Photonic Quantum Convolutional Neural Networks with Adaptive State Injection (arXiv)

Léo Monbroussou, Beatrice Polacchi, Verena Yacoub, Eugenio Caruccio, Giovanni Rodari, Francesco Hoch, Gonzalo Carvacho, Nicolò Spagnolo, Taira Giordani, Mattia Bossi, Abhiram Rajan, Niki Di Giano, Riccardo Albiero, Francesco Ceccarelli, Roberto Osellame, Elham Kashefi, Fabio Sciarrino

Linear optical architectures have been extensively investigated for quantum computing and quantum machine learning applications. Recently, proposals for photonic quantum machine learning have combined linear optics with resource adaptivity, such as adaptive circuit reconfiguration, which promises to enhance expressivity and improve algorithm performances and scalability. Moreover, linear optical platforms preserve some subspaces due to the fixed number of particles during the computation, a property recently exploited to design a novel quantum convolutional neural networks. This last architecture has shown an advantage in terms of running time complexity and of the number of parameters needed with respect to other quantum neural network proposals. In this work, we design and experimentally implement the first photonic quantum convolutional neural network (PQCNN) architecture based on particle-number preserving circuits equipped with state injection, an approach recently proposed to increase the controllability of linear optical circuits. Subsequently, we experimentally validate the PQCNN for a binary image classification on a photonic platform utilizing a semiconductor quantum dot-based single-photon source and programmable integrated photonic interferometers comprising 8 and 12 modes. In order to investigate the scalability of the PQCNN design, we have performed numerical simulations on datasets of different sizes. We highlight the potential utility of a simple adaptive technique for a nonlinear Boson Sampling task, compatible with near-term quantum devices. 

When Quantum and Classical Models Disagree: Learning Beyound Minimum Norm Least Square (arXiv)

Slimane Thabet, Léo Monbroussou, Eliott Z. Mamon, Jonas Landman

We study the convergence properties of Variational Quantum Circuits (VQCs) to investigate how they can differ from their classical counterparts. It is known that a VQC is a linear model in a feature map determined by its architecture. Learning a classical model on the same feature map will lead to a solution called the Minimum Norm Least Square (MNLS) estimator. In this work, we characterize the separation between quantum and classical models by their respective weight vector. We show that a necessary condition for a quantum model to avoid dequantization by its classical surrogate is to have a large weight vector norm. Furthermore, we suggest that this can only happen with a high dimensional feature. Through the study of some common quantum architectures and encoding schemes, we obtain bounds on the norms of the quantum weight vector and the corresponding MNLS weight vector. It is possible to find instances allowing for such separation, but in these cases, concentration issues become another concern. We finally prove that there exists a linear model with large weight vector norm and without concentration, potentially achievable by a quantum circuit. 

Toward Quantum Advantage with Photonic State Injection (Physical Review Research)

Léo Monbroussou, Eliott Z. Mamon, Hugo Thomas, Verena Yacoub, Ulysse Chabaud, Elham Kashefi

We propose a new scheme for near-term photonic quantum device that allows to increase the expressive power of the quantum models beyond what linear optics can do. This scheme relies upon state injection, a measurement-based technique that can produce states that are more controllable, and solve learning tasks that are not believed to be tackled classically. We explain how circuits made of linear optical architectures separated by state injections are keen for experimental implementation. In addition, we give theoretical results on the evolution of the purity of the resulting states, and we discuss how it impacts the distinguishability of the circuit outputs. Finally, we study a computational subroutines of learning algorithms named probability estimation, and we show the state injection scheme we propose may offer a potential quantum advantage in a regime that can be more easily achieved that state-of-the-art adaptive techniques. Our analysis offers new possibilities for near-term advantage that require to tackle fewer experimental difficulties.

Subspace Preserving Quantum Convolutional Neural Network  (Quantum Science and Technology)

Léo Monbroussou, Jonas Landman, Letao Wang, Alex B. Grilo, Elham Kashefi

Subspace preserving quantum circuits are a class of quantum algorithms that, relying on some symmetries in the computation, can offer theoretical guarantees for their training. Those algorithms have gained extensive interest as they can offer polynomial speed-up and can be used to mimic classical machine learning algorithms. In this work, we propose a novel convolutional neural network architecture model based on Hamming weight preserving quantum circuits. In particular, we introduce convolutional layers, and measurement based pooling layers that preserve the symmetries of the quantum states while realizing non-linearity using gates that are not subspace preserving. Our proposal offers significant polynomial running time advantages over classical deep-learning architecture. We provide an open source simulation library for Hamming weight preserving quantum circuits that can simulate our techniques more efficiently with GPU-oriented libraries. Using this code, we provide examples of architectures that highlight great performances on complex image classification tasks with a limited number of qubits, and with fewer parameters than classical deep-learning architectures.

Constrained and Vanishing Expressivity of Quantum Fourier Models  (Quantum)

Hela Mhiri, Léo Monbroussou, Mario Herrero-Gonzalez, Slimane Thabet, Elham Kashefi, Jonas Landman

In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQC) for machine learning. A large class of these models, seen as Fourier Series which frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially when the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for Quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviours are observed in numerical simulations. 

Trainability and Expressivity of Hamming Weight Preserving Quantum Circuits for Machine Learning  (Quantum)

Léo Monbroussou, Eliott Z. Mamon, Jonas Landman, Alex B. Grilo, Romain Kukla, Elham Kashefi

Quantum machine learning has become a promising area for real world applications of quantum computers, but near-term methods and their scalability are still important research topics. In this context, we analyze the trainability and controllability of specific Hamming weight preserving quantum circuits. These circuits use gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight . They are good candidates for mimicking neural networks, by both loading classical data and performing trainable layers. In this work, we first design and prove the feasibility of new heuristic data loaders, performing quantum amplitude encoding of n choose k dimensional vectors by training a n-qubit quantum circuit. Then, we analyze more generally the trainability of Hamming weight preserving circuits, and show that the variance of their gradients is bounded according to the size of the preserved subspace. This proves the conditions of existence of Barren Plateaus for these circuits, and highlights a setting where a recent conjecture on the link between controllability and trainability of variational quantum circuits does not apply.