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Abstract

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.