uniformity of quantum circuit families for error free algorithms in order to establish the computational equivalence between quantum turing machines qtms and quantum circuit families qcfs using yao s quantum circuit simulation of qtms we have previously introduced the class of uniform qcfs based on an infinite set of elementary gates which has been shown to be computationally equivalent to polynomial time qtms up to bounded error simulation however the complexity classes zqp and eqp introduced by bernstein and vazirani for qtms do not appear to equal their counterparts for uniform qcfs recently we have introduced a subclass of uniform qcfs the class of finitely generated uniform qcfs and showed that they are perfectly equivalent to the class of polynomial time qtms in the sense that both classes can be exactly simulated with each other here we further investigate the power of uniform qcfs comparing with that of finitely generated uniform qcfs in detail we obtain the following results i if a permutation mf x f x can be implemented with zero error by a uniform qcf then both f and f 1 can be exactly computed by uniform qcfs ii the quantum fourier transform qft of any order cannot be implemented with zero error by any finitely generated uniform qcf while it has been shown in contrast by mosca and zalka that the qft of any order can be exactly implemented by a uniform qcf

holomorphic quantization on the torus and finite quantum mechanics we explicitly construct the quantization of classical linear maps of on toroidal phase space of arbitrary modulus using the holomorphic chiral version of the metaplectic representation we show that finite quantum mechanics fqm on tori of arbitrary integer discretization is a consistent restriction of the holomorphic quantization of to the subgroup being the principal congruent subgroup on a finite dimensional hilbert space the generators of the rotation group for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates

one way communication complexity and the neciporuk lower bound on formula size in this paper the neciporuk method for proving lower bounds on the size of boolean formulas is reformulated in terms of one way communication complexity we investigate the settings of probabilistic formulas nondeterministic formulas and quantum formulas in all cases we can use results about one way communication complexity to prove lower bounds on formula size the main results regarding formula size are as follows we show a polynomial size gap between probabilistic quantum and deterministic formulas a near quadratic gap between the sizes of nondeterministic formulas with limited access to nondeterministic bits and nondeterministic formulas with access to slightly more such bits and a near quadratic lower bound on quantum formula size furthermore we give a polynomial separation between the sizes of quantum formulas with and without multiple read random inputs the lower bound methods for quantum and probabilistic formulas employ a variant of the neciporuk bound in terms of the vapnik chervonenkis dimension to establish our lower bounds we show optimal separations between one way and two way protocols for limited nondeterministic and quantum communication complexity and we show that zero error quantum one way communication complexity asymptotically equals deterministic one way communication complexity for total functions

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

prime decomposition and correlation measure of finite quantum systems under the name prime decomposition pd a unique decomposition of an arbitrary n dimensional density matrix into a sum of separable density matrices with dimensions determined by the coprime factors of n is introduced for a class of density matrices a complete tensor product factorization is achieved the construction is based on the chinese remainder theorem and the projective unitary representation of by the discrete heisenberg group the pd isomorphism is unitarily implemented and it is shown to be co associative and to act on as comultiplication density matrices with complete pd are interpreted as group like elements of to quantify the distance of from its pd a trace norm correlation index is introduced and its invariance groups are determined

efficient and exact quantum compression we present a divide and conquer based algorithm for optimal quantum compression decompression using o n log 4n loglogn elementary quantum operations our result provides the first quasi linear time algorithm for asymptotically optimal in size and fidelity quantum compression and decompression we also outline the quantum gate array model to bring about this compression in a quantum computer our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of o n 3 for this operation

universality of 2 qudit ternary reversible gates in this paper we investigate the synthesis of ternary reversible circuits in the absence of ancilla bits we demonstrated that 2 qudit ternary swap not and 1 controlled not gates are universal for realization of arbitrary ternary n qudit reversible circuits without ancilla qudits and all even ternary n qudit reversible circuits can be constructed by ternary not and ternary 1 controlled not gates without ancilla qudits the realization approach is constructive this result is significantly different from the binary case

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

quasiclassical computation abstract the chemical kinetic description of time evolution where the phase is random but the states are discrete is discussed as a basis for a computational approach this proposed scheme uses numbers in the entire range of 0 to 1 to represent boolean propositions in the implementation by chemical kinetics these numbers are the mole fractions of different species vibrational relaxation in a mixture of hcl and dci is the physical system that is used to illustrate the approach energy exchange in such a mixture corresponds to two strongly coupled two level systems a search problem previously discussed in the quantum computing literature is solved as an example the solution requires the same number of function evaluations as in the quantal case the action of the oracle is described in detail

