# Photonic Integrated Reconfigurable Linear Processors as Neural Network Accelerators
## 3 Reconfigurable Linear Optical Processors  
The basic element of a linear optical processor is a \(2\times 2\) reconfigurable Mach-Zehnder interferometer (MZI). It consists of two 3 dB couplers (realized by means of \(2\times 2\) multimode interferometers (MMI) or directional couplers) with a phase shifter on one of the internal arms (\(\theta\)) and another phase shifter at one of the outputs after the second coupler (\(\phi\)), as shown in the inset of Figure 1. This MZI can implement any matrix in the special unitary group of degree two (i.e., SU(2)), composed of all complex square matrices whose conjugate transpose is equal to its inverse (unitary) and with a determinant equal to 1 (special unitary) [33]. The MZI can be programmed by adjusting the two phase shifters, \(\theta\) and \(\phi\).  
By properly combining multiple MZIs, an \(N\)-input \(N\)-output structure can be created, which is capable of implementing any matrix in a special unitary group of degree \(N\) (i.e., \(\text{SU}(N)\)) [34]. For instance, the device sketched in Figure 1 is a \(4\times 4\) linear optical processor able to implement any \(\text{SU}(4)\), followed by a diagonal matrix multiplication section. The diagonal section (depicted as \(\Sigma\) in Figure 1) makes it possible to extend the structure, by adding another \(\text{SU}(4)\), and thus carry out vector matrix multiplications with an arbitrary \(4\times 4\) matrix by exploiting a physical implementation of the singular value decomposition (SVD) theorem [5]. These vector matrix multiplications constitute the linear operations performed in a neural network layers, followed by the application of the nonlinear activation function [14].  
Several MZI arrangements, sketched in Figure 2, can be chosen to implement such reconfigurable linear optical processors. MZIs can be organized in a triangular shape, i.e., a \"Reck\" architecture [34], shown in Figure 2a. With \(N\) inputs, this architecture needs \(N(N-1)/2\) MZIs to implement the \(N\times N\) special unitary matrix, with a processor depth, defined as the maximum number of MZIs crossed on any input-output path, equal to \(2N-3\). Alternatively, in the \"Clements\" architecture, the same number of MZIs are arranged in a rectangular shape [35], as exemplified in Figure 2b. With the \"Clements\" architecture, optical path lengths are balanced and the processor depth is reduced to \(N\). Recently, a diamond architecture has been proposed as an extended version of the \"Reck\" topology [36], obtained by adding \((N-1)(N-2)/2\) MZIs to the triangular one. This architecture, depicted in Figure 2c, requires \((N-1)^{2}\) MZIs, and has the same processor depth as the \"Reck\", \(2N-3\).  
Both the \"Clements\" and the diamond architectures are more robust to imperfections, noise, and loss with respect to the \"Reck\" one, translating into higher accuracies when used in neuromorphic applications [35; 36]. However, the diamond mesh requires more MZIs, thus resulting in less power and footprint, and a more complex calibration. We also remark that, for all topologies, the number of phase shifters to be controlled scales quadratically with the size \(N\) of the optical processor (i.e., with the number of input/output ports), which makes the handling of electrical I/O a nontrivial task.  
Figure 1: Scheme of the linear optical processor and measurement setup (MZI: Mach–Zehnder interferometer; DC: directional coupler; PD: photodetector; ESA: electrical spectrum analyzer).  
Figure 2: Topologies for reconfigurable linear optical processors based on MZI basic elements: (a) “Reck” architecture, (b) “Clements” architecture, (c) diamond architecture.