Related papers: Quantum interpolation of polynomials

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…

The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the…

Recently proposed implementations of quantum computer suffer from unavoidable interaction between quantum bits depending upon data being written in them. Novel procedure of avoiding multiqubit errors arising due to uncontrollable…

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…

Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…

It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure…

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on…

We present a quantum probabilistic algorithm which tests with a polynomial computational complexity whether a given composite number is of the Carmichael type. We also suggest a quantum algorithm which could verify a conjecture by…

Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting…

A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical…

Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited…

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition…

We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…

We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…

Let K be F_q((T)), or more generally any field of characteristic p equipped with a valuation having a finite residue field of q elements. Then a polynomial f(x) in K[x] having k+1 nonzero coefficients has at most q^k distinct zeros in K. We…

We propose a realization of quantum computing using polarized photons. The information is coded in two polarization directions of the photons and two-qubit operations are done using conditional Faraday effect. We investigate the performance…

This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…

We present a FFT-based algorithm for the computation of a polynomial's coefficients from its roots, and apply it to obtain the coefficients of interpolation polynomials, to invert Vandermondians and to evaluate the symmetric functions of a…

The purpose this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta function which interpolates multiple q-Euler numbers at negative integers.

We investigate the power of quantum computers when they are required to return an answer that is guaranteed correct after a time that is upper-bounded by a polynomial in the worst case. In an oracle setting, it is shown that such machines…