Quantum Circuit Encodings of Polynomial Chaos Expansions

Abstract: This work investigates the expressive power of quantum circuits in approximating high-dimensional, real-valued functions. We focus on countably-parametric holomorphic maps $u:U\to \mathbb{R}$, where the parameter domain is $U=[-1,1]^{\mathbb{N}}$. We establish dimension-independent quantum circuit approximation rates via the best $n$-term truncations of generalized polynomial chaos (gPC) expansions of these parametric maps, demonstrating that these rates depend solely on the summability exponent of the gPC expansion coefficients. The key to our findings is based on the fact that so-called ``$(\boldsymbol{b},\epsilon)$-holomorphic'' functions, where $\boldsymbol{b}\in (0,1]^\mathbb N \cap \ell^p(\mathbb N)$ for some $p\in(0,1)$, permit structured and sparse gPC expansions. Then, $n$-term truncated gPC expansions are known to admit approximation rates of order $n^{-1/p + 1/2}$ in the $L^2$ norm and of order $n^{-1/p + 1}$ in the $L^\infty$ norm. We show the existence of parameterized quantum circuit (PQC) encodings of these $n$-term truncated gPC expansions, and bound PQC depth and width via (i) tensorization of univariate PQCs that encode Cheby\v{s}ev-polynomials in $[-1,1]$ and (ii) linear combination of unitaries (LCU) to build PQC emulations of $n$-term truncated gPC expansions. The results provide a rigorous mathematical foundation for the use of quantum algorithms in high-dimensional function approximation. As countably-parametric holomorphic maps naturally arise in parametric PDE models and uncertainty quantification (UQ), our results have implications for quantum-enhanced algorithms for a wide range of maps in applications.