Showing papers on "Hadamard transform published in 2018"

Classical Verification of Quantum Computations

Abstract: We present the first protocol allowing a classical computer to interactively verify the result of an efficient quantum computation. We achieve this by constructing a measurement protocol, which enables a classical verifier to use a quantum prover as a trusted measurement device. The protocol forces the prover to behave as follows: the prover must construct an n qubit state of his choice, measure each qubit in the Hadamard or standard basis as directed by the verifier, and report the measurement results to the verifier. The soundness of this protocol is enforced based on the assumption that the learning with errors problem is computationally intractable for efficient quantum machines.

Hadamard triples generate self-affine spectral measures

Abstract:

Let R R be an expanding matrix with integer entries, and let B , L B,L be finite integer digit sets so that ( R , B , L ) (R,B,L) form a Hadamard triple on R d {\\mathbb {R}}^d in the sense that the matrix 1 | det R | [ e 2 π i ⟨ R − 1 b , ℓ ⟩ ] ℓ ∈ L , b ∈ B \\begin{equation*} \\frac {1}{\\sqrt {|\\det R|}}\\left [e^{2\\pi i \\langle R^{-1}b,\\ell \\rangle }\\right ]_{\\ell \\in L,b\\in B} \\end{equation*} is unitary. We prove that the associated fractal self-affine measure μ = μ ( R , B ) \\mu = \\mu (R,B) obtained by an infinite convolution of atomic measures μ ( R , B ) = δ R − 1 B ∗ δ R − 2 B ∗ δ R − 3 B ∗ ⋯ \\begin{equation*} \\mu (R,B) = \\delta _{R^{-1} B}\\ast \\delta _{R^{-2}B}\\ast \\delta _{R^{-3}B}\\ast \\cdots \\end{equation*} is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in L 2 ( μ ) L^2(\\mu ) . This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.

Color image watermarking scheme based on quaternion Hadamard transform and Schur decomposition

Abstract: Based on quaternion Hadamard transform (QHT) and Schur decomposition, a novel color image watermarking scheme is presented. To consider the correlation between different color channels and the significant color information, a new color image processing tool termed as the quaternion Hadamard transform is proposed. Then an efficient method is designed to calculate the QHT of a color image which is represented by quaternion algebra, and the QHT is analyzed for color image watermarking subsequently. With QHT, the host color image is processed in a holistic manner. By use of Schur decomposition, the watermark is embedded into the host color image by modifying the Q matrix. To make the watermarking scheme resistant to geometric attacks, a geometric distortion detection method based upon quaternion Zernike moment is introduced. Thus, all the watermark embedding, the watermark extraction and the geometric distortion parameter estimation employ the color image holistically in the proposed watermarking scheme. By using the detection method, the watermark can be extracted from the geometric distorted color images. Experimental results show that the proposed color image watermarking is not only invisible but also robust against a wide variety of attacks, especially for color attacks and geometric distortions.

New Jensen and Hermite–Hadamard type inequalities for h -convex interval-valued functions

Abstract: In this paper, we introduce the h-convex concept for interval-valued functions. By using the h-convex concept, we present new Jensen and Hermite–Hadamard type inequalities for interval-valued functions. Our inequalities generalize some known results.

Generalized Riemann-Liouville $k$ -Fractional Integrals Associated With Ostrowski Type Inequalities and Error Bounds of Hadamard Inequalities

Abstract: Ostrowski inequality provides the estimation of a function to its integral mean. It is useful in error estimations of quadrature rules in numerical analysis. The objective of this paper is to define a more general form of Riemann–Liouville $k$ -fractional integrals with respect to an increasing function, which are used to obtain fractional integral inequalities of Ostrowski type. A simple and straightforward approach is followed to establish these inequalities. The applications of established results are also briefly discussed and succeeded to get bounds of some fractional Hadamard inequalities.

Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

Abstract: We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, -convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

Hermite-Hadamard Inequalities for Exponentially Convex Functions

Abstract: Abstract: In this paper, we introduce and investigate a new class of con vex functions, which is called exponentially convex functi ons. Several new Hermite-Hadamard type integral inequalities v a exponentially convex functions are established. Some sp ecial cases are discussed as applications of our results. The ideas and tech niques of this paper may be the starting point for further res earch in this field.

