Uniform Approximation by (Quantum) Polynomials

TL;DR

This paper demonstrates how quantum algorithms can re-prove classical approximation theorems, specifically Jackson's Theorem, offering new quantum-based proofs for uniform polynomial approximation of continuous functions.

Contribution

It introduces two quantum algorithm-based proofs of Jackson's Theorem, connecting quantum computing techniques with classical approximation theory.

Findings

Quantum algorithms can effectively re-prove classical approximation theorems.

Two distinct quantum methods—quantum counting and phase estimation—are used.

The proofs provide a quantum perspective on uniform polynomial approximation.

Abstract

We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version of Weierstrass's Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectively on quantum counting and on quantum phase estimation.