computation in finitary stochastic and quantum processes abstract we introduce stochastic and quantum finite state transducers as computation theoretic models of classical stochastic and quantum finitary processes formal process languages representing the distribution over a process behaviors are recognized and generated by suitable specializations we characterize and compare deterministic and nondeterministic versions summarizing their relative computational power in a hierarchy of finitary process languages quantum finite state transducers and generators are a first step toward a computation theoretic analysis of individual repeatedly measured quantum dynamical systems they are explored via several physical systems including an iterated beam splitter an atom in a magnetic field and atoms in an ion trap a special case of which implements the deutsch quantum algorithm we show that these systems behaviors and so their information processing capacity depends sensitively on the measurement protocol

moderate deviations of dependent random variables related to clt this paper consists of three parts in the first part we find a common condition the c 2 regularity both for clt and for moderate deviations we show that this condition is verified in two important situations the lee yang theorem case and the fkg system case in the second part we apply the previous results to the additive functionals of a markov process by means of feynman kac formula and kato s analytic perturbation theory we show that the lee yang theorem holds under the assumption that 1 is an isolated simple and the only eigenvalue with modulus 1 of the operator p 1 acting on an appropriate banach space be c b e l 2 the last part is devoted to some applications to statistical mechanical systems where the c 2 regularity becomes a property of the pressure functionals and the two situations presented above become exactly the lee yang theorem case and the fkg system case we shall discuss in detail the ferromagnetic model and give some general remarks on some other models

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

sos approximants for potts crystal shapes abstract an approximation made when replacing the orientation dependent potts model surface tension by the sos one is rigorously estimated as a result an sos model may be used for an explicit and precise construction of crystal and meniscus shapes for potts models at low temperatures

constructive quantum shannon decomposition from cartan involutions the work presented here extends upon the best known universal quantum circuit the quantum shannon decomposition proposed by shende et al 2006 ieee trans comput aided des integr circuits syst 25 1000 we obtain the basis of the circuit s design in a pair of cartan decompositions this insight gives a simple constructive factoring algorithm in terms of the cartan involutions corresponding to these decompositions

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

almost any quantum logic gate is universal it is shown that if one can apply some hamiltonian repeatedly to a few variables at a time one can in general effect any desired unitary time evolution on an arbitrarily large number of variables as a result almost any quantum logic gate with two or more inputs is computationally universal in that copies of the gate can be wired together to effect any desired logic circuit and to perform any desired unitary transformation on a set of quantum variables

on the convergence of cluster expansions abstract a new simple proof of the convergence of cluster expansions in statistical mechanics is presented in the case of an abstract polymer system

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

physical constraints on hypercomputation many attempts to transcend the fundamental limitations to computability implied by the halting problem for turing machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous as opposed to bounded or discrete computation thus such schemes may include the deployment of actualised rather than potential infinities of physical resources or of physical representations of real numbers to arbitrary precision here we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability

counting by quantum eigenvalue estimation for every computation there corresponds the physical task of manipulating a starting state into an output state with a desired property as the classical theory of physics has been replaced by quantum physics it is interesting to consider the capabilities of a computer that can exploit the distinctive quantum features of nature the extra capabilities seem enormous for example with only an expected o square rootn evaluations of a function f 0 1 n 1 0 1 we can find a solution to f x 1 provided one exists another example is the ability to 1nd efficiently the order of an element g in a group by using a quantum computer to estimate a random eigenvalue of the unitary operator that multiplies by g in the group by using this eigenvalue estimation algorithm to estimate an eigenvalue of the unitary operator used in quantum searching we can approximately count the number of solutions to f x 1 this paper describes this eigenvector approach to quantum counting and related algorithms

prospects of quantum classical optimization for digital design abstract the optimal design of digital systems can be complicated by non linearities that arise from discrete optimization processes and conventional methods often attain suboptimal results quantum computing may be an interesting technology to overcome that issue but so far its effectiveness has been addressed from a most abstract perspective this paper tackles the digital design problem from a practical viewpoint and derives universal criteria for assessing whether and how much quantum technologies can improve over conventional methods the advantage is expressed in terms of computing time required to attain optimal design the general framework is applied to two modern problems i e training support vector machines and building vector quantizers the experimental comparison between conventional and quantum methods addresses the actual performances of the designed systems in the case of vector quantization empirical evidence matches theoretical expectations and supports the design method s consistency

characterizing classes of functions computable by quantum parallelism computation by quantum parallelism involves associating a quantum state v f to each function f z m z n v f is formed from superpositions of states labelled by the values of f the standard choice being an equally weighted superposition of all the values of f a joint property g f of the values f 0 f m 1 is called computable by quantum parallelism qpc if there is an observable g which will reveal the value g f with non zero probability when applied to v f and will never show a false value it is shown that the problem of deciding which g s are qpc can be formulated entirely in terms of the linear relations which exist among the v f s in the case of f z m z 2 g z 2 m z 2 we explicitly describe all properties which are qpc and show that this includes only m 2 m 2 of the 2 2 m such properties by using a suitable nonstandard definition for v f this number can be increased to 2 2 m 2 m 1 for m 1

