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Optics Express

Retrieving space-dependent polarization transformations via near-optimal quantum process tomography

Francesco Di Colandrea, Lorenzo Amato, Roberto Schiattarella, Alexandre Dauphin, Filippo Cardano

Published Jul 7, 2023• 4 citations
Publisher

Abstract

An optical waveplate rotating light polarization can be modeled as a single-qubit unitary operator. This analogy can be exploited to experimentally retrieve a polarization transformation within the paradigm of quantum process tomography. Standard approaches to tomographic problems rely on the maximum-likelihood estimation, providing the most likely transformation to yield the same outcomes as a set of experimental projective measurements. The performances of this method strongly depend on the number of input measurements and the numerical minimization routine that is adopted. Here we investigate the application of genetic and machine learning approaches to this problem, finding that both allow for accurate reconstructions and fast operations when processing a set of projective measurements very close to the minimal one. We apply these techniques to the case of space-dependent polarization transformations, providing an experimental characterization of the optical action of spin-orbit metasurfaces having patterned birefringence. Our efforts thus expand the toolbox of methodologies for optical process tomography. In particular, we find that the neural network-based scheme provides a significant speed-up, that may be critical in applications requiring a characterization in real-time. We expect these results to lay the groundwork for the optimization of tomographic approaches in more general quantum processes, including non-unitary gates and operations in higher-dimensional Hilbert spaces.