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

fault tolerant quantum computation with constant error rate this paper shows that quantum computation can be made fault tolerant against errors and inaccuracies when eta the probability for an error in a qubit or a gate is smaller than a constant threshold eta c this result improves on shor s result proceedings of the 37th symposium on the foundations of computer science ieee los alamitos ca 1996 pp 56 65 which shows how to perform fault tolerant quantum computation when the error rate eta decays polylogarithmically with the size of the computation an assumption which is physically unreasonable the cost of making the quantum circuit fault tolerant in our construction is polylogarithmic in time and space our result holds for a very general local noise model which includes probabilistic errors decoherence amplitude damping depolarization and systematic inaccuracies in the gates moreover we allow exponentially decaying correlations between the errors both in space and in time fault tolerant computation can be performed with any universal set of gates the result also holds for quantum particles with p 2 states namely p qudits and is also generalized to one dimensional quantum computers with only nearest neighbor interactions no measurements or classical operations are required during the quantum computation we estimate the threshold of our construction to be eta c simeq 10 6 in the best case by this we show that local noise is in principle not an obstacle for scalable quantum computation the main ingredient of our proof is the computation on states encoded by a quantum error correcting code qecc to this end we introduce a special class of calderbank shor steane css codes called polynomial codes the quantum analogue of reed solomon codes their nice algebraic structure allows all of the encoded gates to be transversal we also provide another version of the proof which uses more general css codes but its encoded gates are slightly less elegant to achieve fault tolerance we encode the quantum circuit by another circuit by using one of these qeccs this step is repeated polyloglog many times each step slightly improving the effective error rate to achieve the desired reliability the resulting circuit exhibits a hierarchical structure and for the analysis of its robustness we borrow terminology from khalfin and tsirelson found phys 22 1992 pp 879 948 and gacs advances in computing research a research annual randomness and computation jai press greenwich ct 1989 the paper is to a large extent self contained in particular we provide simpler proofs for many of the known results we use such as the fact that it suffices to correct for bit flips and phase flips the correctness of css codes and the fact that two qubit gates are universal together with their extensions to higher dimensional particles we also provide full proofs of the universality of the sets of gates we use the proof of universality was missing in shor s paper this paper thus provides a self contained and complete proof of universal fault tolerant quantum computation in the presence of local noise

multiple regression and inverse moments improve the characterization of the spatial scaling behavior of daily streamflows in the southeast united states understanding the spatial structure of daily streamflow is essential for managing freshwater resources especially in poorly gaged regions spatial scaling assumptions are common in flood frequency prediction e g index flood method and the prediction of continuous streamflow at ungaged sites e g drainage area ratio with simple scaling by drainage area being the most common assumption in this study scaling analyses of daily streamflow from 173 streamgages in the southeastern united states resulted in three important findings first the use of only positive integer moment orders as has been done in most previous studies captures only the probabilistic and spatial scaling behavior of flows above an exceedance probability near the median negative moment orders inverse moments are needed for lower streamflows second assessing scaling by using drainage area alone is shown to result in a high degree of omitted variable bias masking the true spatial scaling behavior multiple regression is shown to mitigate this bias controlling for regional heterogeneity of basin attributes especially those correlated with drainage area previous univariate scaling analyses have neglected the scaling of low flow events and may have produced biased estimates of the spatial scaling exponent third the multiple regression results show that mean flows scale with an exponent of one low flows scale with spatial scaling exponents greater than one and high flows scale with exponents less than one the relationship between scaling exponents and exceedance probabilities may be a fundamental signature of regional streamflow this signature may improve our understanding of the physical processes generating streamflow at different exceedance probabilities

periodic solutions in an inhibitory two neuron network in this paper we consider a delayed network of two neurons with self feedback and interaction described by an all or none threshold function the discontinuity of signal function makes it difficult to apply directly dynamical system we show that the dynamics of the network can be understood in terms of the iterations of a one dimensional map and we obtain the existence and attractivity of periodic solutions moreover because the network is a limiting case of the corresponding smooth system as the parameter tends to infinity the above results can act as the guide to the rich dynamics of the smooth system therefore our results have important significance in both theory and applications

regular languages accepted by quantum automata in this paper we analyze some features of the behaviour of quantum automata in particular we prove that the class of languages recognized by quantum automata with isolated cut point is the class of reversible regular languages as a more general result we give a bound on the inverse error that implies the regularity of the language accepted by a quantum automaton

logic networks with a minimum number of nor nand gates for parity functions of n variables design of logic networks in single rail input logic with a minimum number of nor gates for parity functions of an arbitrary number of variables is described this is partly based on minimum networks for parity functions of a small number of variables which are designed by the integer programming logic design method although it is generally difficult to design minimum networks for functions of an arbitrarily large number of variables we have previously designed minimum networks for adders of an arbitrary number of variables the minimum networks for parity functions of an arbitrary number of variables discussed in this paper is another case many unique properties of minimum nor networks for parity functions are shown minimum networks with nand gates for parity functions can be easily obtained from those with nor gates because of duality relationship between nand and nor

