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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 map.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.

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.

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.

Quantum machine learning (QML) 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 variational quantum circuits (VQCs). These circuits use qubit gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight $k$. In this work, we first design and prove the feasibility of new heuristic data loaders, performing quantum amplitude encoding of $\binom{n}{k}$-dimensional vectors by training an $n$-qubit quantum circuit. These data loaders are obtained using dimensionality reduction techniques, by checking the Quantum Fisher Information Matrix (QFIM)'s rank. Second, we provide a theoretical justification for the fact that the rank of the QFIM of any VQC state is almost-everywhere constant, which is of separate interest. Lastly, we analyze the trainability of Hamming weight preserving circuits, and show that the variance of the $l_2$ cost function gradient is bounded according to the dimension $\binom{n}{k}$ of the subspace. This proves conditions of existence/lack 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.

Many real-world problems, like modelling environment dynamics, physical processes, time series etc., involve solving Partial Differential Equations (PDEs) parameterised by problem-specific conditions. Recently, a deep learning architecture called Fourier Neural Operator (FNO) proved to be capable of learning solutions of given PDE families for any initial conditions as input. However, it results in a time complexity linear in the number of evaluations of the PDEs while testing. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we exploit the running efficiency offered by these and propose quantum algorithms inspired by the classical FNO, which result in time complexity logarithmic in the number of evaluations and are, therefore, expected to be substantially faster than their classical counterpart. At their core, we use the unary encoding paradigm and orthogonal quantum layers and introduce a circuit to perform quantum Fourier transform in the unary basis. We propose three different quantum circuits to perform a quantum FNO. The proposals differ in their depth and their similarity to the classical FNO. We also benchmark our proposed algorithms on three PDE families, namely Burgers' equation, Darcy's flow equation and the Navier-Stokes equation. The results show that our quantum methods are comparable in performance to the classical FNO. We also perform an analysis on small-scale image classification tasks where our proposed algorithms are at par with the performance of classical CNNs, proving their applicability to other domains as well.

Quantum machine learning techniques have been proposed as a way to potentially enhance performance in machine learning applications. In this paper, we introduce two new quantum methods for neural networks. The first one is a quantum orthogonal neural network, which is based on a quantum pyramidal circuit as the building block for implementing orthogonal matrix multiplication. We provide an efficient way for training such orthogonal neural networks; novel algorithms are detailed for both classical and quantum hardware, where both are proven to scale asymptotically better than previously known training algorithms. The second method is quantum-assisted neural networks, where a quantum computer is used to perform inner product estimation for inference and training of classical neural networks. We then present extensive experiments applied to medical image classification tasks using current state of the art quantum hardware, where we compare different quantum methods with classical ones, on both real quantum hardware and simulators. Our results show that quantum and classical neural networks generates similar level of accuracy, supporting the promise that quantum methods can be useful in solving visual tasks, given the advent of better quantum hardware., Comment: arXiv admin note: substantial text overlap with arXiv:2109.01831, arXiv:2106.07198

Many applications of quantum computing in the near term rely on variational quantum circuits (VQCs). They have been showcased as a promising model for reaching a quantum advantage in machine learning with current noisy intermediate scale quantum computers (NISQ). It is often believed that the power of VQCs relies on their exponentially large feature space, and extensive works have explored the expressiveness and trainability of VQCs in that regard. In our work, we propose a classical sampling method that may closely approximate a VQC with Hamiltonian encoding, given only the description of its architecture. It uses the seminal proposal of Random Fourier Features (RFF) and the fact that VQCs can be seen as large Fourier series. We provide general theoretical bounds for classically approximating models built from exponentially large quantum feature space by sampling a few frequencies to build an equivalent low dimensional kernel, and we show experimentally that this approximation is efficient for several encoding strategies. Precisely, we show that the number of required samples grows favorably with the size of the quantum spectrum. This tool therefore questions the hope for quantum advantage from VQCs in many cases, but conversely helps to narrow the conditions for their potential success. We expect VQCs with various and complex encoding Hamiltonians, or with large input dimension, to become more robust to classical approximations.

