AI 中文总结
量子SEDONet是一种用于偏微分方程的谱嵌入量子深度算子网络，通过为坐标匹配边界的谱基降低误差，在四个基准测试中显著减少平均相对L2误差且无额外量子资源成本。
AI 中文摘要
量子DeepONet通过在量子计算机上评估正交参数化网络来加速神经算子推理，在理想模拟中重现经典对应模型的精度，同时实现渐近更低的推理成本。然而，其主干网络接收的查询坐标具有有限的谱结构，需要网络通过非线性特性学习振荡特征。我们提出量子SEDONet（Spectral-Embedded Deep Operator Network，谱嵌入深度算子网络），该模型根据边界条件为每个主干坐标分配谱基：周期坐标采用傅里叶特征，有界非周期坐标采用切比雪夫特征。该基按坐标而非按问题选择，允许单个问题内同时使用两种表示。在一元振幅编码下，当嵌入维度保持在网络宽度范围内时，嵌入不会增加额外量子比特或电路深度，仅使参数数量增加几个百分点。在四个基准测试中，量子SEDONet使平均相对L2误差降低：反导数问题降低54.1%，平流问题降低49.6%，伯格斯问题降低36.0%，混合边界通道泊松问题降低36.2%。量子与经典评估路径全程在10^-8范围内一致。通道泊松问题同时在周期方向使用傅里叶特征、在有界方向使用切比雪夫特征，证明了按坐标匹配边界的谱嵌入无需额外量子资源成本。
英文摘要
Quantum DeepONet accelerates neural-operator inference by evaluating an orthogonally parameterized network on a quantum computer, reproducing in ideal simulation the accuracy of its classical counterpart at asymptotically lower inference cost. Its trunk network, however, receives query coordinates with limited spectral structure, requiring the network to learn oscillatory features through its nonlinearities. We propose Quantum SEDONet (Spectral-Embedded Deep Operator Network), which assigns each trunk coordinate a spectral basis according to its boundary condition: Fourier features for periodic coordinates and Chebyshev features for bounded, non-periodic coordinates. The basis is selected per coordinate rather than per problem, allowing both representations within a single problem. Under unary amplitude encoding, the embedding incurs no additional qubits or circuit depth when its dimension remains within the network width, while increasing the parameter count by only a few percent. Across four benchmarks, Quantum SEDONet reduces the mean relative L2 error by 54.1% for the antiderivative, 49.6% for advection, 36.0% for Burgers, and 36.2% for a mixed-boundary channel Poisson problem. Quantum and classical evaluation paths agree to within 10^-8 throughout. The channel Poisson problem simultaneously uses Fourier features in the periodic direction and Chebyshev features in the bounded direction, demonstrating coordinate-wise boundary-matched spectral embedding without additional quantum-resource cost.