multiscaling and skew separation in regional floods matalas et al 1975 msw found that the simulated values of flood peaks in a region using various common forms of flood frequency distributions did not reproduce the empirical skew statistics for 14 different regions covering the conterminous united states specifically the field data always demonstrated a higher value of standard deviation of skew for a given sample value of mean skew than did the simulations msw termed this difference separation and further showed that it could not be explained either by autocorrelation of flood peaks or as a small sample property in this paper we discuss an explanation of this property using the recently developing scaling theories of regional floods it is shown that in a homogeneous region recently defined by us separation would result from the multiscaling structure of flood peaks separation would not occur if floods obey simple scaling nor would separation necessarily occur with heterogeneity or mixing among different homogeneous regions mixing must be of a particular kind in order to cause separation the use of normalized flood frequencies having mean of zero and variance of 1 in the simulations carried out by msw is shown to be consistent with the assumption of index flood or simple scaling but not multiscaling in the 14 megaregions analyzed by msw mixing among subregions within each megaregion may add to the magnitude of separation the separation in 14 regions is physically interpreted based on different physical mechanisms that have been recently hypothesized by us to be responsible for the presence of simple scaling or multiscaling in floods

hierarchy and equivalence of multi letter quantum finite automata multi letter quantum finite automata qfas are a new one way qfa model proposed recently by belovs rosmanis and smotrovs a belovs a rosmanis j smotrovs multi letter reversible and quantum finite automata in proceedings of the 13th international conference on developments in language theory dlt 2007 harrachov czech republic in lecture notes in computer science vol 4588 springer berlin 2007 pp 60 71 and they showed that multi letter qfas can accept with no error some regular languages a b b that are unacceptable by the one way qfas in this paper we continue to study multi letter qfas we mainly focus on two issues 1 we show that k 1 letter qfas are computationally more powerful than k letter qfas that is k 1 letter qfas can accept some regular languages that are unacceptable by any k letter qfa a comparison with the one way qfas is made by some examples 2 we prove that a k 1 letter qfa a 1 and another k 2 letter qfa a 2 are equivalent if and only if they are n 1 n 2 4 k 1 equivalent and the time complexity of determining the equivalence of two multi letter qfas using this method is o n 1 2 k 2n 4 kn 8 where n 1 and n 2 are the numbers of states of a 1 and a 2 respectively and k max k 1 k 2 some other issues are addressed for further consideration

multidimensional color image storage retrieval and compression based on quantum amplitudes and phases abstract in this study we propose a new representation method for multidimensional color images called an n qubit normal arbitrary superposition state nass where n qubits represent the colors and coordinates of 2 n pixels e g a three dimensional color image of 1024 1024 1024 using only 30 qubits based on nass we present an n 1 qubit normal arbitrary superposition state with relative phases nassrp and an n 2 qubit normal arbitrary superposition state with three components nasstc for lossless and lossy quantum compression respectively we also design three general quantum circuits to generate nass nassrp and nasstc states where we retrieve an image from a quantum system using different projection measurement operators finally we define the quantum compression ratio and analyze lossless and lossy quantum compression algorithms of multidimensional quantum images for the first time we implemented the compression of multidimensional color images on a quantum computer thus we address the theoretical and practical aspects of image processing on a quantum computer

orthologic and quantum logic models and computational elements motivated by a growing need to understand the computational potential of quantum devices we suggest an approach to the relevant issues via quantum logic and its model theory by isolating such notions as quantum parallelism and interference within a model theoretic setting quite divorced from their customary physical trappings we seek to lay bare their logical underpinnings and possible computational ramifications in the first part of the paper a brief account of the relevant model theory is given and some new results are derived in the second part we model the simplest classical gate namely the n gate propose a quantization scheme which translates between classical and quantum models and from which emerges a logical interpretation of the notion of quantum parallelism and apply it to the classical n gate model a class of physical instantiations of the resulting quantum n gate model is also briefly discussed

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

on a new boolean function with applications consider a hypercube of 2 sup n points described by n boolean variables and a subcube of 2 sup m points m spl les n as is well known the boolean function with value 1 in the points of the subcube can be expressed as the product and of n m variables the standard synthesis of arbitrary functions exploits this property we extend the concept of subcube to the more powerful pseudocube the basic set is still composed of 2 sup m points but has a more general form the function with value 1 in a pseudocube called pseudoproduct is expressed as the and of n m exor factors each containing at most m 1 variables subcubes are special cases of pseudocubes and their corresponding pseudoproducts reduce to standard products an arbitrary boolean function can be expressed as a sum of pseudoproducts spp this expression is in general much shorter than the standard sum of products as demonstrated on some known benchmarks the logical network of an n bit adder is designed in spp as a relevant example of application of this new technique a class of symmetric functions is also defined particularly suitable for spp representation

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

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

a theory of computation based on quantum logic i the meta logic underlying classical theory of computation is boolean two valued logic quantum logic was proposed by birkhoff and von neumann as a logic of quantum mechanics more than 60 years ago it is currently understood as a logic whose truth values are taken from an orthomodular lattice the major difference between boolean logic and quantum logic is that the latter does not enjoy distributivity in general the rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic the present paper is the first step toward such a new theory and it focuses on the simplest models of computation namely finite automata we introduce the notion of orthomodular lattice valued quantum automaton various properties of automata are carefully reexamined in the framework of quantum logic by employing an approach of semantic analysis we define the class of regular languages accepted by orthomodular lattice valued automata the acceptance abilities of orthomodular lattice valued nondeterministic automata and their various modifications such as deterministic automata and automata with e moves are compared the closure properties of orthomodular lattice valued regular languages are derived the kleene theorem about equivalence of regular expressions and finite automata is generalized into quantum logic we also present a pumping lemma for orthomodular lattice valued regular languages it is found that the universal validity of many properties for example the kleene theorem the equivalence of deterministic and nondeterministic automata of automata depend heavily upon the distributivity of the underlying logic this indicates that these properties does not universally hold in the realm of quantum logic on the other hand we show that a local validity of them can be recovered by imposing a certain commutativity to the atomic statements about the automata under consideration this reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic

