The Goldilocks Principle of Learning Unitaries by Interlacing Fixed Operators with Programmable Phase Shifters on a Photonic Chip

TL;DR

This work systematically investigates a general family of photonic circuits for realizing arbitrary unitaries based on a simple architecture that interlaces a fixed intervening layer with programmable phase shifter layers and introduces a criterion for the intervening operator that guarantees the universality of this architecture for representing arbitrary N×N.

Abstract

Programmable photonic integrated circuits represent an emerging technology that amalgamates photonics and electronics, paving the way for light-based information processing at high speeds and low power consumption. Programmable photonics provides a flexible platform that can be reconfigured to perform multiple tasks, thereby holding great promise for revolutionizing future optical networks and quantum computing systems. Over the past decade, there has been constant progress in developing several different architectures for realizing programmable photonic circuits that allow for realizing arbitrary discrete unitary operations with light. Here, we systematically investigate a general family of photonic circuits for realizing arbitrary unitaries based on a simple architecture that interlaces a fixed intervening layer with programmable phase shifter layers. We introduce a criterion for the intervening operator that guarantees the universality of this architecture for representing arbitrary $N \times N$ unitary operators with $N+1$ phase layers. We explore this criterion for different photonic components, including photonic waveguide lattices and meshes of directional couplers, which allows the identification of several families of photonic components that can serve as the intervening layers in the interlacing architecture. Our findings pave the way for efficiently designing and realizing novel families of programmable photonic integrated circuits for multipurpose analog information processing.

Figures (5)

• Figure 1: Universal architecture scheme. The proposed architecture involving alternating layers of random unitary matrices $F$ and diagonal phase shifts layers (PL) $\{\phi_{n}^{(p)}\}$, with $p=1,\ldots,N+1$. The upper insets depict the modulus and argument of the potential candidates for the unitary matrix $F$, which have been selected as the DFT, DFrFT, and a random unitary matrix. The lower insets illustrate potential photonic implementations to perform the unitary matrix F.
• Figure 2: Numerical universality test. (a) Architecture depiction (left column) and optimization objective function (right column) for 100 target matrices at various values of M and N. Black boxes denote any possible realization for the $F$ matrix. (b) Multiple trials for $N=8$ and $M=9$ using 250 random $F$ matrices were considered; 250 targets were used for each matrix $F$. Shown is the distribution of the number of LMA runs to achieve a norm lower than the stopping norm of $10^{-10}$, with a maximum of 50 iterations per run. (c) Norm (log${}_{10}$) in terms of the number of iterations for the run with the best norm. Using 100 random matrices $F$, each with a single target matrix.
• Figure 3: Density estimation and performance test. (a) Points $\Vec{R}$ associated with density estimation for the set of unitary matrices $\{e^{A_{j}},e^{D_{j}}\}_{j=1}^{50}$ (left column) and $\{e^{B_{j}},e^{C_{j}}\}_{j=1}^{50}$ (right column). The blue heat maps denote the absolute value, $\widetilde{\mathcal{U}}$, for some particular choices of unitary matrices. (b) Error norm (log$_{10}$) $L$ in \ref{['Q']} for each unitary matrix under consideration with fifty testing targets per matrix. (c) Mean and standard deviation $N\widetilde{\mu}$ and $N\widetilde{\sigma}$, respectively, related to the density estimation for each unitary matrix in (a). The horizontal blue and red lines denote the universality threshold for $N\widetilde{\mu}$ and $N\widetilde{\sigma}$, respectively.
• Figure 4: Photonic lattice realization and their universality. Sketch for the waveguide array associated with the $J_{x}$ lattice (a), homogeneous lattice (b), and homogenous lattice with disorder effects (c). Density criterion as a function of the lattice length $\ell$ (d-f) and the corresponding numerical performance test at the reference lengths $\ell^{(m)}_{j}$ and $\ell^{(M)}_{j}$ (g-i) for the $J_x$ lattice (left column), homogeneous lattice (central column), and disordered lattice (right column).
• Figure 5: Geometric array using power dividers. (a) Power divider array composed of $p$ layers as defined in \ref{['PD']}. Light-shaded and dark-shaded layers denote $\mathcal{L}_{1}=\mathbb{I}_{5}\otimes\mathcal{T}_{0}$ and $\mathcal{L}_{2}=\mathbb{I}_{1}\oplus(\mathbb{I}_{4}\otimes\mathcal{T}_{0})\oplus\mathbb{I}_{1}$, respectively. Density criterion (b) and error norm (log$_{10} L$) (c) of the mesh architecture in (a) as a function of the number of layers $p$.