Approximating Korobov Functions via Quantum Circuits

TL;DR

This paper develops quantum circuits based on QSP and LCU algorithms to approximate functions in Korobov spaces, providing a theoretical foundation for quantum function approximation in scientific computing.

Contribution

It introduces a novel quantum circuit construction for Korobov function approximation using QSP and LCU, with analysis of error rates and complexity.

Findings

Quantum circuits can approximate Korobov functions with quantifiable error bounds.

The proposed circuits leverage Chebyshev polynomial outputs for function approximation.

Complexity analysis shows feasible implementation for high-dimensional functions.

Abstract

Understanding the capacity of quantum circuits through the lens of approximation theory is essential for evaluating the complexity of quantum circuits required to solve various problems in scientific computation. We design quantum circuits capable of approximating d-dimensional functions within the Korobov function space. This is achieved by leveraging the quantum signal processing (QSP) and the linear combination of unitaries (LCU) algorithms to build quantum circuits that output Chebyshev polynomials. We also present a quantitative analysis of the approximation error rates and evaluates the computational complexity of implementing the proposed circuits. Since the Korobov function space is a subspace of the certain Sobolev spaces, our work develops a theoretical foundation for implementing a large class of functions suitable for applications on a quantum computer.