Supervised Online Hashing via Hadamard Codebook Learning

Abstract: In recent years, binary code learning, a.k.a. hashing, has received extensive attention in large-scale multimedia retrieval. It aims to encode high-dimensional data points into binary codes, hence the original high-dimensional metric space can be efficiently approximated via Hamming space. However, most existing hashing methods adopted offline batch learning, which is not suitable to handle incremental datasets with streaming data or new instances. In contrast, the robustness of the existing online hashing remains as an open problem, while the embedding of supervised/semantic information hardly boosts the performance of the online hashing, mainly due to the defect of unknown category numbers in supervised learning. In this paper, we propose an online hashing scheme, termed Hadamard Codebook based Online Hashing (HCOH), which aims to solving the above problems towards robust and supervised online hashing. In particular, we first assign an appropriate high-dimensional binary codes to each class label, which is generated randomly by Hadamard codes. Subsequently, LSH is adopted to reduce the length of such Hadamard codes in accordance with the hash bits, which can adapt the predefined binary codes online, and theoretically guarantee the semantic similarity. Finally, we consider the setting of stochastic data acquisition, which facilitates our method to efficiently learn the corresponding hashing functions via stochastic gradient descend (SGD) online. Notably, the proposed HCOH can be embedded with supervised labels and is not limited to a predefined category number. Extensive experiments on three widely-used benchmarks demonstrate the merits of the proposed scheme over the state-of-the-art methods.

Measuring the Winding Number in a Large-Scale Chiral Quantum Walk.

Abstract: We report the experimental measurement of the winding number in an unitary chiral quantum walk. Fundamentally, the spin-orbit coupling in discrete time quantum walks is implemented via a birefringent crystal collinearly cut based on a time-multiplexing scheme. Our protocol is compact and avoids extra loss, making it suitable for realizing genuine single-photon quantum walks at a large scale. By adopting a heralded single photon as the walker and with a high time resolution technology in single-photon detection, we carry out a 50-step Hadamard discrete-time quantum walk with high fidelity up to 0.948±0.007. Particularly, we can reconstruct the complete wave function of the walker that starts the walk in a single lattice site through the local tomography of each site. Through a Fourier transform, the wave function in quasimomentum space can be obtained. With this ability, we propose and report a method to reconstruct the eigenvectors of the system Hamiltonian in quasimomentum space and directly read out the winding numbers in different topological phases (trivial and nontrivial) in the presence of chiral symmetry. By introducing nonequivalent time frames, we show that whole topological phases in a periodically driven system can also be characterized by two different winding numbers. Our method can also be extended to the high winding number situation.

Efficient Encrypted Images Filtering and Transform Coding With Walsh-Hadamard Transform and Parallelization

Abstract: Since homomorphic encryption operations have high computational complexity, image applications based on homomorphic encryption are often time consuming, which makes them impractical. In this paper, we study efficient encrypted image applications with the encrypted domain Walsh-Hadamard transform (WHT) and parallel algorithms. We first present methods to implement real and complex WHTs in the encrypted domain. We then propose a parallel algorithm to improve the computational efficiency of the encrypted domain WHT. To compare the WHT with the discrete cosine transform (DCT), integer DCT, and Haar transform in the encrypted domain, we conduct theoretical analysis and experimental verification, which reveal that the encrypted domain WHT has the advantages of lower computational complexity and a shorter running time. Our analysis shows that the encrypted WHT can accommodate plaintext data of larger values and has better energy compaction ability on dithered images. We propose two encrypted image applications using the encrypted domain WHT. To accelerate the practical execution, we present two parallelization strategies for the proposed applications. The experimental results show that the speedup of the homomorphic encrypted image application exceeds 12.

Neural-network states for the classical simulation of quantum computing

Abstract: Simulating quantum algorithms with classical resources generally requires exponential resources. However, heuristic classical approaches are often very efficient in approximately simulating special circuit structures, for example with limited entanglement, or based on one-dimensional geometries. Here we introduce a classical approach to the simulation of general quantum circuits based on neural-network quantum states (NQS) representations. Considering a set of universal quantum gates, we derive rules for exactly applying single-qubit and two-qubit Z rotations to NQS, whereas we provide a learning scheme to approximate the action of Hadamard gates. Results are shown for the Hadamard and Fourier transform of entangled initial states for systems sizes and total circuit depths exceeding what can be currently simulated with state-of-the-art brute-force techniques. The overall accuracy obtained by the neural-network states based on Restricted Boltzmann machines is satisfactory, and offers a classical route to simulating highly-entangled circuits. In the test cases considered, we find that our classical simulations are comparable to quantum simulations affected by an incoherent noise level in the hardware of about $10^{-3}$ per gate.

