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.

理解参数化量子电路在机器学习任务中的能力是量子机器学习中最重要的问题之一。本文通过函数逼近的角度分析了参数化量子电路的表达能力，并展示了在逼近连续和平滑函数方面数据重新上传参数化量子电路的明确构造，以及其在宽度、深度和可训练参数数量方面的定量逼近误差界限。此外，我们还将提出的参数化量子电路与近乎最优的深度神经网络进行了比较，在逼近高维平滑函数方面，表明参数化量子电路和深度神经网络之间的模型尺寸比在输入维度方面呈指数级减小，这为展示量子机器学习的量子优势提供了潜在的新途径。

参数化量子电路在函数近似中的可证好处

Provable Advantage of Parameterized Quantum Circuit in Function  Approximation

Training DRL agents is often a time-consuming process as a large number of
samples and environment interactions is required. This effect is even amplified
in the case of Batch RL, where the agent is trained without environment
interactions solely based on a set of previously collected data. Novel
approaches based on quantum computing suggest an advantage compared to
classical approaches in terms of sample efficiency. To investigate this
advantage, we propose a batch RL algorithm leveraging VQC as function
approximators in the discrete BCQ algorithm. Additionally, we present a novel
data re-uploading scheme based on cyclically shifting the input variables'
order in the data encoding layers. We show the efficiency of our algorithm on
the OpenAI CartPole environment and compare its performance to classical neural
network-based discrete BCQ.

该研究论文提出了一种基于量子计算的 VQC 算法结合数据重载方案的批量 RL 算法，并在 OpenAI CartPole 环境下与基于神经网络的离散 BCQ 算法进行了比较，证明了其在效率上的优势。