Continuous Functions as Quantum Operations: a probabilistic approximation

TL;DR

This paper extends the classical Stone-Weierstrass theorem to quantum operations, showing that polynomial quantum operations can probabilistically approximate any continuous function from [0,1]^n to [0,1].

Contribution

It introduces polynomial quantum operations and establishes their capacity to approximate continuous functions within a quantum framework.

Findings

Quantum operations can approximate continuous functions probabilistically.

Polynomial quantum operations form a dense class for function approximation.

The approach bridges classical approximation theory and quantum information.

Abstract

In this note we propose a version of the classical Stone-Weierstrass theorem in the context of quantum operations, by introducing a particular class of quantum operations, dubbed polynomial quantum operations. This result permits to interpret from a probabilistic point of view, and up to a certain approximation, any continuous function from the real cube [0; 1]^n to the real interval [0; 1] as a quantum operation.