Hadamard and Fejér-Hadamard inequalities for extended generalized fractional integrals involving special functions.

Abstract: In this paper we prove the Hadamard and the Fejer–Hadamard inequalities for the extended generalized fractional integral operator involving the extended generalized Mittag-Leffler function. The extended generalized Mittag-Leffler function includes many known special functions. We have several such inequalities corresponding to special cases of the extended generalized Mittag-Leffler function. Also there we note the known results that can be obtained.

Caputo-Hadamard fractional differential equations in banach spaces

Abstract: Abstract This article deals with some existence results for a class of Caputo–Hadamard fractional differential equations. The results are based on the Mönch’s fixed point theorem associated with the technique of measure of noncompactness. Two illustrative examples are presented.

Robust image watermarking scheme using bit-plane of hadamard coefficients

Abstract: Nowadays, due to widespread usage of the Internet, digital contents are distributed quickly and inexpensively throughout the world. Watermarking techniques can help protect authenticity of digital contents by identifying their owners. In a watermarking procedure, owner information may be embedded in the spatial domain or transform domain of host images. Since watermarking algorithms must be tamper resistant and transparent, we present a watermarking method based on a transform domain. In this method, we employ Hadamard transform as it requires simpler operations compared to other transforms such as discrete cosine transform (DCT) and discrete wavelet transform (DWT) while it still attains robustness. We analyze each bit of the Hadamard’s coefficients in terms of robustness and transparency for hiding the watermark information and find a bit-plane that maintains both robustness and transparency. After that, watermark information is hidden redundantly in the selected bit-plane. The proposed extraction algorithm is classified as a blind algorithm since it extracts all versions of the concealed watermark with no information from the host image. The output of the extraction algorithm is a logo obtained by an intelligent voting among all versions of the hidden logo. The experimental results show that the proposed method, while providing transparency, is robust against many image processing attacks such as compression, image cropping and Gaussian filtering.

ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

Abstract: We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.

A 7-D Hyperchaotic System-Based Encryption Scheme for Secure Fast-OFDM-PON

Abstract: In this paper, we propose and experimentally demonstrate a novel two-step I/Q-encryption scheme for improving physical layer security in fast orthogonal frequency division multiplexing passive optical network (fast-OFDM-PON). To meet the requirement of this encryption scheme and further enhance the security, a seven-dimensional hyperchaotic system with five positive Lyapunov exponents is designed. In this system, Walsh–Hadamard transform and discrete cosine transform are chosen to reduce the computation complexity and increase the key space simultaneously. An optional discrete Fourier transform spread is required to achieve the maximum scrambling degree. Encryption of 9.4-Gb/s 16-quadrature amplitude modulation fast-OFDM signals with a total key space of ∼10788 against the exhaustive trial has been successfully implemented over 50-km standard single mode fiber in a fast-OFDM-PON, which shows good resistance against illegal access to optical network units. All the results indicate that this proposed encryption scheme has improved security performance with relatively low computation complexity.

Hermite–Hadamard–Fejér Type Inequalities for p -Convex Functions via Fractional Integrals

Abstract: In this paper, firstly, Hermite–Hadamard–Fejer type inequalities for p-convex functions in fractional integral forms are built. Secondly, an integral identity and some Hermite–Hadamard–Fejer type integral inequalities for p-convex functions in fractional integral forms are obtained. Finally, some Hermite–Hadamard and Hermite–Hadamard–Fejer inequalities for convex, harmonically convex and p-convex functions are given. Many results presented here for p-convex functions provide extensions of others given in earlier works for convex, harmonically convex and p-convex functions.

Complexity classification of conjugated clifford circuits

Abstract: Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and π/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.

Methodology for replacing indirect measurements with direct measurements

Abstract: In quantum computing, the indirect measurement of unitary operators such as the Hadamard test plays a significant role in many algorithms. However, in certain cases, the indirect measurement can be reduced to the direct measurement, where a quantum state is destructively measured. Here we investigate in what cases such a replacement is possible and develop a general methodology for trading an indirect measurement with sequential direct measurements. The results can be applied to construct quantum circuits to evaluate the analytical gradient, metric tensor, Hessian, and even higher order derivatives of a parametrized quantum state. Also, we propose a new method to measure the out-of-time-order correlator based on the presented protocol. Our protocols can reduce the depth of the quantum circuit significantly by making the controlled operation unnecessary and hence are suitable for quantum-classical hybrid algorithms on near-term quantum computers.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Abstract: This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

Separated Boundary Value Problems of Sequential Caputo and Hadamard Fractional Differential Equations

Abstract: In this paper, we discuss the existence and uniqueness of solutions for new classes of separated boundary value problems of Caputo-Hadamard and Hadamard-Caputo sequential fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples.