quantum computing without entanglement it is generally believed that entanglement is essential for quantum computing we present here a few simple examples in which quantum computing without entanglement is better than anything classically achievable in terms of the reliability of the outcome after a fixed number of oracle calls using a separable that is unentangled state we show that the deutsch jozsa problem and the simon problem can be solved more reliably by a quantum computer than by the best possible classical algorithm even probabilistic we conclude that a entanglement is not essential for quantum computing and b some advantage of quantum algorithms over classical algorithms persists even when the quantum state contains an arbitrarily small amount of information that is even when the state is arbitrarily close to being totally mixed

boolean functions classification via fixed polarity reed muller forms in this paper we present a new method to characterize completely specified boolean functions the central theme of the classification is the functional equivalence a k a boolean matching two boolean functions are equivalent if there exists input permutation input negation or output negation that can transform one function to the other we have derived a method that can efficiently identify equivalence classes of boolean functions the well known canonical fixed polarity reed muller fprm forms are used as a powerful analysis tool the necessary transformations to derive one function from the other are inherent in the fprm representations to identify uniquely each equivalence class a set of well known characteristics of boolean functions and their variables including linearity symmetry total symmetry self complement and self duality are employed it is shown that all the equivalence classes of four variable functions are uniquely identified where majority of the classes have a single fprm form as their representative the boolean matching has applications in technology mapping and in design of standard cell libraries

parallelizing quantum circuits we present a novel automated technique for parallelizing quantum circuits via the forward and backward translation to measurement based quantum computing patterns and analyze the trade off in terms of depth and space complexity as a result we distinguish a class of polynomial depth circuits that can be parallelized to logarithmic depth while adding only a polynomial number of auxiliary qubits in particular we provide for the first time a full characterization of patterns with flow of arbitrary depth based on the notion of influencing walks and a simple rewriting system on the angles of the measurement our method provides new insight for constructing parallel circuits and as applications we demonstrate several classes of circuits that can be parallelized to constant or logarithmic depth furthermore we prove a logarithmic separation in terms of quantum depth between the quantum circuit model and the measurement based model

global analysis of a schistosomiasis infection model with biological control in this paper the global stability of a schistosomiasis infection model that involves human and intermediate snail hosts as well as an additional mammalian host and a competitor snail species is studied by constructing lyapunov functions and using properties of k monotone systems we derive the basic reproduction number r 0 for the deterministic model and establish that the global dynamics are completely determined by the values of r 0 we show that the disease can be eradicated when r 0 1 in the case where r 0 1 we prove the existence uniqueness and global asymptotic stability of an endemic steady state this mathematical analysis of the model gives insight about the epidemiological consequences of the introduction of a competitor resistant snail species

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

the geometry of entanglement metrics connections and the geometric phase using the natural connection equivalent to the su 2 yang mills instanton on the quaternionic hopf fibration of s7over the quaternionic projective space hp1 s4 with an su 2 s3 fibre the geometry of entanglement for two qubits is investigated the relationship between base and fibre i e the twisting of the bundle corresponds to the entanglement of the qubits the measure of entanglement can be related to the length of the shortest geodesic with respect to the mannoury fubini study metric on hp1 between an arbitrary entangled state and the separable state nearest to it using this result an interpretation of the standard schmidt decomposition in geometric terms is given schmidt states are the nearest and furthest separable ones obtained by parallel transport along the geodesic passing through the entangled state some examples showing the correspondence between the anholonomy of the connection and entanglement via the geometric phase are shown connections with important notions such as the bures metric and uhlmann s connection the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

pspace has constant round quantum interactive proof systems in this paper we introduce quantum interactive proof systems which are interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages it is proved that every language in pspace has a quantum interactive proof system that requires a total of only three messages to be sent between the prover and verifier and has exponentially small one sided probability of error it follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant round case unless the polynomial time hierarchy collapses to the second level

majority based reversible logic gates reversible logic plays an important role in the synthesis of circuits for quantum computing in this paper we introduce families of reversible gates based on the majority boolean function mbf and we prove their properties in reversible circuit synthesis these gates can be used to synthesize reversible circuits of minimum scratchpad register width for arbitrary reversible functions we show that given a mbf f with 2k 1 inputs f can be implemented by a reversible logic gate with 2k 1 inputs and 2k 1 outputs i e without any constant inputs we also demonstrate new gates from this family with very efficient quantum realizations for majority based applications they can be used to synthesize any reversible function of the same width in conjunction with inverters and feynman 2 qubit controlled not gates the gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software gap

