Quantum Integration Error on Some Classes of Multivariate Functions

This paper presents a certain class of analytic and multivalent functions with linear operator defined on unit. We obtained many geometric properties like, Coefficient’s inequality, Distortion and Growth theorem, Radii of starlikeness and Neighborhood.

A certain class of multivalent functions

The classes of the multivariate functions with bounded moduli on R d and T d are given and their average σ-widths and non-linear n-widths are discussed. The weak asymptotic behaviors are established for the corresponding quantities.

The quantum query complexity of elliptic PDE

We show that the model of quantum computation based on density matrices and superoperators can be decomposed into a pure classical (functional) part and an effectful part modelling probabilities and measurement. The effectful part can be modelled using a generalisation of monads called arrows. We express the resulting executable model of quantum computing in the Haskell programming language using its special syntax for arrow computations. However, the embedding in Haskell is not perfect: a faithful model of quantum computing requires type capabilities that are not directly expressible in Haskell.

Journal Article Complexity of deterministic and randomized methods for multivariate integration problems for the class H p Λ (I d ) Get access Fang Gensun, Fang Gensun **Email: fanggs@bnu.edu.cn Search for other works by this author on: Oxford Academic Google Scholar Ye Peixin Ye Peixin Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 25, Issue 3, July 2005, Pages 473–485, https://doi.org/10.1093/imanum/dri006 Published: 01 July 2005

In this article, we underline and compare different quantum and deterministic models of computation by means of a probabilistic point of view. The main models used for this are deterministic, probabilistic and quantum Turing machines (DTM, PTM and QTM, respectively). Then we suggest that extending PTM with only part of quantum features is not useful to implement real quantum supremacy if we don’t incorporate them fully. This is done reviewing some notorious quantum physical experiments and quantum algorithms, showing that generally PTM does not fit for them, also if in some particular cases it fits. Here we compare probabilistic and quantum models of computation to underline some common features and differences.

In this paper, we introduce and investigate a new class of multivalent functions by using the Hadamard product structure defined in the open unit disk. We obtain coefficient estimates, a distortion theorem, extreme points and criteria of starlikeness and convexity for functions which belong to our class. Relevant connections of the results with those obtained in earlier works are also pointed out.

A Class of Multivalent Functions

The lack of purely Quantum Programming Languages constitutes a hurdle in the general description of quantum computational processes; the implementation is heavily dependent on the considered quantum computational model. To bypass the obstacle, this paper pursues a new direction, investigating the compilation of classical programming paradigms over different quantum computational models: Gate-Based, Measurement-Based and Adiabatic Quantum Computation. Since graphs can be exploited to describe both classical and quantum computations, the problem of graph encoding on quantum hardware is tightly connected to our purposes. As such, it holds a major relevance in our quest for quantum compilation. While studying these topics through the lenses of Graph Theory, declarative programming emerges as the ideal candidate for such endeavour. In this paper we consider some existing quantum computational models and for each of them we identify the main subtleties in the compilation of classical languages. In turn, we break these complexities down into easier problems to stimulate further developments in this area of research. As it turns out, the observations for each model differ widely. Nevertheless, as for the tasks here considered, no model seems to claim supremacy over the others. In contrast, declarative programming maintains the spot as the ideal candidate for quantum compilation, independently of the model.

Adiabatic quantum computing (AQC) is a model of computation that uses quantum mechanical processes operating under adiabatic conditions. As a form of universal quantum computation, AQC employs the principles of superposition, tunneling, and entanglement that manifest in quantum physical systems. The AQC model of quantum computing is distinguished by the use of dynamical evolution that is slow with respect to the time and energy scales of the underlying physical systems. This adiabatic condition enforces the promise that the quantum computational state will remain well-defined and controllable thus enabling the development of new algorithmic approaches. Several notable algorithms developed within the AQC model include methods for solving unstructured search and combinatorial optimization problems. In an idealized setting, the asymptotic complexity analyses of these algorithms indicate computational speed-ups may be possible relative to state-of-the-art conventional methods. However, the presence of non-ideal conditions, including non-adiabatic dynamics, residual thermal excitations, and physical noise complicate the assessment of the potential computational performance. A relaxation of the adiabatic condition is captured in the complementary computational heuristic of quantum annealing, which accommodates physical systems operating at finite temperature and in open environments. While quantum annealing (QA) provides a more accurate model for the behavior of actual quantum physical systems, the possibility of non-adiabatic effects obscures a clear separation with conventional computing complexity. A series of technological advances in the control of quantum physical systems have enabled experimental AQC and QA. Prominent examples include demonstrations using superconducting electronics, which encode quantum information in the magnetic flux induced by a weak current operating at cryogenic temperatures. A family of devices developed specifically for unconstrained optimization problems has been applied to solve problems in specific domains including logistics, finance, material science, machine learning, and numerical analysis. An accompanying infrastructure has also developed to support these experimental demonstrations and to enable access of a broader community of users. Although AQC is most commonly applied in superconducting technologies, alternative approaches include optically trapped neutral atoms and ion-trap systems. The significant progress in the understanding of AQC has revealed several open topics that continue to motivate research into this model of quantum computation. Foremost is the development of methods for fault-tolerant operation that will ensure the scalability of AQC for solving large-scale problems. In addition, unequivocal experimental demonstrations that differentiate the computational power of AQC and its variants from conventional computing approaches are needed. This will also require advances in the fabrication and control of quantum physical systems under the adiabatic restrictions.

The quantity of the precise upper bound of the deviations of the linear methods of summation, which are determined by rectangular number matrix Λ = IIλn,kII on the classes of continuous periodic functions in the uniform metric is given. As possible application of the obtained results, we study the asymptotic behavior of threeharmonic Poisson integrals in the case when the classes W r∞, r ∈ N , are an object of approximation. The asymptotic equalities reveal the theoretical foundations and mathematical features of one of the main problems of approximation theory − the Kolmogorov-Nikol'sky problem. In particular, the problem is solved for the threeharmonic Poisson integrals on the Sobolev classes in the uniform metric. It is found that the threeharmonic Poisson integrals possess approximation properties that are different from the properties of the harmonic and biharmonic Poisson integrals, which were studied previously, and some concepts and techniques of approximation theory can also be useful in studying the spaces of functions with generalized derivatives. An important moment in the solution of this problem is the fact that with the help of the asymptotic equalities, which are studied, a wide range of economic problems can be solved, the solution of which by methods of classical linear algebra and mathematical analysis is a complicated process. Economic modeling and forecasting on the basis of the constructed mathematical model can be used in the analysis of processes of economic dynamics, considering polyharmonic regimes. The purpose of the work is to develop a mathematical apparatus that allows to build mathematical models of periodic economic processes. Modeling serves as a means of analyzing the economy and the phenomena occurring in it, as well as justifying the decisions made, forecasting and managing economic processes and objects. We also analyze some fundamental problem of the modern economy, solved by methods of the approximation theory

This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int construction. Along the way, we discuss some new examples of partially traced categories, mostly arising in the context of quantum computation. The second part deals with the construction of categorical models of higher-order quantum computation. We construct a concrete semantic model of Selinger and Valiron's quantum lambda calculus, which has been an open problem until now. We do this by considering presheaf categories over appropriate base categories arising from first-order quantum computation. The main technical ingredients are Day's convolution theory and Kelly and Freyd's notion of continuity of functors. We first give an abstract description of the properties required of the base categories for the model construction to work; then exhibit a specific example of base categories satisfying these properties.

This chapter presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D 44, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve $\mathit{NP}$-complete problems with only a polynomial number of quantum gates.