scholarly journals Rational approximations and quantum algorithms with postselection

2015 ◽  
Vol 15 (3&4) ◽  
pp. 295-307
Author(s):  
Urmila Mahadev ◽  
Ronald de Wolf
Keyword(s):  
Boolean Function ◽  
Value Function ◽  
Quantum Algorithms ◽  
Rational Functions ◽  
Query Complexity ◽  
Rational Approximations ◽  
Absolute Value ◽  
Majority Function ◽  
Low Degree ◽  
Classic Theorem

We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a factor of 2) the minimal query complexity of the latter. We give optimal (up to constant factors) quantum algorithms with postselection for the Majority function, slightly improving upon an earlier algorithm of Aaronson. Finally we show how Newman's classic theorem about low-degree rational approximation of the absolute-value function follows from these algorithms.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals An Analytic Expression for the Inverse Involute

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Alberto López Rosado ◽  
Federico Prieto Muñoz ◽  
Roberto Alvarez Fernández
Keyword(s):  
Approximation Theory ◽  
Jacobi Polynomials ◽  
Rational Functions ◽  
Rational Approximations ◽  
Efficient Computation ◽  
Remez Algorithm ◽  
Tangent Function ◽  
Analytic Expression ◽  
Involute Curve ◽  
Theoretical Foundations

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

scholarly journals Polar Metric-Weighted Norm-Based Scan Matching for Robot Pose Estimation

2016 ◽  
Vol 2016 ◽  
pp. 1-20
Author(s):  
Guanglei Huo ◽  
Lijun Zhao ◽  
Ke Wang ◽  
Ruifeng Li ◽  
Jianqiang Li
Keyword(s):  
Pose Estimation ◽  
Value Function ◽  
Rotation Angle ◽  
Accurate Estimation ◽  
Weighted Norm ◽  
Map Building ◽  
Scan Matching ◽  
Constant Weight ◽  
Absolute Value ◽  
Point To Point

A novel point-to-point scan matching approach is proposed to address pose estimation and map building issues of mobile robots. Polar Scan Matching (PSM) and Metric-Based Iterative Closest Point (Mb-ICP) are usually employed for point-to-point scan matching tasks. However, due to the facts that PSM considers the distribution similarity of polar radii in irrelevant region of reference and current scans and Mb-ICP assumes a constant weight in the norm about rotation angle, they may lead to a mismatching of the reference and current scan in real-world scenarios. In order to obtain better match results and accurate estimation of the robot pose, we introduce a new metric rule, Polar Metric-Weighted Norm (PMWN), which takes both rotation and translation into account to match the reference and current scan. For robot pose estimation, the heading rotation angle is estimated by correspondences establishing results and further corrected by an absolute-value function, and then the geometric property of PMWN called projected circle is used to estimate the robot translation. The extensive experiments are conducted to evaluate the performance of PMWN-based approach. The results show that the proposed approach outperforms PSM and Mb-ICP in terms of accuracy, efficiency, and loop closure error of mapping.

scholarly journals The equation y′ = fy in zero residue characteristic

1991 ◽  
Vol 33 (2) ◽  
pp. 149-153
Author(s):  
Alain Escassut ◽  
Marie-Claude Sarmant
Keyword(s):  
Uniform Convergence ◽  
Characteristic Zero ◽  
Residue Class ◽  
Rational Functions ◽  
Closed Field ◽  
Class Field ◽  
Absolute Value ◽  
Residue Class Field ◽  
Analytic Elements ◽  
Residue Characteristic

Let K be an algebraically closed field complete with respect to an ultrametric absolute value |.| and let k be its residue class field. We assume k to have characteristic zero (hence K has characteristic zero too).Let D be a clopen bounded infraconnected set [3] in K, let R(D) be the algebra of the rational functions with no pole in D, let ‖.‖D be the norm of uniform convergence on D defined on R(D), and let H(D) be the algebra of the analytic elements on D i.e. the completion of R(D) for the norm ‖.‖D.

scholarly journals Following Forrelation – quantum algorithms in exploring Boolean functions' spectra

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Suman Dutta ◽  
Subhamoy Maitra ◽  
Chandra Sekhar Mukherjee
Keyword(s):  
Boolean Functions ◽  
Cross Correlation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Sampling Technique ◽  
Linear Functions ◽  
Query Complexity ◽  
Walsh Spectrum ◽  
Correlation Spectrum ◽  
The Cross

<p style='text-indent:20px;'>Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Quantum conditional query complexity

2017 ◽  
Vol 17 (7&8) ◽  
pp. 541-567
Author(s):  
Imdad S.B. Sardharwalla ◽  
Sergii Strelchuk ◽  
Richard Jozsa
Keyword(s):  
Boolean Functions ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Query Complexity ◽  
Equivalence Testing ◽  
Conditional Probabilities ◽  
Underlying Distribution ◽  
New Type ◽  
Testing Equivalence ◽  
Oracle Access

We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or e-far from balanced; and (c) give an algorithm, requiring O˜(n/e) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.

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