  Quantum computing has the potential to address challenging problems in
scientific computation. Therefore, it is important to analyze the capability of
quantum circuits in solving computational problems from the perspective of
approximation theory. In this paper, we explicitly construct quantum circuits
that can approximate $d$-dimensional functions in the Korobov function space,
$X^{2, p}([0,1]^d)$. We achieve this goal by leveraging the quantum signal
processing algorithm and the linear combinations of unitaries technique to
construct quantum circuits that implement Chebyshev polynomials which can
approximate functions in $X^{2, p}([0,1]^d)$. Our work provides quantitative
approximation bounds and estimates the complexity of implementing the proposed
quantum circuits. Since $X^{2, p}(\Omega)$ is a subspace of Sobolev spaces,
$W^{k,p}([0,1]^d)$, for $\max_{1 \leq i \leq d} k_i = 2$, our works develops a
theoretical foundation to implement a large class of functions on a quantum
computer. Our research adds to discussions about merging quantum computing
techniques with scientific computing, suggesting promising paths for using
quantum algorithms to solve challenging computational problems more
efficiently.
