and investigate the difference between pure and mixed state controllabilities introduced in previous work we provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group we also address the problem of which states can be prepared if the dynamical lie group is not sufficiently large to allow the system to be controllable

a universal two bit gate for quantum computation we prove the existence of a class of two input two output gates any one of which is universal for quantum computation this is done by explicitly constructing the three bit gate introduced by deutsch proc r soc lond a 425 73 1989 as a network consisting of replicas of a single two bit gate

rapid solution of problems by quantum computation a class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method the quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation

detection of rotationally symmetric shape orientations by fold invariant shape specific points abstract two new types of shape specific points called fold invariant centroid fic and fold invariant radius weighted mean firwm are introduced for detecting the orientations of rotationally symmetric shapes a given rotationally symmetric shape is first transformed into a new fold invariant shape the centroid and the radius weighted mean of the new shape are then transformed back to obtain the proposed fic and firwm respectively fic and firwm points are proved to be invariant to translation rotation and scaling and so can be used for matching rotationally symmetric shapes using the proposed fold invariant shape specific points to detect the orientations of rotationally symmetric shapes is easier to implement than that of using the fold principal axes some illustrative examples and a method to avoid the case where the proposed fold invariant shape specific points are too close to the centroid are also given

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

manifold algorithmic errors in quantum computers with static internal imperfections the inevitable existence of static internal imperfections and residual interactions in some quantum computer architectures results in internal decoherence dissipation and destructive unitary shifts of active algorithms by exact numerical simulations we determine the relative importance and origin of these errors for a josephson charge qubit quantum computer in particular we determine that the dynamics of a cnot gate interacting with its idle neighboring qubits via native residual coupling behaves much like a perturbed kicked top in the exponential decay regime where fidelity decay is only weakly dependent on perturbation strength this means that retroactive removal of gate errors whether unitary or non unitary may not be possible and that effective error correction schemes must operate concurrently with the implementation of subcomponents of the gate

automatic rotational symmetry determination for shape analysis abstract the determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision applications in this paper a simple but effective algorithm to analyse the rotational symmetry of a given closed curve shape s is proposed a circle c with the centroid of s as the circle center and the average radius of s as the circle radius is superimposed on s resulting in the intersection of c and s at a set of points by theoretical analysis the relationship between the order n s of the rotational symmetry of s and the number of intersection points between c and s is established all the possible values for n s are determined and finally n s is determined by evaluating the similarity between s and its rotated versions in the proposed algorithm only simple pixel operations and second order moment function computation are involved several problems caused by the use of discrete image coordinates are analysed and solved for correct decision making in the algorithm good experimental results are also included

nonparametric regression sinogram smoothing using a roughness penalized poisson likelihood objective function the authors develop and investigate an approach to tomographic image reconstruction in which nonparametric regression using a roughness penalized poisson likelihood objective function is used to smooth each projection independently prior to reconstruction by unapodized filtered backprojection fbp as an added generalization the roughness penalty is expressed in terms of a monotonic transform known as the link function of the projections the approach is compared to shift invariant projection filtering through the use of a hanning window as well as to a related nonparametric regression approach that makes use of an objective function based on weighted least squares wls rather than the poisson likelihood the approach is found to lead to improvements in resolution noise tradeoffs over the hanning filter as well as over the wls approach the authors also investigate the resolution and noise effects of three different link functions the identity square root and logarithm links the choice of link function is found to influence the resolution uniformity and isotropy properties of the reconstructed images in particular in the case of an idealized imaging system with intrinsically uniform and isotropic resolution the choice of a square root link function yields the desirable outcome of essentially uniform and isotropic resolution in reconstructed images with noise performance still superior to that of the hanning filter as well as that of the wls approach

quantum implicit computational complexity we introduce a quantum lambda calculus inspired by lafont s soft linear logic and capturing the polynomial quantum complexity classes eqp bqp and zqp the calculus is based on the classical control and quantum data paradigm this is the first example of a formal system capturing quantum complexity classes in the spirit of implicit computational complexity it is machine free and no explicit bound e g polynomials appears in its syntax