cosmological lower bound on the circuit complexity of a small problem in logic an exponential lower bound on the circuit complexity of deciding the weak monadic second order theory of one successor ws1s is proved circuits are built from binary operations or 2 input gates which compute arbitrary boolean functions in particular to decide the truth of logical formulas of length at most 610 in this second order language requires a circuit containing at least 10125 gates so even if each gate were the size of a proton the circuit would not fit in the known universe this result and its proof due to both authors originally appeared in 1974 in the ph d thesis of the first author in this article the proof is given the result is put in historical perspective and the result is extended to probabilistic circuits

statistical self similarity in river networks parameterized by elevation a stochastic theory is developed to describe spatial variability in link heights in topologically random river networks both the systematic and random spatial variability in link heights are reflected in a scaling invariance property of their probability distributions with drainage area serving as a scale parameter tests of theoretical predictions against empirical observations show that link height distributions exhibit scaling invariance under magnification or reduction of the scale parameter this invariance property is referred to as statistical self similarity it provides a fundamental theoretical basis for some existing empirical relationships on gradients and other river geometries in channel networks and points to important research directions in river basin hydrology

generalized reed muller forms as a tool to detect symmetries in this paper we present a new method for detecting groups of symmetric variables of completely specified boolean functions the canonical generalized reed muller grm forms are used as a powerful analysis tool to reduce the search space we have developed a set of signatures that allow us to identify quickly sets of potentially symmetric variables our approach allows for detecting symmetries of any number of inputs simultaneously totally symmetric functions can be detected very quickly the traditional definitions of symmetry have also been extended to include more types this extension has the advantage of grouping input variables into more classes experiments have been performed on mcnc benchmark cases and the results verify the efficiency of our method

systems of ordinary differential equations which generate an order preserving flow a survey of results this article consists of a survey of results concerning the qualitative behavior of solutions of systems of ordinary differential equations which generate an order preserving flow we restrict our consideration to partial orderings on r n induced by any one of its orthants a flow preserves ordering if any two solutions x t and y t are ordered x t leqq y t for all t 0 whenever x 0 leqq y 0 many of the important results for such systems have only recently been obtained principally by m w hirsch who pointed out the tendency of their solutions to converge to equilibrium less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria a particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibrium one of our main goals is to show that by allowing partial orderings on r n generated by orthants

symmetry constraints on temporal order in measurement based quantum computation we discuss the interdependence of resource state measurement setting and temporal order in measurement based quantum computation the possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation we provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting in terms of a matroid conversely we show that classical processing relations necessary for turning the local measurement outcomes into computational output determine the resource state and measurement setting up to local equivalence further we find a symmetry transformation related to local complementation that leaves the temporal relations invariant

realizing ternary quantum switching networks without ancilla bits this paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits the results we established are twofold the first shows that ternary swap ternary not and ternary toffoli gates are universal for the realization of arbitrary n n ternary quantum switching networks without ancilla bits the second result proves that all n n quantum ternary networks can be generated by not controlled not multiply two and toffoli gates our approach is constructive

the quantum reverse shannon theorem based on one shot information theory the quantum reverse shannon theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel s entanglement assisted classical capacity in this paper we provide a new proof of this theorem which has previously been proved by bennett devetak harrow shor and winter our proof has a clear structure being based on two recent information theoretic results one shot quantum state merging and the post selection technique for quantum channels

quantum formulas a lower bound and simulation we show that nechiporuk s method i wegener the complexity of boolean functions teubner wiley new york 1987 for proving lower bounds for boolean formulas can be extended to the quantum case this leads to an omega n 2 log 2 n lower bound for quantum formulas computing an explicit function the only known previous explicit lower bound for quantum formulas a yao proceedings of 34th ieee symposium on foundations of computer science ieee computer society press los alamitos ca 1993 pp 352 361 states that the majority function does not have a linear size quantum formula we also show that quantum formulas can be simulated by boolean circuits of almost the same size

universal quantum computation with unlabelled qubits we show that an nth root of the walsh hadamard transform obtained from the hadamard gate and a cyclic permutation of the qubits together with two diagonal matrices namely a local qubit flip for a fixed but arbitrary qubit and a non local phase flip for a fixed but arbitrary coefficient can do universal quantum computation on n qubits a quantum computation making use of n qubits and based on these operations is then a word of variable length but whose letters are always taken from an alphabet of cardinality three therefore in contrast with other universal sets no choice of qubit lines is needed for the application of the operations described here a quantum algorithm based on this set can be interpreted as a discrete diffusion of a quantum particle on a de bruijn graph corrected on the fly by auxiliary modifications of the phases associated with the arcs

