The data-driven approach shows great potential for designing new materials with unprecedented properties by using machine learning and optimization. Recently, a data-driven framework was successfully applied to design a unit cell of metamaterial achieving super-compressibility, despite being built out of brittle base material. The key element of the framework is the algorithm called Gaussian processes regression (GPs) – a unique machine learning method that provides with the uncertainty of the prediction, which can be used during the design process to account for inherent material imperfections, ensuring the robustness of the design. Despite their superior predictive performance, however, GPs suffer from scalability issues, which limit their application to relatively small design problems. In the future, those limitations could be surpassed by quantum computing. This research aims at demonstrating how quantum computing could enhance the computational design of materials, specifically by replacing the expensive machine learning step of the data-driven design framework with an exponentially faster quantum algorithm for Gaussian processes (QGP). This objective is achieved in two steps. First, the QGP algorithm was Implementation and simulated within a quantum computing framework (Qiskit), which allowed to understand and control its performance (in particular its accuracy), proving the feasibility for practical applications. Furthermore, the numerical tests exposed a mechanism for inducing a low-rank approximation, which allows for additional speed-up, making the QGP algorithm similar to classical sparse Gaussian processes, which rely on low-rank approximations to improve the scalability of full GPs. In the second part of this research, the implemented QGP algorithm was integrated within a computational framework for the data-driven design of materials and applied to two example design problems of optimization a unit cell for a super-compressible metamaterial. The results obtained with the QGP were comparable to those obtained with classical methods (also from literature), which proved the feasibility of the concept.
The data-driven approach shows great potential for designing new materials with unprecedented properties by using machine learning and optimization. Recently, a data-driven framework was successfully applied to design a unit cell of metamaterial achieving super-compressibility, despite being built out of brittle base material. The key element of the framework is the algorithm called Gaussian processes regression (GPs) – a unique machine learning method that provides with the uncertainty of the prediction, which can be used during the design process to account for inherent material imperfections, ensuring the robustness of the design. Despite their superior predictive performance, however, GPs suffer from scalability issues, which limit their application to relatively small design problems. In the future, those limitations could be surpassed by quantum computing. This research aims at demonstrating how quantum computing could enhance the computational design of materials, specifically by replacing the expensive machine learning step of the data-driven design framework with an exponentially faster quantum algorithm for Gaussian processes (QGP). This objective is achieved in two steps. First, the QGP algorithm was Implementation and simulated within a quantum computing framework (Qiskit), which allowed to understand and control its performance (in particular its accuracy), proving the feasibility for practical applications. Furthermore, the numerical tests exposed a mechanism for inducing a low-rank approximation, which allows for additional speed-up, making the QGP algorithm similar to classical sparse Gaussian processes, which rely on low-rank approximations to improve the scalability of full GPs. In the second part of this research, the implemented QGP algorithm was integrated within a computational framework for the data-driven design of materials and applied to two example design problems of optimization a unit cell for a super-compressible metamaterial. The results obtained with the QGP were comparable to those obtained with classical methods (also from literature), which proved the feasibility of the concept.
Abstract Gaussian processes are well-established Bayesian machine learning algorithms with significant merits, despite a strong limitation: lack of scalability. Clever solutions address this issue by inducing sparsity through low-rank approximations, often based on the Nystrom method. Here, we propose a different method to achieve better scalability and higher accuracy using quantum computing, outperforming classical Bayesian neural networks for large datasets significantly. Unlike other approaches to quantum machine learning, the computationally expensive linear algebra operations are not just replaced with their quantum counterparts. Instead, we start from a recent study that proposed a quantum circuit for implementing quantum Gaussian processes and then we use quantum phase estimation to induce a low-rank approximation analogous to that in classical sparse Gaussian processes. We provide evidence through numerical tests, mathematical error bound estimation, and complexity analysis that the method can address the “curse of dimensionality,” where each additional input parameter no longer leads to an exponential growth of the computational cost. This is also demonstrated by applying the algorithm in a practical setting and using it in the data-driven design of a recently proposed metamaterial. The algorithm, however, requires significant quantum computing hardware improvements before quantum advantage can be achieved.
Gaussian process regression (GPR) is a nonparametric supervised learning method, widely valued for its ability to quantify uncertainty. Despite its advantages and broad applications, classical GPR implementations face significant scalability challenges, as they involve matrix operations with a cubic complexity in relation to the dataset size. This computational challenge is further compounded by the demand of optimizing the Gaussian Process (GP) model over its hyperparameters, rendering the total computational cost prohibitive for data intensive problems. To address this issue, we propose a quantum gradient descent algorithm to optimize the GP model. Taking advantage of recent advances in quantum algorithms for linear algebra, our algorithm achieves exponential speedup in computing the gradients of the log marginal likelihood. The entire gradient descent process is integrated into the quantum circuit. Through runtime analysis and error bounds, we demonstrate that our approach significantly improves the scalability of GPR model optimization, making it computationally feasible for large-scale applications.
