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.
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.
Capturing the correlation emerging between constituents of many-body systems accurately is one of the key challenges for the appropriate description of various systems whose properties are underpinned by quantum mechanical fundamentals. This thesis discusses novel tools and techniques for the (classical) modelling of quantum many-body wavefunctions with the ultimate goal to introduce a universal framework for finding accurate representations from which system properties can be extracted efficiently. It is outlined how synergies with standard machine learning approaches can be exploited to enable an automated inference of the most relevant intrinsic characteristics through rigorous Bayesian regression techniques. Based on the probabilistic framework forming the foundation of the introduced ansatz, coined the Gaussian Process State, different compression techniques are explored to extract numerically feasible representations of relevant target states within stochastic schemes. By following intuitively motivated design principles, the resulting model carries a high degree of interpretability and offers an easily applicable tool for the numerical study of quantum systems, including ones which are notoriously difficult to simulate due to a strong intrinsic correlation. The practical applicability of the Gaussian Process States framework is demonstrated within several benchmark applications, in particular, ground state approximations for prototypical quantum lattice models, Fermi-Hubbard models and J1−J2 models, as well as simple ab-initio quantum chemical systems.
The quantum many-electron problem is not just at the heart of condensed matter phenomena, but also essential for first-principles simulation of chemical phenomena. Strong correlation in chemical systems are prevalent and present a formidable challenge in the simulation of these systems, while predictive phenomena in this domain often also requires a demanding level of accuracy to inform chemical behavior. Efficient representations of the many-electron states of chemical systems are therefore also being inspired by machine learning principles to provide an alternative to established approaches. In this chapter, we review recent progress in this endeavor for quantum chemical problems represented in second quantization, and the particular challenges present in this field. In particular, we focus on the application of Gaussian Process States emerging from efficient representations of the many-body wavefunction with rigorous Bayesian modeling frameworks, allowing for the unification of multiple paradigms under a common umbrella. We show how such models (and other representations derived from machine learning) can be used as novel tools to compute ab initio chemical properties, while in turn also informing the design of machine learning models to extract correlation patterns in classical data.
In this thesis, I present several results on quantum statistical inference in the following two directions. Firstly, I demonstrate that quantum algorithms can be applied to enhance the computing and training of Gaussian processes (GPs), a powerful model widely used in classical statistical inference and supervised machine learning. A crucial component of the quantum GP algorithm is solving linear systems with quantum computers, for which I present a novel algorithm that achieves a provable advantage over previously known methods. I will also explicitly address the task of encoding the classical data into a quantum state for machine learning applications. I then apply the quantum enhanced GPs to Bayesian deep learning and present an experimental demonstration on contemporary hardware and simulators. Secondly, I look into the notion of quantum causality and apply it to inferring spatial and temporal quantum correlations, and present an analytical toolkit for causal inference in quantum data. I will also make the connection between causality and quantum communications, and present a general bound for the quantum capacity of noisy communication channels.
Despite rapid recent advances in quantum machine learning, the field is in many ways stuck. Existing approaches can exhibit serious limitations, and we still lack learning frameworks that are simple, interpretable, scalable, and naturally suited to quantum data. To address this, here we introduce quantum Gaussian processes, a Bayesian framework for learning from quantum systems through priors over unknown quantum transformations. We show that, under suitable conditions, unitary quantum stochastic processes define Gaussian processes, thereby enabling regression, classification, and Bayesian optimization directly on quantum data. The key ingredient in this framework is sufficient knowledge of a quantum process's structure and symmetries to define an informative prior through its corresponding quantum kernel, effectively injecting a strong, physics-informed inductive bias into the learning model. We then prove that matchgate, or free-fermionic, evolutions give rise to provable and scalable quantum Gaussian processes, providing the first family in our framework where the unknown unitary acts non-trivially on all qubits. Finally, we demonstrate accurate long-range extrapolation, phase-diagram learning in many-body systems, and sample-efficient Bayesian optimization in a quantum sensing task. Our results identify quantum Gaussian processes as a promising route toward simpler and more structured forms of quantum learning.
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.
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.