causal inference using the algorithmic markov condition inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure markovian here we argue why causal inference is also possible when the sample size is one we develop a theory how to generate causal graphs explaining similarities between single objects to this end we replace the notion of conditional stochastic independence in the causal markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules we explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities making it possible to select among markov equivalent causal graphs this insight provides a theoretical foundation of a heuristic principle proposed in earlier work we also sketch some ideas on how to replace kolmogorov complexity with decidable complexity criteria this can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution

scaling and better approximating quantum fourier transform by higher radices quantum fourier transform qft plays a principal role in the development of efficient quantum algorithms since the number of quantum bits that can be built is limited while many quantum technologies are inherently three or more valued we consider extending the reach of the realistic quantum systems by building a qft over ternary quantum digits compared to traditional binary qft the q valued transform improves approximation properties and increases the state space by a factor of q 2 n further we use nonbinary qft derivation to generalize and improve the approximation bounds for qft

message passing theory for percolation models on multiplex networks with link overlap multiplex networks describe a large variety of complex systems including infrastructures transportation networks and biological systems most of these networks feature a significant link overlap it is therefore of particular importance to characterize the mutually connected giant component in these networks here we provide a message passing theory for characterizing the percolation transition in multiplex networks with link overlap and an arbitrary number of layers m specifically we propose and compare two message passing algorithms that generalize the algorithm widely used to study the percolation transition in multiplex networks without link overlap the first algorithm describes a directed percolation transition and admits an epidemic spreading interpretation the second algorithm describes the emergence of the mutually connected giant component that is the percolation transition but does not preserve the epidemic spreading interpretation we obtain the phase diagrams for the percolation and directed percolation transition in simple representative cases we demonstrate that for the same multiplex network structure in which the directed percolation transition has nontrivial tricritical points the percolation transition has a discontinuous phase transition with the exception of the trivial case in which all the layers completely overlap

higher order boltzmann machines and entropy bounds we examine some aspects of the interface area between mathematical statistics and statistical physics relevant to the study of boltzmann machines the boltzmann machine learning algorithm is based on a variational principle gibbs lemma for relative entropy this fact suggests the possibility of a scheme of successive approximations here we consider successive approximations parametrized by the order of many body interactions among individual units we prove bounds on the gain in relative entropy in the crucial step of adding and estimating by hebb s rule a new parameter we address the problem of providing on the basis of local observations upper and lower bounds on the entropy while upper bounds are easily obtained by subadditivity lower bounds involve localization of hirschman bounds on a dual quantum system

critical behavior of k core percolation numerical studies k core percolation has served as a paradigmatic model of discontinuous percolation for a long time recently it was revealed that the order parameter of k core percolation of random networks additionally exhibits critical behavior thus k core percolation exhibits a hybrid phase transition unlike the critical behaviors of ordinary percolation that are well understood those of hybrid percolation transitions have not been thoroughly understood yet here we investigate the critical behavior of k core percolation of erd ifmmode mbox h o else h o fi s r enyi networks we find numerically that the fluctuations of the order parameter and the mean avalanche size diverge in different ways thus we classify the critical exponents into two types those associated with the order parameter and those with finite avalanches the conventional scaling relations hold within each set however these two critical exponents are coupled finally we discuss some universal features of the critical behaviors of k core percolation and the cascade failure model on multiplex networks

entropy and quantum kolmogorov complexity a quantum brudno s theorem in classical information theory entropy rate and algorithmic complexity per symbol are related by a theorem of brudno in this paper we prove a quantum version of this theorem connecting the von neumann entropy rate and two notions of quantum kolmogorov complexity both based on the shortest qubit descriptions of qubit strings that run by a universal quantum turing machine reproduce them as outputs

computing power of turing machines in the framework of unsharp quantum logic we present recursion theoretical characterizations of the computational power of turing machines in the framework of unsharp quantum logic for unsharp quantum logic let a lattice ordered quantum multiple valued qmv algebra be its truth value lattice when lattice ordered qmv algebras satisfy a locally finite condition that is every non zero element has a finite order and its meet operation is computable we prove that 1 0 1 0 l d t e or l w t e 2 0 where l d t e respectively l w t e denotes the class of languages accepted by these turing machines in the depth first method respectively the width first method for sharp quantum logic similar results are obtained for a general orthomodular truth value lattice