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

strategies for designing geometric transformations on quantum images three design strategies for constructing new geometric transformations on quantum images from other transformations are proposed the strategies focus on the affected areas in the images the separability and smoothness of the transformations by exploiting a representation of images on quantum computers extensively the complexity in terms of the number of basic gates and the depth of the corresponding circuits are analyzed to understand the advantages and disadvantages of each strategy examples to demonstrate the applicability of the proposed strategies are presented the strategies provide high level tools to explore and analyze transformations which are necessary to build practical image processing applications on quantum computers

l moment diagrams should replace product moment diagrams it is well known that product moment ratio estimators of the coefficient of variation c skewness and kurtosis exhibit substantial bias and variance for the small n 100 samples normally encountered in hydrologic applications consequently l moment ratio estimators termed l coefficient of variation 2 l skewness 3 and l kurtosis 4 are now advocated because they are nearly unbiased for all underlying distributions the advantages of l moment ratio estimators over product moment ratio estimators are not limited to small samples monte carlo experiments reveal that product moment estimators of c and are also remarkably biased for extremely large samples n 1000 from highly skewed distributions a case study using large samples n 5000 of average daily streamflow in massachusetts reveals that conventional moment diagrams based on estimates of product moments c and reveal almost no information about the distributional properties of daily streamflow whereas l moment diagrams based on estimators of 2 3 and 4 enabled us to discriminate among alternate distributional hypotheses

aspects of sine gordon solitons defects and gates it was recently noted how the classical sine gordon theory can support discontinuities or defects and yet maintain integrability by preserving sufficiently many conservation laws since soliton number is not preserved by a defect a possible application to the construction of logical gates is suggested

hypercomputation with quantum adiabatic processes despite the recursive non computability of hilbert s tenth problem we outline and argue for a quantum algorithm that is based on the quantum adiabatic theorem it is explained how this algorithm can solve hilbert s tenth problem the algorithm is then considered in the context of several no go arguments against such hypercomputation logical arguments are usually based on cantor s diagonal technique used for proving non computability of the turing halting problem which is related to hilbert s tenth problem physical arguments are related to the limited computability of a class of quantum computation based on qubits and dimensionally finite quantum logical gates

quantum conservative many valued computing we extend the basic principles and results of conservative logic to include the main features of many valued logics with a finite number of truth values different approaches to many valued logics are examined in order to determine some possible functionally complete sets of logic connectives as a result we describe some possible finite valued gates which realize a functionally complete set of fundamental connectives one of the purposes of this work is to show that the framework of reversible and conservative computation can be extended toward some non classical reasoning environments originally proposed to deal with propositions which embed imprecise and or uncertain information that are usually based upon many valued and modal logics we also describe a possible quantum realization of the proposed gates using creation and annihilation operators in such realization the gates are expressed as formulas that are obtained using three techniques a brute force technique an extension of the conditional quantum control method introduced by barenco deutsch ekert and jozsa conditional quantum control and logic gates phys rev lett 74 1995 4083 and a new technique that we call the constants method we show the constants method allows one to reduce the number of local operators in the formulas which correspond to the proposed gates

implementing an arbitrary reversible logic gate the 2w reversible logic gates of width w i e reversible logic gates with w inputs and w outputs together with the action of cascading form a group r we define a subgroup k consisting of the switched gates there are 2w 1 2 such gates they partition the group r into 2w 1 1 double cosets this allows us to decompose an arbitrary reversible gate into the cascade of three gates a switched gate an upside down simple control gate and a second switched gate we present an algorithm to perform this factorization and thus provide a method of implementing an arbitrary reversible gate into hardware the algorithm can be used to automate the implementation of a reversible function in future c mos technologies realizing low cost computing

quantum communication and complexity in the setting of communication complexity two distributed parties want to compute a function depending on both their inputs using as little communication as possible the required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication we survey the main results of the young area of quantum communication complexity its relation to teleportation and dense coding the main examples of fast quantum communication protocols lower bounds and some applications

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

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

generating the group of reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to have an insight into the structure of the group formed by the reversible logic gates and their cascading into reversible circuits such insight is gained from constructing chains of maximal subgroups the subgroup of control gates plays a prominent role as it is a sylow 2 subgroup

quantum associative memory abstract this paper combines quantum computation with classical neural network theory to produce a quantum computational learning algorithm quantum computation qc uses microscopic quantum level effects to perform computational tasks and has produced results that in some cases are exponentially faster than their classical counterparts the unique characteristics of quantum theory may also be used to create a quantum associative memory quam with a capacity exponential in the number of neurons this paper combines two quantum computational algorithms to produce such a quantum associative memory the result is an exponential increase in the capacity of the memory when compared to traditional associative memories such as the hopfield network the paper covers necessary high level quantum mechanical and quantum computational ideas and introduces a quam theoretical analysis proves the utility of the memory and it is noted that a small version should be physically realizable in the near future

the decomposition of an arbitrary reversible logic circuit the 2w reversible logic circuits of width w i e reversible logic circuits with w inputs and w outputs together with the action of cascading form a group g isomorphic to the symmetric group we define two conjugate subgroups g1 and g2 together they partition the group g into 2w 1 1 double cosets these allow us to decompose an arbitrary member of g into a cascade of three simpler members this decomposition is a far relative of the well known lu decomposition of a square matrix

