Noise-tolerant tomography of multimode linear optical interferometers with single photons
Yu. A. Biriukov, R. D. Morozov, I. V. Dyakonov, Maxim V. Rakhlin, A. I. Galimov, Grigorii V. Klimko, Sergey Vladimirovich Sorokin, I. V. Sedova, Marina M Kulagina, Yu. M. Zadiranov, Alexey A. Toropov, Alexander A. Korneev, S. P. Kulik, Stanislav S. Straupe
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 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.
Multimode optical interferometers represent the most viable platforms for the successful implementation of several quantum information schemes that take advantage of optical processing. Examples range from quantum communication and sensing, to computation, including optical neural networks, optical reservoir computing, or simulation of complex physical systems. The realization of such routines requires high levels of control and tunability of the parameters that define the operations carried out by the device. This requirement becomes particularly crucial in light of recent technological improvements in integrated photonic technologies, which enable the implementation of progressively larger circuits embedding a considerable amount of tunable parameters. We formulate efficient procedures for the characterization of optical circuits in the presence of imperfections that typically occur in physical experiments, such as unbalanced losses and phase instabilities in the input and output collection stages. The algorithm aims at reconstructing the transfer matrix that represents the optical interferometer without making any strong assumptions about its internal structure and encoding. We show the viability of this approach in an experimentally relevant scenario, defined by a tunable integrated photonic circuit, and we demonstrate the effectiveness and robustness of our method. Our findings can find application in a wide range of optical setups, based on both bulk and integrated configurations.
Abstract Tunable linear-optical interferometers are a key element in the implementation of both classical and quantum information technologies. At the same time, during the production process, these systems are subject to various instrumental imperfections and it is necessary to be able to determine with what accuracy and what transformations the manufactured device is capable to reproduce. For integrated devices, such tasks are difficult to perform due to the difficulties in obtaining direct access to individual optical elements of the circuit. Existing methods for reconstructing the transfer matrix of tunable linear-optical interferometers require measuring the phases of its matrix elements at different values of control signals in the circuit, which is also an experimentally time-consuming task. However, in some cases, it is possible to construct a mathematical model of the circuit that works for any values of its control signals, measuring only partial amplitude transmission coefficients at the outputs of the circuit, which significantly simplifies reconstruction and is relevant in a number of applications. In this paper, we performed numerical and experimental approbation of the method on the example of a linear-optical interferometer with four inputs and four outputs with a universal architecture based on mixing layers. The average fidelity value of the reconstruction of the reduced transfer matrix of the optical interferometer, taking into account realistic noise in the experiment, was 99.9% and can be improved due to the proven stability of the model to them.
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.
Abstract Programmable linear optical interferometers are promising for classical and quantum applications. Their integrated design makes it possible to create more scalable and stable devices. To use them in practice, one has to reconstruct the whole device model taking the manufacturing errors into account. The inability to address individual interferometer elements complicates the reconstruction problem. A naive approach is to train the model via some complex optimization procedure. A faster optimization-free algorithm has been recently proposed (Bantysh et al 2023 Opt. Express 31 16729–42). However, it requires the full transfer matrix tomography while a more practical setup measures only the fields intensities at the interferometer output. In this paper, we propose the modification of the fast algorithm, which uses additional set of interferometer configurations in order to reconstruct the model in the case of intensity-only measurements. We show that it performs slightly worse than the original fast algorithm but it is more practical and still does not require intensive numerical optimization.
Multimode Gaussian states are a versatile resource for quantum information technologies and have been realized across a wide range of physical platforms. Recent progress in the large-scale generation of such states provides a key ingredient for scalable quantum technologies. Despite the importance of accurately characterizing these states, conventional tomography methods are often impractical because they require large sample sizes and can yield unphysical states. Here we present a reliable and efficient tomography method for multimode Gaussian states based on maximum-likelihood estimation. By directly operating on covariance matrices, the method avoids the exponential overhead associated with density-matrix reconstruction. We consider two commonly used detection schemes--single and joint homodyne detection--and systematically analyze the reconstruction performance. Our method outperforms conventional approaches by ensuring physical covariance matrices and achieving better agreement with the true states. To demonstrate the experimental applicability of the method, we experimentally generate various multipartite entangled states--six-mode graph states with different connectivity, a six-mode GHZ state, and a fully connected ten-mode graph state--and reconstruct their covariance matrices. Using the reconstructed covariance matrices, we quantify fidelities, detect entanglement, and reveal the multimode structure of squeezing and noise. Our technique offers a practical diagnostic tool for developing scalable quantum technologies.
Transport of information through a multimode optical fibre raises challenges when one wants to increase the data traffic using many spatial modes due to modal cross-talk and dispersion. Instead of considering those complex mixing of modes as a detrimental process, in this dissertation, we harness its mode mixing to process quantum optical information. We implement a reconfigurable linear optical network, a fundamental building block for scalable quantum technologies, based on an inverse photonic approach exploiting the technology of wavefront shaping. We experimentally demonstrate manipulation of two-photon quantum interference on various linear optical networks across both spatial and polarization degrees of freedom. In particular, we experimentally show the zero-transmission law in Fourier and Sylvester interferometers, which are used to certificate the degree of indistinguishability of an input state. Moreover, thanks to the ability to implement a non-unitary network, we observe the photon anti-coalescence effect in all output configurations, as well as the realization of a tunable coherent absorption experiment. Therefore, we demonstrate the reconfigurability, accuracy, scalability and robustness of the implemented linear optical networks for quantum information processing. Furthermore, we study the statistical properties of one-and two-photon speckles generated from various ground-truth states of light after propagating through a multimode fibre. These statistical properties of speckles can be used to extract information about the dimensionality, purity, and indistinguishability of an unknown input state of light, therefore allowing for state classification. Our results highlight the potential of complex media combined with wavefront shaping for quantum information processing.