Approximating Korobov Functions via Quantum Circuits

• URL: http://arxiv.org/abs/2404.14570v2
• Date: Sat, 28 Sep 2024 21:42:02 GMT
• Title: Approximating Korobov Functions via Quantum Circuits
• Authors: Junaid Aftab, Haizhao Yang,
• Abstract summary: We explicitly construct quantum circuits that can approximate $d$-dimensional functions in the Korobov function space. Our work provides quantitative approximation error bounds and estimates the complexity of implementing the proposed quantum circuits.
• Score: 6.460951804337735
• License:
• Abstract: Quantum computing has the potential to tackle large-scale problems in scientific computation, including high-dimensional partial differential equations (PDE). Therefore, understanding the capability of quantum circuits through the lens of approximation theory is essential for evaluating the complexity needed for these circuits to solve such problems. In this paper, we explicitly construct quantum circuits that can approximate $d$-dimensional functions in the Korobov function space. We accomplish this by utilizing the quantum signal processing algorithm and the linear combinations of unitaries technique to build quantum circuits that implement Chebyshev polynomials, which are capable of approximating functions in the Korobov function space. Our work provides quantitative approximation error bounds and estimates the complexity of implementing the proposed quantum circuits. Since the Korobov function space is a subspace of the certain Sobolev spaces which are ubiquitous in studying solutions to high-dimensional PDE, our work develops a theoretical foundation for implementing a large class of functions suitable for applications on a quantum computer.

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