dipole quadrupole decomposition of two coupled spin 1 systems using dipole and quadrupole operators as the orthogonal basis of su 3 algebra and successive cartan decompositions the decomposition of matrices for two coupled spin 1 systems is investigated so as to meet the requirements of some realistic quantum systems finally this kind of decomposition for the ternary swap gate is given specifically this method can be used to investigate the realization of two qutrit logic gates

a framework for structured quantum search abstract a quantum algorithm for general combinatorial search that uses the underlying structure of the search space to increase the probability of finding a solution is presented this algorithm shows how coherent quantum systems can be matched to the underlying structure of abstract search spaces the algorithm is evaluated empirically with a variety of search problems and shown to be particularly effective for searches with many constraints furthermore the algorithm provides a simple framework for utilizing search heuristics it also exhibits the same phase transition in search difficulty as found for sophisticated classical search methods indicating that it is effectively using the problem structure

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

analogies and differences between quantum and stochastic automata abstract we analyze some features of the behaviour of quantum automata providing analogies and differences with the corresponding stochastic models in particular we prove there is a quantum automaton where the change of state depends on unitary transformations defined by matrices with nonnull amplitudes that accepts a non regular language with cut point zero and inverse error polynomially bounded stochastic automata with matrices having nonnull elements and with polynomial bounds on the inverse error recognize only regular languages the class of stochastic languages contains the class of quantum languages quantum languages are empty or contain an infinite number of words the class of quantum languages is not closed under complementation

attractors in strongly monotone flows where f is a c vector field in r and all the off diagonal terms of its jacobian matrix are nonnegative the well known kamke theorem implies that the flow 4 i ir0 generated by 1 has the property it preserves the partial ordering 0 another system which has this property is the semilinear parabolic equation on a compact n dimensional submanifold solutions of its dirichlet or neumann problem determine a monotone flow in appropriate function space see 11 or 51 in general a flow which preserves a partial ordering on the state space is called monotone m w hirsch has studied this flow in a series of papers 14 io he 2 31 has studied the limiting behavior and convergence of solutions of system 1 he 1 4 lo has investigated this flow in the considerably abstract state space in chapter iii of cl he has used two sections to discuss the properties of attractors for monotone flows and obtained some very important results he has shown that a monotone flow cannot have an attracting cycle see l corollary 2 41 and that every attractor must contain an equilibrium see cl theorem 3 11 in particular if the flow is strongly monotone he has shown that every attractor k contains an order stable equilibrium see 1 theorem 4 11 meanwhile if we suppose z is attracted to k but is not quasiconvergent then k contains two order stable equilibria p q such that p o z 4 q see cl theorem 4 31 the purpose of this paper is further to discuss the properties of attractors for strongly monotone flows we shall study when an o limit set can be an 210 0022 247x 91 3 00

r universal reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to know which reversible logic gates can be used as building block for the reversible implementation of an arbitrary boolean function and which cannot

a note on quantum sequential machines quantum sequential machines qsms are a quantum version of stochastic sequential machines ssms recently we showed that two qsms m 1 and m 2 with n 1 and n 2 states respectively are equivalent iff they are n 1 n 2 2 equivalent l z li d w qiu determination of equivalence between quantum sequential machines theoretical computer science 358 2006 65 74 however using this result to check the equivalence is likely to need exponential expected time in this note we consider the time complexity of deciding the equivalence between qsms and related problems the main results are as follows 1 we present a polynomial time algorithm for deciding the equivalence between qsms and if two qsms are not equivalent this algorithm will produce an input output pair with length not more than n 1 n 2 2 2 we improve the bound for the equivalence between qsms from n 1 n 2 2 to n 1 2 n 2 2 1 by employing moore and crutchfield s method c moore j p crutchfield quantum automata and quantum grammars theoretical computer science 237 2000 275 306 also quant ph 9707031 1997 in addition by viewing mo 1qfas as a special case of qsms we briefly discuss the equivalence between mo 1qfas where the method used and the result obtained are slightly different from those given by koshiba t koshiba polynomial time algorithms for the equivalence for one way quantum finite automata in proceedings of the 12th international symposium on algorithms and computation isaac 2001 christchurch new zealand in lecture notes in computer science vol 2223 springer berlin 2001 pp 268 278

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

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

realization and synthesis of reversible functions reversible circuits play an important role in quantum computing this paper studies the realization problem of reversible circuits for any n bit reversible function we present a constructive synthesis algorithm given any n bit reversible function there are n distinct input patterns different from their corresponding outputs where email protected 2 n and the other 2 n n input patterns will be the same as their outputs we show that this circuit can be synthesized by at most email protected n n 1 cnot gates and 4n email protected n not gates the time and space complexities of the algorithm are w email protected 4 n and w email protected 2 n respectively the computational complexity of our synthesis algorithm is exponentially lower than that of breadth first search based synthesis algorithms

