Towards the Information-Theoretic Limit of Programmable Photonics

TL;DR

This work establishes an information-theoretic limit on the average phase shift required for universal programmable photonic circuits, showing a fundamental $O(1/\sqrt{N})$ scaling with circuit size. It then demonstrates that a 3-MZI mesh can approach this limit within a factor of about 2, achieving a practical $\sim$10× reduction in average phase shift compared to traditional MZI meshes, and proves that non-unitary (Gaussian) targets can saturate the bound using crossbar architectures. The authors also show that optical neural networks can be trained with all phase shifters constrained to $\lesssim 0.2$ radians without accuracy loss, highlighting the method’s potential for scalable photonic computing. Collectively, the results provide near-optimal, phase-efficient designs for large-scale photonic circuits and new routes for phase-constrained photonic learning and processing.

Abstract

The scalability of many programmable photonic circuits is limited by the $2π$ tuning range needed for the constituent phase shifters. To address this problem, we introduce the concept of a phase-efficient circuit architecture, where the average phase shift is $\ll 2π$. We derive a universal information-theoretic limit to the phase-shift efficiency of universal multiport interferometers, and propose a "3-MZI" architecture that approaches this limit to within a factor of $2\times$, approximately a $10\times$ reduction in average phase shift over the prior art, where the average phase shift scales inversely with system size as $O(1/\sqrt{N})$. For non-unitary circuits, we show that the 3-MZI saturates the theoretical bound for Gaussian-distributed target matrices. Using this architecture, we show optical neural network training with all phase shifters constrained to $\lesssim 0.2$ radians without loss of accuracy.

Figures (8)

• Figure 1: (a) Schematic of a generic multiport interferometer, with four concrete implementations: Reck triangle, Clements rectangle, MPLC, and programmable MMI (clockwise from bottom left). (b) Representation of a programmable circuit consisting of discrete phase shifters (colored) and fixed coupler unitaries (gray), with (c) the corresponding design of a Clements mesh, and (d) the mathematical description of the unitary as a product of phase-shift $D(\psi_i)$ and coupler $U_i$ matrices. (e) Related push-pull construction for mesh-like multiport interferometers.
• Figure 2: Visualization of the map $f: \vec{\psi} \rightarrow U$ implemented by a multiport interferometer, where the unit volume ${\rm d} V_\psi$ is mapped to a parallelepiped spanned by the gradient vectors $\nabla_{\psi_m} f$ and has a volume ${\rm d} V_U = |\partial(U)/\partial(\psi)| {\rm d} V_\psi$.
• Figure 3: (a) $8\times 8$ Clements mesh, which consists of a rectangular array of tunable crossings (the phase screen is omitted for clarity). (b) MZI crossing and its representation by a $2\times 2$ matrix $T$ (Eq. (\ref{['eq:tmat']})) with splitting ratio $s$. (c) $P(s)$ for MZIs of rank 1, 3, 5, and 7 for Haar-uniform unitaries, showing how the distribution becomes concentrated as one moves to the center of the mesh.
• Figure 4: (a) Standard MZI crossing, which maps $(\theta, \phi) \rightarrow s$ via polar coordinates. The probability $P(s)$ for an $N = 16$ mesh is plotted to show the concentration near $s = 0$. (b) Probability $P(\theta, \phi)$ as a function of mesh size, showing confinement in $\theta$ but not $\phi$. (c) 3-MZI crossing, whose mapping is locally Cartesian near $s = 0$ and which confines both $\theta$ and $\phi$. (d) Use of fabrication-induced phase offsets to shift the distribution $P(\theta, \phi)$ to center around zero.
• Figure 5: (a) Visualization of meshes of size $N = 4$--1024, programmed to realize specific Haar-uniform sampled matrices, where MZIs are colored according to their average phase shift $|\psi| \equiv \tfrac{1}{2}(|\theta| + |\phi|)$. (b) Plot of the phase-shift moments as a function of mesh size (left) and the MZI/3-MZI ratio for each moment type, illustrating the advantage of the latter mesh type (right). (c) Entropy per degree of freedom of the MZI and 3-MZI meshes, as compared to the information-theoretic bound. Both generic and push-pull (PP) bounds are plotted. (d) Phase-shift moments as compared to the bound.