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arXiv:2312.05648 · Physical Review Research

Learning arbitrary complex matrices by interlacing amplitude and phase masks with fixed unitary operations

Matthew Markowitz, Mohammad‐Ali Miri, Alexey Ovchinnikov, Kevin Zelaya

Published Sep 3, 2024• 14 citations
arXivHTMLPDF

Abstract

Programmable photonic integrated circuits are an emerging technology that amalgamates photonics and electronics, paving the way for light-based information processing at high speeds and low power consumption. Considering its wide range of applications as one of the most fundamental mathematical operations, there has been considerable interest in developing reliably implementable programmable circuit architectures that perform matrix-vector multiplication. Recently, it was shown that discrete unitary operations can be parameterized by interlacing fixed operators with diagonal phase parameters realized with phase shifter arrays. We show that these decompositions are a special case of a broader class of factorizations that enable parametrization of arbitrary complex matrices. The proposed representation of an $N\ifmmode\times\else\texttimes\fi{}N$ matrix is given by $N+1$ amplitude-and-phase-modulation layers interlaced with a fixed unitary layer that can be implemented, for example, via a coupled waveguide array. Thus, we introduce an architecture for physically implementing discrete linear operations, enabling the development of novel families of programmable photonic circuits for on-chip analog information processing.

DOI
10.1103/4vth-6n7q

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Programmable photonic integrated circuits are an emerging technology that amalgamates photonics and electronics, paving the way for light-based information processing at high speeds and low power consumption. Considering its wide range of applications as one of the most fundamental mathematical operations, there has been considerable interest in developing reliably implementable programmable circuit architectures that perform matrix-vector multiplication. Recently, it was shown that discrete unitary operations can be parameterized by interlacing fixed operators with diagonal phase parameters realized with phase shifter arrays. We show that these decompositions are a special case of a broader class of factorizations that enable parametrization of arbitrary complex matrices. The proposed representation of an N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN matrix is given by N+1N+1N+1 amplitude-and-phase-modulation layers interlaced with a fixed unitary layer that can be implemented, for example, via a coupled waveguide array. Thus, we introduce an architecture for physically implementing discrete linear operations, enabling the development of novel families of programmable photonic circuits for on-chip analog information processing.
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×NN \times NN×N N × N unitary operators with N+1N+1N+1 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.
We introduce a novel parameterization of complex unitary matrices, which allows for the efficient photonic implementation of arbitrary linear discrete unitary operators. The proposed architecture is built on factorizing an N×NN \times NN×N unitary matrix into interlaced discrete fractional Fourier transforms and NNN-parameter diagonal phase shifts. We show that such a configuration can represent arbitrary unitary operators with N+1N+1N+1 phase layers. We discuss a gradient-based algorithm for finding the optimal phase parameters for implementing a given unitary matrix. By increasing the number of phase layers beyond the critical value of N+1N+1N+1, the optimization consistently converges faster as the system becomes over-determined. We propose an integrated photonic circuit realization of this architecture with coupled waveguide arrays and reconfigurable phase modulators. The proposed architecture can pave the way for developing novel families of programmable photonic circuits for optical classical and quantum information processing.
Recent investigations suggest that the discrete linear unitary group U(N)U(N)U(N) can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical justifications, no formal proof has been provided. Here, we show that elements of U(N)U(N)U(N) can be decomposed into a sequence of NNN-parameter phases alternating with 111-parameter propagators of a lattice Hamiltonian. The proof is based on building a Lie group by alternating these two operators and showing its completeness to represent U(N)U(N)U(N) for a finite number of layers, which is numerically found to be exactly NNN. This architecture can be implemented using elementary optical components and can successfully reconstruct arbitrary unitary matrices. We propose example devices such as optical logic gates, which perform logic gate operations using a single-layer lossless and passive optical circuit design.
In recent years, there has been growing interest in analog information processing and unconventional computing using Photonic Integrated Circuits (PICs) to address the increasing demands of next-generation computing. This has driven the development of on-chip architectures capable of implementing fundamental linear operations such as matrix-vector multiplication (MVM), which plays a central role in fields such as quantum information processing and the acceleration of deep neural networks. Within the feed-forward paradigm, interleaved meshes of Mach–Zehnder Interferometers (MZIs) have emerged as the leading architecture for programmable integrated photonics (PIP), enabling real-time implementation of arbitrary linear transformations using tunable phase shifter elements. More recently, an alternative class of programmable photonic architectures based on Multi-Plane Light Conversion (MPLC) has been proposed, consisting of alternating diagonal phase-shifter layers and multimode mixing elements. Although these architectures lack a deterministic synthesis algorithm for the phase actuators, they have been shown to offer several advantages over conventional MZI-based architectures. In this thesis we investigate the properties and universal capabilities of MPLC architectures utilizing unequally spaced coupled waveguide arrays (CWAs), with a particular emphasis on the well-known Jx-lattice. Defect tolerance and self-calibration properties are studied, demonstrating substantially greater robustness to imperfections than conventional MZI meshes. We further introduce several extensions of the MPLC architecture that demonstrate its flexibility, such as the redistribution of amplitude modulators and their interchangeable use with phase shifters to realize universal transformations with a near-minimal number of tunable units. Collectively, these results provide a broader understanding of the capabilities of MPLC architectures and establish a foundation for their application as a flexible alternative and complement to MZI-based designs.
Programmable photonic meshes provide a promising platform for analog matrix-vector multiplication, but their scalability is often limited by the large phase-shift ranges required in universal interferometer circuits. We introduce a perturbative programming method that operates the circuit near a fixed reference configuration and realizes the target transformation through interferometric subtraction, thereby reducing the required programmable phase excursion. We develop this approach for photonic matrix-vector multiplication architectures based on universal unitary meshes, and low-depth non-unitary constructions based on sums of unitaries. We identify favorable reference configurations through a local conditioning criterion, analyze the phase statistics obtained for random target matrices, and show that perturbative programming produces phase distribution shrinking as the matrix size increases. Identified reference point allows for the subtraction to be carried out electronically after detection, avoiding the interferometric subtraction loss. We further quantify the trade-off between reduced phase range and the intrinsic overhead introduced by the subtraction architecture, and show that for sufficiently lossy phase shifters the reduced phase range can compensate for this penalty. These results identify perturbative programming as a conditional but potentially useful route toward more scalable programmable photonic matrix processors.
Abstract Implementing arbitrary unitary transformations is crucial for applications in quantum computing, signal processing, and machine learning. Unitaries govern quantum state evolution, enabling reversible transformations critical in quantum tasks like cryptography and simulation and playing key roles in classical domains such as dimensionality reduction and signal compression. Integrated optical waveguide arrays have emerged as a promising platform for these transformations, offering scalability for both quantum and classical systems. However, scalable and efficient methods for implementing arbitrary unitaries remain challenging. Here, we present a theoretical framework for realizing arbitrary unitary matrices through programmable waveguide arrays (PWAs). We provide a mathematical proof demonstrating that cascaded PWAs can implement any unitary matrix within practical constraints, along with a numerical optimization method for customized PWA designs. Our results establish PWAs as a universal and scalable architecture for quantum photonic computing, effectively bridging quantum and classical applications, and positioning PWAs as an enabling technology for advancements in quantum simulation, machine learning, secure communication, and signal processing.