With gates of a quantum computer designed to encode multi-dimensional vectors, projections of quantum computer states onto specific qubit states can produce kernels of reproducing kernel Hilbert spaces. We show that quantum kernels obtained with a fixed ansatz implementable on current quantum computers can be used for accurate regression models of global potential energy surfaces (PESs) for polyatomic molecules. To obtain accurate regression models, we apply Bayesian optimization to maximize marginal likelihood by varying the parameters of the quantum gates. This yields Gaussian process models with quantum kernels. We illustrate the effect of qubit entanglement in the quantum kernels and explore the generalization performance of quantum Gaussian processes by extrapolating global six-dimensional PESs in the energy domain.
With gates of a quantum computer designed to encode multi-dimensional vectors, projections of quantum computer states onto specific qubit states can produce kernels of reproducing kernel Hilbert spaces. We show that quantum kernels obtained with a fixed ansatz implementable on current quantum computers can be used for accurate regression models of global potential energy surfaces (PESs) for polyatomic molecules. To obtain accurate regression models, we apply Bayesian optimization to maximize marginal likelihood by varying the parameters of the quantum gates. This yields Gaussian process models with quantum kernels. We illustrate the effect of qubit entanglement in the quantum kernels and explore the generalization performance of quantum Gaussian processes by extrapolating global six-dimensional PESs in the energy domain.
Abstract Quantum computing algorithms have been shown to produce performant quantum kernels for machine-learning classification problems. Here, we examine the performance of quantum kernels for regression problems of practical interest. For an unbiased benchmarking of quantum kernels, it is necessary to construct the most optimal functional form of the classical kernels and the most optimal quantum kernels for each given data set. We develop an algorithm that uses an analog of the Bayesian information criterion to optimize the sequence of quantum gates used to estimate quantum kernels for Gaussian process models. The algorithm increases the complexity of the quantum circuits incrementally, while improving the performance of the resulting kernels, and is shown to yield much higher model accuracy with fewer quantum gates than a fixed quantum circuit ansatz. We demonstrate that quantum kernels thus obtained can be used to build accurate models of global potential energy surfaces (PES) for polyatomic molecules. The average interpolation error of the six-dimensional PES obtained with a random distribution of 2000 energy points is 16 cm−1 for H3O+, 15 cm−1 for H2CO and 88 cm−1 for HNO2. We show that a compositional optimization of classical kernels for Gaussian process regression converges to the same errors. This indicates that quantum kernels can achieve the same, though not better, expressivity as classical kernels for regression problems.
This thesis illustrates the use of machine learning algorithms and exact numerical methods to study quantum observables for different systems. The first part of this thesis depicts how to construct accurate potential energy surfaces (PESs) using supervised learning algorithms such as Gaussian Process (GP) regression. PESs have a leading part in quantum chemistry since they are used to study chemical reaction dynamics. Constructing the PES from quantum reactive scattering calculations, as the reaction probability, is known as the inverse scattering problem. Here, we illustrate a possible solution to the inverse scattering problem with a two-tiered GP model one GP model interpolates the PES and the second in Bayesian optimization (BO) algorithm. The end result is an accurate PES constructed with a GP with fewer points than with standard methods previously used for PES. BO is an optimization algorithm for black-box functions that use GP regression as an approximation of the interrogative function. We applied BO to find the optimal parameters of hybrid-density functionals. Quantum observables can differ between phases of matter. GP models with kernel combinations can extrapolate quantum observables such as the polaron dispersion energy between different phases and discover phases of matter. The same algorithm can predict quantum observables where standard numerical techniques lack convergence. In the second half of the dissertation, we studied the evolution of quantum walks in various graphs with Hamiltonians permitting particle number changes. We showed that particle number-changing interactions accelerate quantum walks for any of the graph considered. Quantum simulators to study many-body physics is an active research field. We proposed the use of magnetic atoms trapped in optical lattices to experimentally mimic Bose-Hubbard type models by preparing atoms in different Zeeman states.
Abstract Gaussian process regression is a well-established Bayesian machine learning method. We propose a new approach to Gaussian process regression using quantum kernels based on parameterized quantum circuits. By employing a hardware-efficient feature map and careful regularization of the Gram matrix, we demonstrate that the variance information of the resulting quantum Gaussian process can be preserved. We also show that quantum Gaussian processes can be used as a surrogate model for Bayesian optimization, a task that critically relies on the variance of the surrogate model. To demonstrate the performance of this quantum Bayesian optimization algorithm, we apply it to the hyperparameter optimization of a machine learning model which performs regression on a real-world dataset. We benchmark the quantum Bayesian optimization against its classical counterpart and show that quantum version can match its performance.
Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.
Gaussian processes (GPs) are flexible nonparametric Bayesian priors over functions and are widely used in regression, time-series modelling, and Bayesian optimisation. Yet their performance is often bottlenecked by the choice of covariance kernel, especially when the relevant correlations are unknown or inherently non-stationary. In this work, we put forward a NISQ-compatible framework that uses a quantum device as a generator of local quantum kernels for GP learning. At a high level, shallow, geometry-aware randomly parametrized circuits with area-law entanglement encode inputs into many-qubit states, and few-body Pauli measurements-estimated efficiently via classical shadows-produce quantum feature vectors that are subsequently wrapped into classical scalar kernels. A multiplekernel learning layer then fits a convex mixture of these quantum kernels by maximizing the GP marginal likelihood, so that all heavy optimisation remains on the classical side, while the quantum hardware is used solely for kernel estimation. On synthetic regression tasks, we find that the learned mixture can approximate both stationary and non-stationary covariances, achieving high kernel alignment and accurate predictions. Taken together, these results suggest that local quantum kernels can in principle form a family of flexible GP covariance learners within realistic NISQ resource budgets.
Constructing accurate global potential energy surfaces (PESs) describing chemically reactive molecule-molecule collisions of alkali metal dimers presents a major challenge. To be suitable for quantum scattering calculations, such PESs must represent accurately three- and four-body interactions, describe conical intersections, and have a proper asymptotic form at the long range. Here, we demonstrate that such global potentials can be obtained by Gaussian Process (GP) regression merged with the analytic asymptotic expansions at the long range. We propose an efficient sampling technique, which allows us to construct an accurate global PES accounting for different chemical arrangements with <2500 ab initio calculations. We apply this method to (NaK)2 and obtain the first global PES for a system of four alkali metal atoms. The resulting surface exhibits a complex landscape including a pair and a quartet of symmetrically equivalent local minima and a seam of conical intersections. The dissociation energy found from our ab initio calculations is 4534 cm−1. This result is reproduced by the GP models with an error of less than 3%. The GP models of the PES allow us to analyze the features of the global PES, representative of general alkali metal four-atom interactions. Understanding these interactions is of key importance in the field of ultracold chemistry.