Existence of periodic solutions for nonlinear implicit Hadamard’s fractional differential equations

Abstract: In this paper, by applying the coincidence degree theory which was first introduced by Mawhin, we obtain an existence result for a class of problem for nonlinear implicit fractional differential equations (IFDE for short) with Hadamard fractional derivative. We present two examples to show the applicability of our results.

Thwarting Adversarial Examples: An $L_0$-Robust Sparse Fourier Transform

Abstract: We give a new algorithm for approximating the Discrete Fourier transform of an approximately sparse signal that is robust to worst-case $L_0$ corruptions, namely that some coordinates of the signal can be corrupt arbitrarily Our techniques generalize to a wide range of linear transformations that are used in data analysis such as the Discrete Cosine and Sine transforms, the Hadamard transform, and their high-dimensional analogs We use our algorithm to successfully defend against worst-case $L_0$ adversaries in the setting of image classification We give experimental results on the Jacobian-based Saliency Map Attack (JSMA) and the CW $L_0$ attack on the MNIST and Fashion-MNIST datasets as well as the Adversarial Patch on the ImageNet dataset

Certain Gronwall type inequalities associated with Riemann-Liouville k-and Hadamard k-fractional derivatives and their applications

Abstract: We aim to establish certain Gronwall type inequalities associated with Riemann-Liouville k-and Hadamard k-fractional derivatives. The results presented here are sure to be new and potentially useful, in particular, in analyzing dependence solutions of certain k-fractional differential equations of arbitrary real order with initial conditions. Some interesting special cases of our main results are also considered.

Asymptotics for Sketching in Least Squares Regression

Abstract: We consider a least squares regression problem where the data has been generated from a linear model, and we are interested to learn the unknown regression parameters. We consider "sketch-and-solve" methods that randomly project the data first, and do regression after. Previous works have analyzed the statistical and computational performance of such methods. However, the existing analysis is not fine-grained enough to show the fundamental differences between various methods, such as the Subsampled Randomized Hadamard Transform (SRHT) and Gaussian projections. In this paper, we make progress on this problem, working in an asymptotic framework where the number of datapoints and dimension of features goes to infinity. We find the limits of the accuracy loss (for estimation and test error) incurred by popular sketching methods. We show separation between different methods, so that SRHT is better than Gaussian projections. Our theoretical results are verified on both real and synthetic data. The analysis of SRHT relies on novel methods from random matrix theory that may be of independent interest.

Communication Efficient, Sample Optimal, Linear Time Locally Private Discrete Distribution Estimation.

Abstract: We consider discrete distribution estimation over $k$ elements under $\varepsilon$-local differential privacy from $n$ samples The samples are distributed across users who send privatized versions of their sample to the server All previously known sample optimal algorithms require linear (in $k$) communication complexity in the high privacy regime $(\varepsilon<1)$, and have a running time that grows as $n\cdot k$, which can be prohibitive for large domain size $k$ We study the task simultaneously under four resource constraints, privacy, sample complexity, computational complexity, and communication complexity We propose \emph{Hadamard Response (HR)}, a local non-interactive privatization mechanism with order optimal sample complexity (for all privacy regimes), a communication complexity of $\log k+2$ bits, and runs in nearly linear time Our encoding and decoding mechanisms are based on Hadamard matrices, and are simple to implement The gain in sample complexity comes from the large Hamming distance between rows of Hadamard matrices, and the gain in time complexity is achieved by using the Fast Walsh-Hadamard transform We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), theoretically, and experimentally For $k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR

Old and new challenges in Hadamard spaces

Abstract: Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and nonlinear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kahler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and imaging. We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs.

Solutions for Integral Boundary Value Problems of Nonlinear Hadamard Fractional Differential Equations

Abstract: In this paper using fixed point methods we establish some existence theorems of positive (nontrivial) solutions for integral boundary value problems of nonlinear Hadamard fractional differential equations.