In this work, quantum transformers are designed and analysed in detail by extending the state-of-the-art classical transformer neural network architectures known to be very performant in natural language processing and image analysis. Building upon the previous work, which uses parametrised quantum circuits for data loading and orthogonal neural layers, we introduce three types of quantum transformers for training and inference, including a quantum transformer based on compound matrices, which guarantees a theoretical advantage of the quantum attention mechanism compared to their classical counterpart both in terms of asymptotic run time and the number of model parameters. These quantum architectures can be built using shallow quantum circuits and produce qualitatively different classification models. The three proposed quantum attention layers vary on the spectrum between closely following the classical transformers and exhibiting more quantum characteristics. As building blocks of the quantum transformer, we propose a novel method for loading a matrix as quantum states as well as two new trainable quantum orthogonal layers adaptable to different levels of connectivity and quality of quantum computers. We performed extensive simulations of the quantum transformers on standard medical image datasets that showed competitively, and at times better performance compared to the classical benchmarks, including the best-in-class classical vision transformers. The quantum transformers we trained on these small-scale datasets require fewer parameters compared to standard classical benchmarks. Finally, we implemented our quantum transformers on superconducting quantum computers and obtained encouraging results for up to six qubit experiments., Comment: 20 pages

In this thesis, we investigate whether quantum algorithms can be used in the field of machine learning for both long and near term quantum computers. We will first recall the fundamentals of machine learning and quantum computing and then describe more precisely how to link them through linear algebra: we introduce quantum algorithms to efficiently solve tasks such as matrix product or distance estimation. These results are then used to develop new quantum algorithms for unsupervised machine learning, such as k-means and spectral clustering. This allows us to define many fundamental procedures, in particular in vector and graph analysis. We will also present new quantum algorithms for neural networks, or deep learning. For this, we introduce an algorithm to perform a quantum convolution product on images, as well as a new way to perform a fast tomography on quantum states. We prove that these quantum algorithms are faster versions of equivalent classical algorithms, but exhibit random effects due to the quantum nature of the computation. Many simulations have been carried out to study these effects and measure their learning accuracy on real data. Finally, we will present a quantum orthogonal neural network circuit adapted to the currently available small and imperfect quantum computers. This allows us to perform real experiments to test our theory., Comment: PhD Thesis; several papers are included this manuscript; may overlap with: 1812.0358, 1911.01117, 2007.00280, 2106.07198, 2109.01831

Machine Learning provides powerful tools for a variety of applications, including disease diagnosis through medical image classification. In recent years, quantum machine learning techniques have been put forward as a way to potentially enhance performance in machine learning applications, both through quantum algorithms for linear algebra and quantum neural networks. In this work, we study two different quantum neural network techniques for medical image classification: first by employing quantum circuits in training of classical neural networks, and second, by designing and training quantum orthogonal neural networks. We benchmark our techniques on two different imaging modalities, retinal color fundus images and chest X-rays. The results show the promises of such techniques and the limitations of current quantum hardware.

Quantum machine learning promises great speedups over classical algorithms, but it often requires repeated computations to achieve a desired level of accuracy for its point estimates. Bayesian learning focuses more on sampling from posterior distributions than on point estimation, thus it might be more forgiving in the face of additional quantum noise. We propose a quantum algorithm for Bayesian neural network inference, drawing on recent advances in quantum deep learning, and simulate its empirical performance on several tasks. We find that already for small numbers of qubits, our algorithm approximates the true posterior well, while it does not require any repeated computations and thus fully realizes the quantum speedups., Comment: 17 pages, 11 figures

Orthogonal neural networks have recently been introduced as a new type of neural networks imposing orthogonality on the weight matrices. They could achieve higher accuracy and avoid evanescent or explosive gradients for deep architectures. Several classical gradient descent methods have been proposed to preserve orthogonality while updating the weight matrices, but these techniques suffer from long running times or provide only approximate orthogonality. In this paper, we introduce a new type of neural network layer called Pyramidal Circuit, which implements an orthogonal matrix multiplication. It allows for gradient descent with perfect orthogonality with the same asymptotic running time as a standard layer. This algorithm is inspired by quantum computing and can therefore be applied on a classical computer as well as on a near term quantum computer. It could become the building block for quantum neural networks and faster orthogonal neural networks.

Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with non convex or nested structures. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a low-dimensional space where clustering is more efficient. Despite its success in clustering tasks, spectral clustering suffers in practice from a fast-growing running time of $O(n^3)$, where $n$ is the number of points in the dataset. In this work we propose an end-to-end quantum algorithm performing spectral clustering, extending a number of works in quantum machine learning. The quantum algorithm is composed of two parts: the first is the efficient creation of the quantum state corresponding to the projected Laplacian matrix, and the second consists of applying the existing quantum analogue of the $k$-means algorithm. Both steps depend polynomially on the number of clusters, as well as precision and data parameters arising from quantum procedures, and polylogarithmically on the dimension of the input vectors. Our numerical simulations show an asymptotic linear growth with $n$ when all terms are taken into account, significantly better than the classical cubic growth. This work opens the path to other graph-based quantum machine learning algorithms, as it provides routines for efficient computation and quantum access to the Incidence, Adjacency, and projected Laplacian matrices of a graph.

Quantum computing is a new computational paradigm that promises applications in several fields, including machine learning. In the last decade, deep learning, and in particular Convolutional neural networks (CNN), have become essential for applications in signal processing and image recognition. Quantum deep learning, however remains a challenging problem, as it is difficult to implement non linearities with quantum unitaries. In this paper we propose a quantum algorithm for applying and training deep convolutional neural networks with a potential speedup. The quantum CNN (QCNN) is a shallow circuit, reproducing completely the classical CNN, by allowing non linearities and pooling operations. The QCNN is particularly interesting for deep networks and could allow new frontiers in image recognition, by using more or larger convolution kernels, larger or deeper inputs. We introduce a new quantum tomography algorithm with $\ell_{\infty}$ norm guarantees, and new applications of probabilistic sampling in the context of information processing. We also present numerical simulations for the classification of the MNIST dataset to provide practical evidence for the efficiency of the QCNN.

Quantum machine learning is one of the most promising applications of a full-scale quantum computer. Over the past few years, many quantum machine learning algorithms have been proposed that can potentially offer considerable speedups over the corresponding classical algorithms. In this paper, we introduce q-means, a new quantum algorithm for clustering which is a canonical problem in unsupervised machine learning. The $q$-means algorithm has convergence and precision guarantees similar to $k$-means, and it outputs with high probability a good approximation of the $k$ cluster centroids like the classical algorithm. Given a dataset of $N$ $d$-dimensional vectors $v_i$ (seen as a matrix $V \in \mathbb{R}^{N \times d})$ stored in QRAM, the running time of q-means is $\widetilde{O}\left( k d \frac{\eta}{\delta^2}\kappa(V)(\mu(V) + k \frac{\eta}{\delta}) + k^2 \frac{\eta^{1.5}}{\delta^2} \kappa(V)\mu(V) \right)$ per iteration, where $\kappa(V)$ is the condition number, $\mu(V)$ is a parameter that appears in quantum linear algebra procedures and $\eta = \max_{i} ||v_{i}||^{2}$. For a natural notion of well-clusterable datasets, the running time becomes $\widetilde{O}\left( k^2 d \frac{\eta^{2.5}}{\delta^3} + k^{2.5} \frac{\eta^2}{\delta^3} \right)$ per iteration, which is linear in the number of features $d$, and polynomial in the rank $k$, the maximum square norm $\eta$ and the error parameter $\delta$. Both running times are only polylogarithmic in the number of datapoints $N$. Our algorithm provides substantial savings compared to the classical $k$-means algorithm that runs in time $O(kdN)$ per iteration, particularly for the case of large datasets.