DOI
10.1364/oe.491518

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An optical waveplate rotating light polarization can be modeled as a single-qubit unitary operator. This analogy can be exploited to experimentally retrieve a polarization transformation within the paradigm of quantum process tomography. Standard approaches to tomographic problems rely on the maximum-likelihood estimation, providing the most likely transformation to yield the same outcomes as a set of experimental projective measurements. The performances of this method strongly depend on the number of input measurements and the numerical minimization routine that is adopted. Here we investigate the application of genetic and machine learning approaches to this problem, finding that both allow for accurate reconstructions and fast operations when processing a set of projective measurements very close to the minimal one. We apply these techniques to the case of space-dependent polarization transformations, providing an experimental characterization of the optical action of spin-orbit metasurfaces having patterned birefringence. Our efforts thus expand the toolbox of methodologies for optical process tomography. In particular, we find that the neural network-based scheme provides a significant speed-up, that may be critical in applications requiring a characterization in real-time. We expect these results to lay the groundwork for the optimization of tomographic approaches in more general quantum processes, including non-unitary gates and operations in higher-dimensional Hilbert spaces.
An optical waveplate rotating light polarization can be modeled as a single-qubit unitary operator, whose action can be experimentally determined via quantum process tomography. Standard approaches to tomographic problems rely on the maximum-likelihood estimation, providing the most likely transformation to yield the same outcomes as a set of experimental projective measurements. The performances of this method strongly depend on the number of input measurements and the numerical minimization routine that is adopted. Here we investigate the application of genetic and machine learning approaches to this problem, finding that both allow for accurate reconstructions and fast operations when processing a set of projective measurements very close to the minimal one. We apply these techniques to the case of space-dependent polarization transformations, providing an experimental characterization of the optical action of spin-orbit metasurfaces having patterned birefringence. Our efforts thus expand the toolbox of methodologies for optical process tomography. In particular, we find that the neural network-based scheme provides a significant speed-up, that may be critical in applications requiring a characterization in real-time. We expect these results to lay the groundwork for the optimization of tomographic approaches in more general quantum processes, including non-unitary gates and operations in higher-dimensional Hilbert spaces.
Abstract Efficient and accurate characterization of an experimental setup is a critical requirement in any physical setting. In the quantum realm, the characterization of an unknown operator is experimentally accomplished via Quantum Process Tomography (QPT). This technique combines the outcomes of different projective measurements to reconstruct the underlying process matrix, typically extracted from maximum-likelihood estimation. Here, we exploit the logical correspondence between optical polarization and two-level quantum systems to retrieve the complex action of structured metasurfaces within a QPT-inspired context. In particular, we investigate a deep-learning approach that allows for fast and accurate reconstructions of space-dependent SU(2) operators by only processing a minimal set of measurements. We train a convolutional neural network based on a scalable U-Net architecture to process entire experimental images in parallel. Synthetic processes are reconstructed with average fidelity above 90%. The performance of our routine is experimentally validated in the case of space-dependent polarization transformations acting on a classical laser beam. Our approach further expands the toolbox of data-driven approaches to QPT and shows promise in the real-time characterization of complex optical gates.
Abstract The characterization of a quantum device is a crucial step in the development of quantum experiments. This is accomplished via Quantum Process Tomography, which combines the outcomes of different projective measurements to deliver a possible reconstruction of the underlying process. The tomography is typically performed by processing an overcomplete set of measurements and extracting the process matrix from maximum-likelihood estimation. Here, we introduce Fourier Quantum Process Tomography, a technique which requires a reduced number of measurements, and benchmark its performance against the standard maximum-likelihood approach. Fourier Quantum Process Tomography is based on measuring probability distributions in two conjugate spaces for different state preparations and projections. Exploiting the concept of phase retrieval, our scheme achieves a complete and robust characterization of the setup by processing a near-minimal set of measurements. We experimentally test the technique on different space-dependent polarization transformations, reporting average fidelities higher than 90% and significant computational advantage.
Quantum science and technology have seen rapid progress in recent decades, finding applications in computation, information processing, communication, and sensing. Accurate characterization and manipulation of quantum states is essential to the successful implementation of these applications. Moreover, optimal quantum state estimation is closely related to fundamental questions, such as limits on how fast information can be extracted about quantum systems. In this thesis, we present several contributions to efficiently estimating and transforming quantum states. We discuss optimal quantum state estimation, both in the regime of few available quantum systems as well as the asymptotic regime. We numerically investigate how assuming a pure-state can affect the performance of tomographic methods, and where adaptive quantum state tomography is in fact advantageous. While collective measurements have been formally considered as optimal solutions to various problems in quantum information, their implementation has been largely neglected due to experimental challenges. We present a proof-of-concept experiment realizing Massar and Popescu's optimal collective measurement for quantum state estimation, utilizing two-photon interference and polarization-dependent loss to achieve non-maximally entangled state projections. We demonstrate that collective strategies can outperform the best separable measurements. Furthermore, we introduce and experimentally implement a fast and automated optical polarization compensation scheme using liquid crystal variable retarders and rotating quarter-wave plate tomography. Desired accuracy thresholds can be achieved within a few compensation steps. Finally, we use a matrix-model approach to investigate the realization of spatial unitary transformations via sequences of thin phase gratings and free-space propagation. We numerically demonstrate that the grating parameters and propagation distances can be optimized to achieve various two-beam and three-beam unitary transformations, such as Pauli gates and the discrete Fourier transform. Through experimental and numerical approaches, this thesis offers new perspectives on methods for quantum state estimation and unitary transformation.
The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using several orders of magnitude fewer (single-qubit) measurement shots than traditional tomographic techniques. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.
Quantum state tomography is an essential tool for characterizing quantum systems and underpins nearly every experimental realization of quantum technologies. Conventional tomography relies on performing a sequence of projective measurements on many identical copies of a quantum state, requiring the measurement apparatus to be reconfigured between successive acquisitions. As the Hilbert-space dimension increases, the number of required measurements grows rapidly; in practice, additional overcomplete measurements are often performed to improve robustness to experimental imperfections. Here, we introduce a tomography platform based on structured anisotropic media that performs informationally complete measurements of photonic polarization qubits within a single acquisition. The approach employs three liquid-crystal metasurfaces with spatially varying optic-axis orientations that transform the input polarization into a far-field distribution of discrete transverse-momentum modes. Each diffraction pattern uniquely determines the polarization state, enabling its reconstruction without sequential changes to the measurement apparatus. Unlike previous implementations, our scheme is intrinsically photon-number independent: the same optical device operates identically for arbitrary photon numbers, while the desired photon-number sector can be selected afterwards through post-selection of the corresponding nnn-fold coincidence events. We experimentally demonstrate single-frame quantum state tomography of both single- and two-photon polarization states, providing a simple and scalable route toward efficient quantum-state characterization.