算子学习与变分量子电路
Operator Learning with Variational Quantum Circuits
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中文总结 AI 辅助
提出一种基于变分量子电路扩展DeepONet框架的量子机器学习方法，用于学习微分方程解算子，在浅电路下误差低且无贫瘠高原，误差界优于经典方法。
中文摘要 AI 辅助
我们开发了一种新的量子机器学习方法，使变分量子电路能够学习微分方程的线性和非线性解算子。这是通过扩展DeepONet（一种用于建模微分方程的技术）的框架来实现的。分支网络和主干网络被替换为变分量子电路，利用了量子通用逼近定理。该方法在多个微分方程上进行了测试，即使在较浅的电路深度下也能获得较低的逼近误差和泛化误差。此外，在训练过程中不会遇到贫瘠高原问题。与经典DeepONet相比，该模型在误差和参数缩放方面表现出改进。这是由于逼近误差取决于输入函数空间的维度而非传感器位置的数量这一内在特性。我们建立了一个理论框架，以显式误差界的形式衡量截断、分支和主干电路误差对总算子逼近误差的贡献。在$L^2$、$L^p$、$C^0$和Sobolev $H^k$范数下给出了误差界的完整推导，并进行了经典-量子对比分析（作为补充材料）。
英文摘要
A new methodology is developed for quantum machine learning which enables variational quantum circuits to learn linear and non-linear solution operators to differential equations. This is devised by extending the framework of the DeepONet, a technique for modeling differential equations. The branch and trunk neural networks are replaced by variational quantum circuits, leveraging the quantum universal approximation theorem. The methodology is tested on several differential equations and yields low approximation & generalization errors even with shallow circuit depths. In addition, it does not encounter barren plateaus during training. The model displays improved error and parameter scaling relative to the classical DeepONet. This is due to the intrinsic property that the approximation error depends on the dimension of the input function space, rather than the number of sensor locations. A theoretical framework is established in the form of explicit error bounds, which measure the contributions of the truncation, branch, and trunk circuit errors to the total operator approximation error. Full derivations for the error bounds are developed in the norms $L^2$, $L^p$, $C^0$, and Sobolev $H^k$, along with a classical-quantum comparative analysis (as Supplemental Materials).
发表机构
- University of Pittsburgh（匹兹堡大学）
机构由 AI 辅助整理，请以论文原文为准。
