发表机构
Millennium Institute for Research in Optics (MIRO); Departamento de Física, Facultad de Ciencias, Universidad de Chile; Departamento de Física, Universidad de Concepción; Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile（智利光学研究千年研究所; 智利大学理学院物理系; 孔塞普西翁大学物理系; 智利大学物理与数学科学学院物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出通过玻色子哈密顿量在光子集成回路中实现N维离散傅里叶变换，利用单级多模演化替代干涉仪级联，实现O(N log log N)的缩放，显著降低器件复杂度。
AI 中文摘要
离散傅里叶变换（DFT）支撑着许多经典算法，也是量子信息处理中的基本酉算子。在光子集成回路（PICs）中实现$N$维DFT受到当前架构所需的马赫-曾德尔干涉仪级联的限制。这里我们提出玻色子哈密顿量，通过单级多模演化实现$N$维DFT，仅需在相互作用区域前后辅以移相器，其几何结构适用于3D波导。将系统建模为图，其中边对应耦合，顶点为波导，我们获得了完全图$N\leq 6$的解析解和$N\leq 31$的数值解。对于非完全图，哈密顿量需要不同的传播常数。我们报告了$N\leq 8$的所有解、$N=9$的部分探索和$N=10$的选定情况，以及三个指导$N\geq 11$数值搜索的猜想。这些配置规避了波导间距导致的倏逝耦合强度消失问题，一个闭式灵敏度准则选择那些可作为波导布局且对制造误差最不敏感的解。我们还揭示了6维DFT缺失的非仿射参数，并表明用我们的构建块实现$N$维DFT的缩放律为$\mathcal{O}(N\log\log{N})$。这使得仅用2625个干涉仪即可组装2520维DFT，而Reck和Clements架构需要约$3\times 10^6$个干涉仪。
英文摘要
The discrete Fourier transform (DFT) underpins many classical algorithms and is a fundamental unitary operator for quantum information processing. Implementing the $N$-dimensional DFT in photonic integrated circuits (PICs) is limited by the cascades of Mach-Zehnder interferometers that current architectures require. Here we propose bosonic Hamiltonians that realize the $N$-dimensional DFT through a single stage of multimode evolution, complemented only by phase shifters before and after the interaction region, in a geometry suited to 3D waveguides. Modeling the system as a graph, where edges correspond to couplings and the vertices are the waveguides, we obtain analytical solutions for complete graphs up to $N=6$ and numerical solutions up to $N=31$. For non-complete graphs, different propagation constants are required in the Hamiltonian. We report all solutions for $N\leq 8$, partial exploration for $N=9$, and selected cases for $N=10$, together with three conjectures that guide the numerical search for $N \geq 11$. These configurations circumvent the vanishing evanescent coupling strength imposed by the waveguide separation, and a closed-form sensitivity criterion selects those that are admissible as a waveguide layout and least sensitive to fabrication error. We also uncover the missing non-affine parameters of the $6$-dimensional DFT, and show that the scaling law for implementing the $N$-dimensional DFT with our building blocks is $\mathcal{O}(N\log\log{N})$. This allows assembling the $2520$-dimensional DFT with only $2625$ interferometers, in contrast to the $\approx 3\times 10^6$ of Reck and Clements architectures.
Comments21 pages, 13 figures
