发表机构
Laboratory of Signals and Systems, CentraleSupélec, CNRS, Paris-Saclay University; Fédération de Mathématiques de CentraleSupélec, CentraleSupélec, CNRS, Paris-Saclay University（信号与系统实验室，中央苏佩莱克学院，法国国家科学研究中心，巴黎萨克雷大学; 中央苏佩莱克数学联合会，中央苏佩莱克学院，法国国家科学研究中心，巴黎萨克雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过傅里叶表示分析对称性对量子傅里叶模型表达能力的影响，提出对称化与随机编码方法，并在量子物理信息神经网络中验证其有效性，降低误差。
AI 中文摘要
几何量子机器学习将对称性融入量子模型，但对称性如何塑造其表达能力并指导有效的模型设计仍未被充分理解。我们通过量子傅里叶模型（QFMs）的傅里叶表示来研究这一问题。对称性将频率谱组织成轨道，并将每个轨道内的傅里叶系数求和为一个对称化系数。当QFM可训练层形成独立的精确2-设计时，每个对称化系数的方差等于其轨道内系数方差之和。对于具有ε-近似2-设计可训练层的单层QFM，我们限定了与此恒等式的偏差。由此得到的单个傅里叶系数的界限可能比现有界限指数级地更紧。超八面体群提供了轨道增长的例子，可以缓解对称化系数表达能力的消失。在基本阿贝尔2-群上的QFM对称化也产生纯多元切比雪夫多项式基函数。我们引入了随机编码，它在不借助辅助量子比特或额外电路深度的情况下实现不变模型，用于量子扭曲。我们将这些模型作为量子物理信息神经网络（QPINNs）评估于二维屏蔽泊松和稳态粘性哈密顿-雅可比方程。在硬边界约束下，使用精确对称化和随机编码的QPINNs分别在两个基准测试中实现了最低的平均误差。
英文摘要
Geometric quantum machine learning incorporates symmetry into quantum models, but how symmetry shapes their expressivity and guides effective model design remains insufficiently understood. We address this question through the Fourier representation of quantum Fourier models (QFMs). Symmetry organizes the frequency spectrum into orbits and sums the Fourier coefficients within each orbit into a symmetrized coefficient. When QFM trainable layers form independent exact 2-designs, the variance of each symmetrized coefficient equals the sum of the coefficient variances in its orbit. For single-layer QFMs with $\varepsilon$-approximate 2-design trainable layers, we bound the deviation from this identity. The resulting bound for individual Fourier coefficients can be exponentially tighter than an existing bound. The hyperoctahedral group provides an example of orbit growth that can mitigate vanishing expressivity of the symmetrized coefficients. Symmetrization of QFMs over an elementary abelian 2-group also yields pure multivariate Chebyshev polynomial basis functions. We introduce randomized encoding, which implements invariant models without ancilla qubits or the additional circuit depth for quantum twirling. We evaluate the models as quantum physics-informed neural networks (QPINNs) on two-dimensional screened Poisson and stationary viscous Hamilton-Jacobi equations. Under hard boundary constraints, QPINNs using exact symmetrization and randomized encoding achieve the lowest mean errors in the two benchmarks, respectively.
Comments47 pages, 9 figures
