The Goldilocks Principle of Learning Unitaries with Photonic Circuits

Introduction

Photonics, the science of light, has revolutionized the way we process and transmit information. With the advent of programmable photonic integrated circuits, we can now harness the unique properties of light to perform complex computations at high speeds and low power consumption. One of the key challenges in this field is the realization of arbitrary unitary operations, which are fundamental building blocks for various applications, including quantum computing, optical signal processing, and neuromorphic computing.

In this article, we will explore a novel architecture for realizing arbitrary unitary operations using photonic circuits. Specifically, we will delve into the concept of the "Goldilocks principle," which provides a systematic approach to designing and optimizing these circuits. We will also discuss the potential photonic platforms and feasible realizations, paving the way for the development of efficient and robust programmable photonic devices.

The Interlaced Architecture

The proposed architecture is based on a layered configuration, where programmable phase shifter layers are interlaced with a fixed intervening layer, represented by a unitary matrix F. Mathematically, this architecture can be expressed as:

Here, U is the desired arbitrary N × N unitary matrix, and Pk = e^(iDk) are diagonal phase matrices, where Dk = diag(φ1^(k), ..., φN^(k)) and φn^(k) ∈ (0, 2π]. The phase shifters φn^(k) are the tunable parameters that can be adjusted to reconstruct the target unitary matrix Ut with minimal error.

The key to achieving universality, the ability to represent any unitary matrix, lies in the careful selection of the intervening matrix F. Numerical simulations have shown that generating Haar random matrices for F leads to the desired universality in most cases.

The Goldilocks Principle

While Haar random matrices perform well, generating and testing them for universality can be computationally expensive, especially for large architectures. To address this challenge, we introduce the "Goldilocks principle," a density criterion that allows us to preselect suitable matrices F without extensive optimization.

The density criterion is based on the observation that sparse matrices, such as diagonal or block-diagonal matrices, fail to achieve universality. On the other hand, maximally dense matrices, like the Discrete Fourier Transform (DFT) matrix, are ideal candidates. The Goldilocks principle aims to identify matrices that strike a balance between these two extremes, ensuring the required density for universality.

To quantify the density of a unitary matrix F, we introduce the vector R = (N μ~, N σ~), where μ~ and σ~ are the mean and standard deviation of the variances associated with the modulus of the columns (or rows) of F. By analyzing the distribution of R, we can define a "Goldilocks region" in the (N μ~, N σ~) plane, where matrices F are expected to exhibit universality.

The boundaries of this region are determined by considering the extremal cases of sparse and dense matrices, as well as additional reference points derived from block-diagonal matrices. Matrices whose R vectors fall within the Goldilocks region are considered suitable candidates for the intervening layer F, allowing for efficient preselection and design optimization.

Figure 1 presents a universal architecture scheme featuring alternating layers of random unitary matrices (F) and diagonal phase shift layers (PL) with indices p ranging from 1 to N+1. The upper insets show the modulus and argument of potential candidates for the unitary matrix F, selected as the DFT, DFrFT, and a random unitary matrix. The lower insets demonstrate possible photonic implementations for the unitary matrix F.

Numerical Results and Performance Evaluation

To validate the Goldilocks principle, we performed extensive numerical simulations using various photonic platforms and feasible realizations for the intervening matrix F. The results demonstrate the effectiveness of the proposed approach and provide insights into the design considerations for universal photonic circuits.

Figure 2 illustrates the numerical universality test for different values of M (the number of phase shifter layers) and N (the number of ports). The error norm L, defined as ||U - Ut||^2 / N^2, exhibits a phase transition between M = N and M = N + 1, confirming the requirement of at least N + 1 phase layers for universality.

Figure 3 showcases the density estimation and performance test for randomly generated unitary matrices. The shaded blue area represents the Goldilocks region, where universality is expected for N = 6. The heat maps depict the modulus of selected unitary matrices, and the corresponding error norms and density measures (Nμ~ and Nσ~) are plotted. The results demonstrate that matrices within the Goldilocks region indeed lead to the desired universality, validating the effectiveness of the density criterion.

Photonic Platforms and Feasible Realizations

The proposed architecture can be implemented using various photonic platforms and components. One promising approach is the use of waveguide arrays, which can be modeled using coupled-mode theory. Figure 4 illustrates different waveguide array configurations, including the Jx lattice, homogeneous lattice, and homogeneous lattice with disorder effects. The density criterion is evaluated for each configuration, and the corresponding performance tests are presented, identifying suitable lattice lengths for achieving universality.

Figure 4 depicts various photonic lattice configurations and their characteristics. It includes sketches of the Jx lattice (a), a homogeneous lattice (d), and a homogeneous lattice with disorder effects (e). It also presents a density criterion as a function of lattice length ℓ for N = 10 across Jx (b), homogeneous (e), and disordered (h) lattices, along with numerical performance tests for these lattices at reference lengths ℓ(m) for N = 10.

Another feasible realization involves meshes of directional couplers, which can be constructed using two-port passive elements called power dividers (3-dB directional couplers). Figure 5 depicts a 10-port array composed of power dividers interconnected through different layers. By analyzing the density criterion and performing numerical optimizations, we determine the optimal number of layers required for universality in this architecture.

Figure 5 illustrates a geometric array for the passive matrix F using power dividers (3-dB directional coupler). Part (a) shows a power divider array with p layers, where light-shaded layers represent L1 = I5⊗T0 and dark-shaded layers represent L2 = I1 ⊕ (I4 ⊗T0) ⊕ I1. Parts (b) and (c) display the density criterion and error norm (log10L) of the mesh architecture in (a), respectively, as functions of the number of layers p.

Conclusion

In this article, we have explored a novel approach to realizing arbitrary unitary operations using programmable photonic circuits. The proposed architecture, based on interlacing phase shifter layers with a fixed intervening matrix F, offers a flexible and efficient platform for various applications in classical and quantum computing.

The introduction of the Goldilocks principle provides a systematic method for selecting suitable matrices F, ensuring the universality of the architecture. By quantifying the density of unitary matrices and identifying the Goldilocks region, we can preselect candidate matrices without extensive optimization, streamlining the design process.

Numerical simulations and performance evaluations have validated the effectiveness of the Goldilocks principle, demonstrating its applicability to various photonic platforms and feasible realizations, including waveguide arrays and meshes of directional couplers.

As we continue to explore the vast potential of photonics in information processing, the Goldilocks principle and the proposed universal architecture pave the way for the development of efficient and robust programmable photonic devices, enabling a wide range of applications that harness the unique properties of light.

Reference

[2] Kevin Zelaya, Matthew Markowitz, Mohammad-Ali Miri, "The Goldilocks principle of learning unitaries by interlacing fixed operators with programmable phase shifters on a photonic chip," in Scientific Reports, vol. 14, no. 10950, 2024. https://doi.org/10.1038/s41598-024-60700-8