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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Genetic algorithms to learn an unknown linear transformation

Nicolò Spagnolo, Enrico Maiorino, Chiara Vitelli, Marco Bentivegna, Andrea Crespi, Roberta Ramponi, Paolo Mataloni, Roberto Osellame, Fabio Sciarrino

Published Jun 1, 2017• 0 citations
Publisher

Abstract

Integrated photonics enabled the capability to implement complex interferometers with a large number of modes. The applications of this experimental toolbox can be found in different quantum information tasks, including quantum walks and simulation [1], quantum metrology [2], up to the implementation of Boson Sampling [3-6], one of the most promising candidates to reach experimentally the regime where a quantum device solves a computational problem faster than its classical counterpart. Both from a fundamental aspect and for characterization purposes, it becomes crucial to identify tools allowing to learn the action of an unknown transformation.

DOI
10.1109/cleoe-eqec.2017.8087443

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Integrated photonics enabled the capability to implement complex interferometers with a large number of modes. The applications of this experimental toolbox can be found in different quantum information tasks, including quantum walks and simulation [1], quantum metrology [2], up to the implementation of Boson Sampling [3-6], one of the most promising candidates to reach experimentally the regime where a quantum device solves a computational problem faster than its classical counterpart. Both from a fundamental aspect and for characterization purposes, it becomes crucial to identify tools allowing to learn the action of an unknown transformation.
Recent developments in integrated photonics technology are opening the way to the fabrication of complex linear optical interferometers. The application of this platform is ubiquitous in quantum information science, from quantum simulation to quantum metrology, including the quest for quantum supremacy via the boson sampling problem. Within these contexts, the capability to learn efficiently the unitary operation of the implemented interferometers becomes a crucial requirement. In this letter we develop a reconstruction algorithm based on a genetic approach, which can be adopted as a tool to characterize an unknown linear optical network. We report an experimental test of the described method by performing the reconstruction of a 7-mode interferometer implemented via the femtosecond laser writing technique. Further applications of genetic approaches can be found in other contexts, such as quantum metrology or learning unknown general Hamiltonian evolutions.
It is strongly believed that the evolution of bosons undergoing arbitrary linear unitary transformations quickly becomes hard to predict using classical computers as we increase the number of particles and modes. Photons propagating in a multiport interferometer naturally solve this so-called boson sampling problem: sample the outputs of a multi-photon experiment in a linear-optical interferometer. This motivates the development of technologies that enable precise control of multiphoton interference in large interferometers.
Linear optical networks play a paramount role in current implementations of quantum information protocols using photons. A paradigmatic example of their potential is given by boson sampling experiments, which recently have reached the quantum advantage regime [1], [2]. Such linear devices are essentiallyNNN-mode interferometers, whose operation is entirely captured by anN×NN\times NN×Nunitary matrixUUUwith complex entriesuji=∣uji∣eθjiu_{j i}=\left\vert u_{j i}\right\vert e^{\theta_{j i}}uji​=∣uji​∣eθji​. This matrix characterizes, in a classical framework, the transformation of the phasorsεj(in)\varepsilon_{j}^{(in)}εj(in)​of the input field across the different modes into the phasorsεj(out)\varepsilon_{j}^{(out)}εj(out)​of the output fields according to the linear relationεi(out)=∑iujiεi(in)\varepsilon_i^{({out })}=\sum_i u_{j i} \varepsilon_i^{({in })}εi(out)​=∑i​uji​εi(in)​).
Photonic platforms represent a promising technology for the realization of several quantum communication protocols and for experiments of quantum simulation. Moreover, large-scale integrated interferometers have recently gained a relevant role in quantum computing, specifically with Boson Sampling devices and the race for quantum supremacy. Indeed, various linear optical schemes have been proposed for the implementation of unitary transformations, each one suitable for a specific task. Notwithstanding, so far a comprehensive analysis of the state of the art under broader and realistic conditions is still lacking. In the present work we fill this gap, providing in a unified framework a quantitative comparison of the three main photonic architectures, namely the ones with triangular and square designs and the so-called fast transformations. All layouts have been analyzed in presence of losses and imperfect control over the internal reflectivities and phases, showing that the square design outperforms the triangular scheme in most operational conditions. Our results represent a further step ahead towards the implementation of quantum information protocols on large-scale integrated photonic devices.
Quantum metrology is one of the most relevant applications of quantum information theory to quantum technologies.Here, quantum probes are exploited to overcome classical bounds in the estimation of unknown parameters.In this context, phase estimation, where the unknown parameter is a phase shift between two modes of a quantum system, is a fundamental problem.In practical and realistic applications, it is necessary to devise methods to optimally estimate an unknown phase shift by using a limited number of probes.Here we introduce and experimentally demonstrate a machine learning-based approach for the adaptive estimation of a phase shift in a Mach-Zehnder interferometer, tailored for optimal performances with limited resources.The employed technique is a genetic algorithm used to devise the optimal feedback phases employed during the estimation in an offline fashion.The results show the capability to retrieve the true value of the phase by using few photons, and to reach the sensitivity bounds in such small probe regime.We finally investigate the robustness of the protocol with respect to common experimental errors, showing that the protocol can be adapted to a noisy scenario.Such approach promises to be a useful tool for more complex and general tasks where optimization of feedback parameters is required.