Quantum algorithm for multivariate polynomial interpolation

Quantum algorithm for multivariate polynomial interpolation

How many quantum queries are required to determine the coefficients of a degree-d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields [Formula: see text], [Formula: see text] and [Formula: see text]. We show that [Formula: see text] and [Formula: see text] queries suffice to achieve probability 1 for [Formul...

Optimal Quantum Algorithm for Polynomial Interpolation

We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over Fq. A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2 + 1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower b...

On multivariate polynomial interpolation

We provide a map Θ 7→ ΠΘ which associates each finite set Θ of points in C with a polynomial space ΠΘ from which interpolation to arbitrary data given at the points in Θ is possible and uniquely so. Among all polynomial spaces Q from which interpolation at Θ is uniquely possible, our ΠΘ is of smallest degree. It is also Dand scale-invariant. Our map is monotone, thus providing a Newton form for...

Lectures on Multivariate Polynomial Interpolation

On Error Formulas for Multivariate Polynomial Interpolation

In this paper we prove that the existence of an error formula of a form suggested in [2] leads to some very specific restrictions on an ideal basis that can be used in such formulas. As an application, we provide a negative answer to one version of the question posed by Carl de Boor (cf. [2]) regarding the existence of certain minimal error formulas for multivariate interpolation. §