标题： 量子电路编码的多项式混沌展开 显示英文标题

标题： Quantum Circuit Encodings of Polynomial Chaos Expansions

摘要： 本工作研究了量子电路在逼近高维实值函数方面的表达能力。 我们专注于可数参数的全纯映射 $u:U\to \mathbb{R}$，其中参数域为 $U=[-1,1]^{\mathbb{N}}$。 通过这些参数映射的广义多项式混沌（gPC）展开的最佳 $n$-项截断，我们得出了与维度无关的量子电路逼近速率，表明这些速率仅依赖于gPC展开系数的可求和指数。 我们的发现的关键在于所谓的“$(\boldsymbol{b},\epsilon)$-全纯”函数，其中 $\boldsymbol{b}\in (0,1]^\mathbb N \cap \ell^p(\mathbb N)$对某个 $p\in(0,1)$成立，允许结构化且稀疏的gPC展开。 然后，已知$n$项截断的 gPC 展开式在$L^2$范数下的逼近阶为$n^{-1/p + 1/2}$，在$L^\infty$范数下的逼近阶为$n^{-1/p + 1}$。 我们展示了这些$n$-项截断gPC展开式具有参数化量子电路（PQC）编码的存在性，并通过以下两种方法界定了PQC的深度和宽度：(i) 对在$[-1,1]$中编码切比雪夫多项式的单变量PQC进行张量化，以及(ii) 利用幺正算子的线性组合（LCU）构建$n$-项截断gPC展开式的PQC模拟。 这些结果为高维函数逼近中使用量子算法提供了严格的数学基础。 由于可数参数化的全纯映射在参数化偏微分方程模型和不确定性量化（UQ）中自然出现，我们的成果对应用领域中广泛映射的量子增强算法具有重要意义。 显示英文摘要

摘要： 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.