Approximating Korobov Functions via Quantum Circuits

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.