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

computing affine equivalence classes of boolean functions by group isomorphism affine equivalence classification of boolean functions has significant applications in logic synthesis and cryptography previous studies for classification have been limited by the large set of boolean functions and the complex operations on the affine group although there are many research on affine equivalence classification for parts of boolean functions in recent years there are very few results for the entire set of boolean functions the best existing result has been achieved by harrison with 15768919 affine equivalence classes for 6 variable boolean functions this paper presents a concise formula for affine equivalence classification of the entire set of boolean functions as well as a formula for affine classification of boolean functions with distinct on set size respectively the method outlined in this paper greatly simplifies the affine group s action by constructing an isomorphism mapping from the affine group to a permutation group by this method we can compute the affine equivalence classes for up to 10 variables experiment results indicate that our scheme for calculating the affine equivalence classes for more than 6 variables is a significant advancement over previous published methods

on the symmetry of logic embedding logical operations in non dissipative physical processes requires the use of reversible logic following feynman s 1986 approach the sixteen distinct truth tables of classical logic are shown to be contained in the 8 reversible logic operations covered by the symmetric group s8 which permute the eight values of three logical variables small subgroups of s8 are shown to cover respectively reversible logic reversible switching and reversible arithmetic a new universal primitive is found which generates a covering group of reversible logic it is shown that the octahedral group in four dimensions covers both reversible logic and switching and hence that the orthogonal group o 4 provides a covering group for quantum gates

qualitative theory of compartmental systems dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations many nonlinear that are called compartmental systems in this paper the authors define compartmental systems specify their relations to other nonnegative systems and discuss examples of applications the authors review the qualitative results on linear and nonlinear compartmental systems including their relation to cooperative systems they review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems providing a framework for unifying them under a few general theorems in the course of that they complete the solution of a problem posed by bellman and show that closed nonlinear autonomous n compartment systems can show the full gamut of possible behaviors of systems of odes finally to provide additional structure to this study the authors show how to partiti

charged string tensor networks tensor networks in physics can be traced back to a 1971 paper by penrose 1 such network diagrams appear in digital circuit theory and they form the foundations of quantum computing starting with the work of feynman and others in the 1980s 2 and further extended by deutsch in his quantum computational network model 3 building on a series of results 4 6 liu et al 7 recently developed a topological variant of tensor networks that among other results led to their discovery of an elegant charged string braiding for the controlled not gate also known as the feynman gate fig 1 lafont s 2003 categorical model of quantum circuits included the bialgebra d and hopf g relations between the building blocks needed to form a controlled not gate the practical utility of using baez dolan categories 10 to describe quantum circuits 6 9 is that category theory provides a graphical language that fully dictates the types of admissible relationships and transformations to reason about the interaction of network components a associativity b gate unit laws c symmetry e copy laws f unit scalar given as a blank on the page redrawn from ref 9 category theory is a branch of mathematics well suited to describe a wide range of networks 8 quantum circuits were first given a categorical model in pioneering work by lafont in 2003 9 and dagger compact closed categories 10 also called baez dolan categories were first derived to describe standard quantum theory as well as classes of topological quantum field theories in seminal work published in 1995 10 see ref 8 for a well written review of categorical quantum mechanics liu et al 7 formulated their topological model in part using category theory for practical purposes the graphical language turns out to be mathematically equivalent to 1 1email jacob biamonte at qubit org 1 xref corresp 1 1

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

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

input variable assignment and output phase optimization of pla s a pla minimization system having the following features is presented 1 minimization of both two level pla s and pla s with two bit decoders 2 optimal input variable assignment to the decoders 3 optimal output phase assignment and 4 essential prime implicants detection without generating all the prime implicants

self routing quantum sparse crossbar packet concentrators quantum switching networks are derived from conventional switching networks by replacing the classical switches by quantum switches we give the quantum circuit design and routing of an n m network called a quantum concentrator that can direct quantum bit packets in arbitrary quantum states from any of its k inputs to some of its k outputs where 1 k m n our designs are based on sparse crossbars which are rectangular grids of 2 2 crosspoints sparse crossbar concentrator structures with theoretically minimum crosspoint complexities for any values of n and m are well known but no self routing algorithms have been reported for such concentrators we transform two such families of optimal concentrators called fat slim and banded sparse crossbars into quantum networks and provide self routing algorithms for these families of concentrators in this process we extend the notion of packet concentration to a quantum network and design self routing quantum crosspoints from quantum gates we address issues critical to quantum operation like reversibility and localized self routing and give a rigorous proof that quantum fat slim and banded sparse crossbar concentrators are self routable the self routing algorithms described in the paper can be used for both quantum and classical sparse crossbar concentrators by the linearity property of all quantum systems

quantum matchgate computations and linear threshold gates the theory of matchgates is of interest in various areas in physics and computer science matchgates occur for example in the study of fermions and spin chains in the theory of holographic algorithms and in several recent works in quantum computation in this paper we completely characterize the class of boolean functions computable by unitary two qubit matchgate circuits with some probability of success we show that this class precisely coincides with that of the linear threshold gates the latter is a fundamental family that appears in several fields such as the study of neural networks using the above characterization we further show that the power of matchgate circuits is surprisingly trivial in those cases where the computation is to succeed with high probability in particular the only functions that are matchgate computable with success probability greater than 3 4 are functions depending on only a single bit of the input