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Quantum Computing, Communication, and Simulation

Simple measurements and efficient optimization of linear optical networks

Shannon M. Ray, A. Matthew Smith, Daniel Koch, Sam Ellis, Christopher C. Tison, Paul M. Alsing

Published Mar 19, 2025• 0 citations
Publisher

Abstract

In this proceeding, we describe a simulated model for the characterization of arbitrary unitary matrices implemented by an reconfigurable linear optical network. We build on the method of Hoch et. al, which itself builds on Liang et. al, for characterizing the arbitrary unitary transformation, U, implemented on such devices as an equivalent real bordered form Urb. We disscuss how the measurement scheme requires fewer and simpler measurements than. We describe a solution that utilizes a novel algorithm to efficiently recover the complex signs of Urb under measurement noise and scaling with system size N.

DOI
10.1117/12.3043503

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•Jul 30, 2021•0 citations
In this proceeding, we describe a simulated model for the characterization of arbitrary unitary matrices implemented by an reconfigurable linear optical network. We build on the method of Hoch et. al, which itself builds on Liang et. al, for characterizing the arbitrary unitary transformation, U, implemented on such devices as an equivalent real bordered form Urb. We disscuss how the measurement scheme requires fewer and simpler measurements than. We describe a solution that utilizes a novel algorithm to efficiently recover the complex signs of Urb under measurement noise and scaling with system size N.
We present a simulated model to study the characterization of arbitrary unitary matrices implemented in an NxN integrated photonic Mach–Zehnder interferometer (MZI) mesh. We extend the measurement method of Hoch et al., which itself generalizes Liang et al., for the practical characterization of an arbitrary unitary transformation, U, implemented on such devices as an equivalent real bordered form Urb. The measurement scheme and reconstruction shown here require fewer and simpler measurements than Liang et al. and are more complete than Hoch et al. Completing the method of Hoch using our hybrid greedy-Tabu and Breadth First Search (BFS) algorithm makes our scheme practical in the sense that only simple classical resources are needed. This allows for practical characterization of the phases of Urb outside of a laboratory setting. We examine the performance of our reconstruction under simulated measurement noise and its scaling in time as a function of system size N.
We propose a matrix characterization and optimization method using the wavefront matching algorithm. Simulation results indicate that this method can achieve a matrix size of121×121121\times 121121×121with a fidelity of over 99%.
Linear optical networks are fundamental to the advancement of quantum technologies, including quantum computing, communication, and sensing. The accurate characterization of these networks, described by unitary matrices, is crucial to their effective utilization and scalability. In this work, we present the method for reconstructing the transfer matrix of a linear optical interferometer based on the analysis of cross-correlation functions of photon counts between pairs of output modes. Our approach accounts for losses and photon indistinguishability, making it robust to experimental imperfections. By minimizing the requirements for the input states, the method simplifies the experimental implementation. We demonstrate the effectiveness of our technique through theoretical modeling and experimental validation in a 4-mode programmable integrated optical interferometer. The results show high fidelity in matrix reconstruction and successful application in boson sampling experiments. In addition, we provide a comprehensive formalism for correlation functions and discuss the robustness of the method to measurement errors. This work offers a practical and efficient solution for characterizing linear-optical networks, paving the way for scaling up photonic quantum technologies.
Linear-optical circuits are elementary building blocks for classical and quantum information processing with light. In particular, due to its monolithic structure, integrated photonics offers great phase-stability and can rely on the large scale manufacturability provided by the semiconductor industry. New devices, based on such optical circuits, hold the promise of faster and energy-efficient computations in machine learning applications and even implementing quantum algorithms intractable for classical computers. However, this technological revolution requires accurate and scalable certification protocols for devices that can be comprised of thousands of optical modes. Here, we present a novel technique to reconstruct the transfer matrix of linear optical networks that is based on the recent advances in low-rank matrix recovery and convex optimisation problems known as PhaseLift algorithms. Conveniently, our characterisation protocol can be performed with a coherent classical light source and photodiodes. We prove that this method is robust to noise and scales efficiently with the number of modes. We experimentally tested the proposed characterisation protocol on a programmable integrated interferometer designed for quantum information processing. We compared the transfer matrix reconstruction obtained with our method against the one provided by a more demanding reconstruction scheme based on two-photon quantum interference. For 5-dimensional random unitaries, the average circuit fidelity between the matrices obtained from the two reconstructions is 0.993.
Programmable linear optical interferometers are important for classical and quantum information technologies, as well as for building hardware-accelerated artificial neural networks. Recent results showed the possibility of constructing optical interferometers that could implement arbitrary transformations of input fields even in the case of high manufacturing errors. The building of detailed models of such devices drastically increases the efficiency of their practical use. The integral design of interferometers complicates its reconstruction since the internal elements are hard to address. This problem can be approached by using optimization algorithms [Opt. Express29, 38429 (2021)10.1364/OE.432481]. In this paper, we present what we believe to be a novel efficient algorithm based on linear algebra only, which does not use computationally expensive optimization procedures. We show that this approach makes it possible to perform fast and accurate characterization of high-dimensional programmable integrated interferometers. Moreover, the method provides access to the physical characteristics of individual interferometer layers.
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.