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

k core bootstrap percolation on complex networks critical phenomena and nonlocal effects we develop the theory of the k core bootstrap percolation on uncorrelated random networks with arbitrary degree distributions we show that the k core percolation is an unusual hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous transition we describe the properties of the k core explain the meaning of the order parameter for the k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role corona is a subset of vertices in the k core which have exactly k neighbors in the k core it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we show that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of the k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

quantum computation based on retarded and advanced propagation abstract computation is currently seen as a forward propagator that evolves retards a completely defined initial vector into a corresponding final vector initial and final vectors map the logical input and output of a reversible boolean network respectively whereas forward propagation maps a one way propagation of logical implication from input to output conversely hard np complete problems are characteriied by a two way propagation of logical implication from input to output and vice versa given that both are partly defined from the beginning logical implication can be propagated forward and backward in a computation by constructing the gate array corresponding to the entire reversible boolean network and constraining output bits as well as input bits the possibility of modeling the physical process undergone by such a network by using a retarded and advanced in time propagation scheme is investigated

quantum ground state computation with kinematical gates we develop a computation model for solving boolean networks that implements wires dynamically through quantum ground state computation and implements gates kinematically through identities following from angular momentum algebra and statistics the gates are extra dynamical in the sense that they contribute nothing to the hamiltonian and hold as constants of the motion only the wires are dynamical just as a spin makes an ideal 1 bit memory element a spin 1 makes an ideal 3 bit gate such gates cost no computation time relaxing the wires alone solves the network we compare computation time heuristically with that of an easier boolean network where all the gate constraints are simply removed this computation model is robust with respect to decoherence and yields a generalized quantum speed up for all np non deterministic polynomial problems we give an ising model for such a logical network composed of spins of 1 with bilinear couplings

geometric transformations of multidimensional color images based on nass we present quantum algorithms to realize geometric transformations two point swappings symmetric flips local flips orthogonal rotations and translations based on an n qubit normal arbitrary superposition state nass these transformations are implemented using quantum circuits consisting of basic quantum gates which are constructed with polynomial numbers of single qubit and two qubit gates complexity analysis shows that the global operators symmetric flips local flips orthogonal rotations can be implemented with o n gates the proposed geometric transformations are used to facilitate applications of quantum images with low complexity

color avoiding percolation many real world networks have groups of similar nodes which are vulnerable to the same failure or adversary nodes can be colored in such a way that colors encode the shared vulnerabilities using multiple paths to avoid these vulnerabilities can greatly improve network robustness if such paths exist color avoiding percolation provides a theoretical framework for analyzing this scenario focusing on the maximal set of nodes which can be connected via multiple color avoiding paths in this paper we extend the basic theory of color avoiding percolation that was published in s m krause et al phys rev x 6 041022 2016 2160 330810 1103 physrevx 6 041022 we explicitly account for the fact that the same particular link can be part of different paths avoiding different colors this fact was previously accounted for with a heuristic approximation here we propose a better method for solving this problem which is substantially more accurate for many avoided colors further we formulate our method with differentiated node functions either as senders and receivers or as transmitters in both functions nodes can be explicitly trusted or avoided with only one avoided color we obtain standard percolation avoiding additional colors one by one we can understand the critical behavior of color avoiding percolation for unequal color frequencies we find that the colors with the largest frequencies control the critical threshold and exponent colors of small frequencies have only a minor influence on color avoiding connectivity thus allowing for approximations

the case for hypercomputation abstract the weight of evidence supporting the case for hypercomputation is compelling we examine some 20 physical and mathematical models of computation that are either known or suspected to have super turing or hypercomputational capabilities and argue that there is nothing in principle to prevent the physical implementation of hypercomputational systems hypercomputation may indeed be intrinsic to physics recursion emerges from hypercomputation in the same way that classical physics emerges from quantum theory as scale increases furthermore even if hypercomputation were one day shown to be physically infeasible there would still remain a role for hypercomputation as an organising principle for advanced research

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

automata theory based on quantum logic reversibilities and pushdown automata automata theory based on quantum logic called l valued finite automata l vfas may be viewed as a logical approach to quantum computing this work is mainly divided into two parts one part deals with reversibility of l vfas and the other establishes a basic framework of l valued pushdown automata l vpdas first we provide some preliminaries concerning quantum logic and l vfas and we prove a useful property of l valued successor and source operators then we clarify the relationships between various reversibilities closely related to quantum finite automata in the literature in particular we define a reversibility of l vfas which is termed as retrievability and we clarify the relationships between a number of different fashions regarding retrievability of l vfas we prove that some of them are equivalent but for the others to be equivalent the truth value set is required to satisfy a certain condition this is an essential difference from the classical situation afterwards we introduce l vpdas and show that the class of the languages accepted by l vpdas by empty stack coincides with that accepted by l vpdas by final state finally we provide some examples of l vfas and conclude with some remarks

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

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

surface tension of binary mixtures an exact relation within sos like models we investigate the statistical mechanics of binary mixtures within the framework of a two component sos model where we obtain an exact formula for the free energy this allows for an explicit expression linking the surface tension of a binary mixture to those of its components which is valid at all temperatures answering an open question asked by de coninck et al 2005 j stat phys 119 597 642