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Optical implementation of 2x2 universal unitary matrix transformations

Andrés Macho, Daniel Pérez, J. Capmany

Published Jul 30, 2021• 0 citations
Publisher

Abstract

Unitary operations are a specific class of linear transformations that have become an essential ingredient for the realization of classical and quantum information processing. The ability of implementing any n-dimensional unitary signal transformation by using a reconfigurable optical hardware has recently led to the pioneering concept of programmable linear optical processor, whose basic building block (BB) must be correctly designed to guarantee that the whole system is able to perform n × n universal (i.e., arbitrary) unitary matrix transformations. Here, it is demonstrated that the present architectures of the BB do not fulfil the universal unitary property (at least) in 2 × 2 optical processors, limiting the number of unitary matrix transformations that may be generated. Aiming to solve this fundamental constraint, the theoretical tools required to analyze and design 2 × 2 universal unitary optical circuits and their corresponding BBs are presented. The consequences of this mathematical framework are explored, obtaining a simple route to implement different BB architectures, all of them guaranteeing a true universal unitary functionality in the resulting 2 × 2 optical processors. These findings may pave the way to revisit the design of high-dimensional unitary optical processors, unleashing the potential of programmable integrated photonics technology.

DOI
10.5281/zenodo.5148127

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Unitary operations are a specific class of linear transformations that have become an essential ingredient for the realization of classical and quantum information processing. The ability of implementing any n-dimensional unitary signal transformation by using a reconfigurable optical hardware has recently led to the pioneering concept of programmable linear optical processor, whose basic building block (BB) must be correctly designed to guarantee that the whole system is able to perform n × n universal (i.e., arbitrary) unitary matrix transformations. Here, it is demonstrated that the present architectures of the BB do not fulfil the universal unitary property (at least) in 2 × 2 optical processors, limiting the number of unitary matrix transformations that may be generated. Aiming to solve this fundamental constraint, the theoretical tools required to analyze and design 2 × 2 universal unitary optical circuits and their corresponding BBs are presented. The consequences of this mathematical framework are explored, obtaining a simple route to implement different BB architectures, all of them guaranteeing a true universal unitary functionality in the resulting 2 × 2 optical processors. These findings may pave the way to revisit the design of high-dimensional unitary optical processors, unleashing the potential of programmable integrated photonics technology.
Abstract Unitary operations are a specific class of linear transformations that have become an essential ingredient for the realization of classical and quantum information processing. The ability of implementing any n‐dimensional unitary signal transformation by using a reconfigurable optical hardware has recently led to the pioneering concept of programmable linear optical processor, whose basic building block (BB) must be correctly designed to guarantee that the whole system is able to perform universal (i.e., arbitrary) unitary matrix transformations. Here, it is demonstrated that the present architectures of the BB do not fulfil the universal unitary property (at least) in optical processors, limiting the number of unitary matrix transformations that may be generated. Aiming to solve this fundamental constraint, the theoretical tools required to analyze and design universal unitary optical circuits and their corresponding BBs are presented. The consequences of this mathematical framework are explored, obtaining a simple route to implement different BB architectures, all of them guaranteeing a true universal unitary functionality in the resulting optical processors. These findings may pave the way to revisit the design of high‐dimensional unitary optical processors, unleashing the potential of programmable integrated photonics technology.
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.
General-purpose programmable photonic processors provide a versatile platform for integrating diverse functionalities on a single chip. Leveraging a two-dimensional hexagonal waveguide mesh of Mach-Zehnder interferometers, these systems have demonstrated significant potential in microwave photonics applications. Additionally, they are a promising platform for creating unitary linear transformations, which are key elements in quantum computing and photonic neural networks. However, a general procedure for implementing these transformations on such systems has not been established yet. This work demonstrates the programming of universal unitary transformations on a general-purpose programmable photonic circuit with a hexagonal topology. We detail the steps to split the light on-chip, demonstrate that an equivalent structure to the Mach-Zehnder interferometer with one internal and one external phase shifter can be built in the hexagonal mesh, and program both the triangular and rectangular architectures for matrix multiplication. We recalibrate the system to account for passive phase deviations. Experimental programming of 3x3 and 4x4 random unitary matrices yields fidelities > 98% and bit precisions over 5 bits. To the best of our knowledge, this is the first time that random unitary matrices are demonstrated on a general-purpose photonic processor and pave the way for the implementation of programmable photonic circuits in optical computing and signal processing systems.
General-purpose programmable photonic processors provide a versatile platform for integrating diverse functionalities on a single chip. Leveraging a two-dimensional hexagonal waveguide mesh of Mach–Zehnder interferometers, these systems have demonstrated significant potential in microwave photonic applications. Additionally, they are a promising platform for creating unitary linear transformations, which are key elements in quantum computing and photonic neural networks. However, a general procedure for implementing these transformations on such systems has not been established yet. This work demonstrates the programming of universal unitary transformations on a general-purpose programmable photonic circuit with a hexagonal topology. We detail the steps to split the light on-chip, demonstrate that an equivalent structure to the Mach–Zehnder interferometer with one internal and one external phase shifter can be built in the hexagonal mesh, and program both the triangular and rectangular architectures for matrix multiplication. We recalibrate the system to account for passive phase deviations. Experimental programming of 3 × 3 and 4 × 4 random unitary matrices yields fidelities >98% and bit precisions over five bits. To the best of our knowledge, this is the first time that random unitary matrices are demonstrated on a general-purpose photonic processor and pave the way for the implementation of programmable photonic circuits in optical computing and signal processing systems.
Integrated multi-port interferometers are the second step in programmable photonics. Initially conceived and designed to implement fixed linear optics unitary transformations between N input and N output ports to support mode transformations in quantum optics, the rapid development of this field and the possibility of applying these structures to other areas stimulated development of devices capable of supporting arbitrary linear transformations by suitable programming and surpassing use in square matrix transformations and unitary operators. This chapter reviews some basic concepts on unitary N×N matrices and then describes in detail the basic design principles and integrated optics implementations of triangular and rectangular designs for multi-port interferometers. It then considers more specific interferometer designs, which do not provide possible implementation of general linear transformations but are optimised for performing important operations, e.g. the Fourier transform. Lastly, it covers the linear universal component, which integrates both universal beam couplers and integrated multi-port interferometers.
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.