Quantum Machine Learning (QML) presents a transformative paradigm for artificial intelligence, offering the potential to overcome limitations of classical machine learning algorithms. This paper explores a novel quantum machine learning approach, specifically designed for non-linear data-driven learning, aimed at enhancing model generalization. The core of this research focuses on employing a non-linear data transformation coupled with a quantum algorithm to capture intricate data patterns. We present a design for a quantum-enhanced learning process that surpasses traditional methods in handling complex datasets. The proposed method leverages the inherent properties of quantum mechanics to facilitate efficient pattern recognition and representation, ultimately yielding improved predictive accuracy and robustness. This work investigates the potential of QML to address challenges in data analysis and model optimization, and provides a foundation for future research in this rapidly evolving field. This paper provides a comprehensive overview of the core concepts, proposed methodology, and potential future directions for quantum machine learning.
This thesis investigates the interplay between machine learning (ML) and quantum mechanics to advance quantum machine learning (QML) and quantum calculations, addressing computational challenges in quantum machine learning, materials science, and condensed matter physics. The high dimensionality of quantum systems often renders traditional methods computationally intensive. By integrating classical and quantum ML techniques, innovative algorithms are developed to enhance efficiency and accuracy. The first part investigates the design and optimization of quantum circuit architectures for quantum machine learning, with a focus on quantum support vector machines (QSVM). I develop a Bayesian algorithm that adaptively constructs quantum kernels, achieving superior performance over classical kernels in small-data classification tasks. I also establish an isomorphism between quantum circuits and polyatomic molecules, which enables the restriction of the search space for compositional optimization of quantum circuits. Additionally, I introduce a flexible scheme that maps quantum circuits into graph representations compatible with the neutral atom quantum hardware. Using this circuit-graph mapping scheme, I develop a Gaussian process regression model with quantum evolution kernels and a classical graph neural network to interpolate QSVM accuracy on graph-structured data. The second part applies ML to accelerate quantum calculations. I develop a variational algorithm based on eigenvector continuation to compute ground state energies of particle-phonon systems. This algorithm constructs the lowest eigenvalue and eigenvector for extended lattices by solving eigenvalue problems for small, independent lattice segments, with a lower computational cost. This enables the application of variational quantum eigensolvers to particle-phonon interactions in large and disordered lattices, reducing the demands on quantum resources. I also developed Gaussian process models to accelerate Green’s function calculations for electron-phonon systems, achieving high accuracy by extrapolating from restricted Hilbert spaces to larger systems with lower computational cost. Ultimately, this thesis demonstrates that these advancements pave the way for applications, including the design of materials, enhanced drug discovery, and the development of scalable quantum computing technologies, thereby positioning the integration of ML and quantum physics as a promising approach for future scientific breakthroughs.
This thesis investigates the interplay between machine learning (ML) and quantum mechanics to advance quantum machine learning (QML) and quantum calculations, addressing computational challenges in quantum machine learning, materials science, and condensed matter physics. The high dimensionality of quantum systems often renders traditional methods computationally intensive. By integrating classical and quantum ML techniques, innovative algorithms are developed to enhance efficiency and accuracy. The first part investigates the design and optimization of quantum circuit architectures for quantum machine learning, with a focus on quantum support vector machines (QSVM). I develop a Bayesian algorithm that adaptively constructs quantum kernels, achieving superior performance over classical kernels in small-data classification tasks. I also establish an isomorphism between quantum circuits and polyatomic molecules, which enables the restriction of the search space for compositional optimization of quantum circuits. Additionally, I introduce a flexible scheme that maps quantum circuits into graph representations compatible with the neutral atom quantum hardware. Using this circuit-graph mapping scheme, I develop a Gaussian process regression model with quantum evolution kernels and a classical graph neural network to interpolate QSVM accuracy on graph-structured data. The second part applies ML to accelerate quantum calculations. I develop a variational algorithm based on eigenvector continuation to compute ground state energies of particle-phonon systems. This algorithm constructs the lowest eigenvalue and eigenvector for extended lattices by solving eigenvalue problems for small, independent lattice segments, with a lower computational cost. This enables the application of variational quantum eigensolvers to particle-phonon interactions in large and disordered lattices, reducing the demands on quantum resources. I also developed Gaussian process models to accelerate Green’s function calculations for electron-phonon systems, achieving high accuracy by extrapolating from restricted Hilbert spaces to larger systems with lower computational cost. Ultimately, this thesis demonstrates that these advancements pave the way for applications, including the design of materials, enhanced drug discovery, and the development of scalable quantum computing technologies, thereby positioning the integration of ML and quantum physics as a promising approach for future scientific breakthroughs.
The recently introduced Gaussian Process State (GPS) provides a highly flexible, compact, and physically insightful representation of quantum many-body states based on ideas from the zoo of machine learning approaches. In this work, we give a comprehensive description of how such a state can be learned from given samples of a potentially unknown target state and show how regression approaches based on Bayesian inference can be used to compress a target state into a highly compact and accurate GPS representation. By application of a type II maximum likelihood method based on relevance vector machines, we are able to extract many-body configurations from the underlying Hilbert space, which are particularly relevant for the description of the target state, as support points to define the GPS. Together with an introduced optimization scheme for the hyperparameters of the model characterizing the weighting of modeled correlation features, this makes it possible to easily extract physical characteristics of the state such as the relative importance of particular correlation properties. We apply the Bayesian learning scheme to the problem of modeling ground states of small Fermi-Hubbard chains and show that the found solutions represent a systematically improvable trade-off between sparsity and accuracy of the model. Moreover, we show how the learned hyperparameters and the extracted relevant configurations, characterizing the correlation of the wave function, depend on the interaction strength of the Hubbard model and the target accuracy of the representation.