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Chapman & Hall/CRC eBooks

General Quantum Polynomial Interpolation

Svetlin G. Georgiev, Khaled Zennir

Published Apr 3, 2024• 0 citations
Publisher

Abstract

In this chapter we consider the problem of general quantum polynomial interpolation for finding a polynomial that agrees with some information that we have for a given real-valued function f of a single real-variable x . We construct the Lagrange, β -Lagrange, Hermite and β -Hermite interpolation polynomial for a given real-valued function f .

DOI
10.1201/9781003472131-4

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In this chapter we consider the problem of general quantum polynomial interpolation for finding a polynomial that agrees with some information that we have for a given real-valued function f of a single real-variable x . We construct the Lagrange, β -Lagrange, Hermite and β -Hermite interpolation polynomial for a given real-valued function f .
Can a quantum computer efficiently interpolate polynomials? We consider black-box algorithms that seek to learn information about a polynomial fff from input/output pairs (xi,f(xi))(x_i, f(x_i))(xi​,f(xi​)). We define a more general class of \emph{(d,S)(d,S)(d,S)-independent} function properties, where, outside of a set SSS of exceptions, knowing ddd input values does not help one predict the answer. There are essentially two strategies to computing such a function: query d+1d+1d+1 random input values, or search for one of the ∣S∣|S|∣S∣ exceptions. We show that, up to constant factors, we cannot beat these two approaches.
The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.
Can a quantum computer efficiently interpolate polynomials? We consider black-boxalgorithms that seek to learn information about a polynomial f from input/output pairs(xi, f(xi)). We define a more general class of (d, S)-independent function properties,where, outside of a set S of exceptions, knowing d input values does not help one predictthe answer. There are essentially two strategies to computing such a function: queryd + 1 random input values, or search for one of the |S| exceptions. We show that, up toconstant factors, we cannot beat these two approaches.
Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.
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 F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q