Understanding the power of parameterized quantum circuits (PQCs) in accomplishing machine learning tasks is one of the most important questions in quantum machine learning. In this paper, we analyze the expressivity of PQCs through the lens of function approximation. Previously established universal approximation theorems for PQCs are mainly nonconstructive, leading us to the following question: How large do the PQCs need to be to approximate the target function up to a given error? We exhibit explicit constructions of data re-uploading PQCs for approximating continuous and smooth functions and establish quantitative approximation error bounds in terms of the width, the depth and the number of trainable parameters of the PQCs. To achieve this, we utilize techniques from quantum signal processing and linear combinations of unitaries to construct PQCs that implement multivariate polynomials. We implement global and local approximation techniques using Bernstein polynomials and local Taylor expansion and analyze their performances in the quantum setting. We also compare our proposed PQCs to nearly optimal deep neural networks in approximating high-dimensional smooth functions, showing that the ratio between model sizes of PQC and deep neural networks is exponentially small with respect to the input dimension. This suggests a potentially novel avenue for showcasing quantum advantages in quantum machine learning.

翻译：理解参数化量子电路在完成机器学习任务中的能力是量子机器学习中最重要的课题之一。本文从函数逼近的角度分析了参数化量子电路的表示能力。先前建立的参数化量子电路通用逼近定理主要基于非构造性方法，这引发了以下问题：为实现目标函数的给定误差逼近，参数化量子电路需要达到多大规模？我们针对连续和光滑函数的逼近，明确构造了数据重上传参数化量子电路，并建立了关于电路宽度、深度和可训练参数数量的定量逼近误差界。为此，我们利用量子信号处理和酉线性组合技术，构建了实现多元多项式的参数化量子电路。采用伯恩斯坦多项式和局部泰勒展开实现了全局与局部逼近技术，并分析了它们在量子场景中的性能。此外，我们还将所提出的参数化量子电路与近似最优的深度神经网络在高维光滑函数逼近方面进行比较，结果表明参数化量子电路与深度神经网络的模型规模之比随输入维度呈指数级缩小。这为展示量子机器学习中的量子优势提供了潜在的新路径。