We introduce procedures for decomposing N N unitary matrices into smaller M M unitary matrices. Our procedures enable designing modular and optimal architectures for implementing arbitrary discrete unitary transformations on light. Such architectures rely on systematically combining the M-mode linear optical interferometers together to implement a given N-mode transformation. Thus this work enables the implementation of large linear optical transformations using smaller modules that act on the spatial or the internal degrees of freedom of light such as polarization, time or orbital angular momentum. The architectures lead to a rectangular gate structure, which is optimal in the sense that realizing arbitrary transformations on these architectures needs a minimal number of optical elements and minimal circuit depth. Moreover, the rectangular structure ensures that each of the different optical modes incurs balanced optical losses, so the architectures promise substantially enhanced process fidelities as compared to existing schemes.

我们介绍了将N N个ary矩阵分解为较小的M M个ary矩阵的过程。我们的程序可以设计模块化和最佳架构，以实现对灯光的任意离散discrete变换。这种架构依赖于将M型线性光学干涉仪系统地组合在一起以实现给定的N模式转换。因此，这项工作使得能够使用较小的模块来实现大型线性光学变换，这些模块作用于光的空间或内部自由度，例如偏振，时间或轨道角动量。这些架构导致了矩形门结构，这是最佳的，因为在这些架构上实现任意转换需要最少的光学元件和最少的电路深度。此外，矩形结构确保了每个不同的光学模式都会产生平衡的光学损耗，因此与现有方案相比，该体系结构有望显着提高工艺保真度。