Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Extrapolation of quantum observables with Gaussian processes

Rodrigo A. Vargas–Hernández, Roman V. Krems

Published Nov 21, 2018• 0 citations
OpenAlex

Abstract

For applications in chemistry and physics, machine learning (ML) is generally used to solve one of three problems: interpolation, classification or clustering. These problems use information about physical systems in a certain range of parameters or variables in order to make predictions at unknown values of these variables within the same range. The present work illustrates the application of ML to prediction of physical properties outside the range of the training parameters. We define `physical extrapolation' to refer to accurate predictions $y(\mathbf{x^\ast})$ of a given physical property at a point $\mathbf{x^\ast} = [ x^\ast_1, ..., x^\ast_{\cal D} ]$ in the $\cal D$-dimensional space, if, at least, one of the variables $x^\ast_i \in \left [ x^\ast_1, ..., x^\ast_{\cal D} \right ]$ is {\it outside} of the range covering the training data. We show that Gaussian processes (GPs) can be used to build ML models capable of physical extrapolation of quantum properties of complex systems across quantum phase transitions. The approach is based on training GP models of variable complexity by the evolution of the physical functions. We show that, as the complexity of the models increases, they become capable of predicting new transitions. We also show that, where the evolution of the physical functions is analytic and relatively simple, GP models with simple kernels already yield accurate generalization results, allowing for accurate predictions of quantum properties in a different quantum phase. For more complex problems, it is necessary to build models with complex kernels. The complexity of the kernels is increased using the Bayesian Information Criterion (BIC). We illustrate the importance of the BIC by comparing the results with random kernels of various complexity and illustrate a method to obtain meaningful extrapolation results without direct validation in the extrapolated region.

Related papers

Applications of machine learning for solving complex quantum problems

Rodrigo A. Vargas–Hernández

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.

•Jan 1, 2019•0 citations

Provable and scalable quantum Gaussian processes for quantum learning

Jonas Jäger, Paolo Braccia, Pablo Bermejo +3 more

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.

quant-pharXiv:2605.00099•May 4, 2026•0 citations

Extrapolating Quantum Observables with Machine Learning: Inferring Multiple Phase Transitions from Properties of a Single Phase

Rodrigo A. Vargas–Hernández, John Sous, Mona Berciu +1 more

We present a machine-learning method for predicting sharp transitions in a Hamiltonian phase diagram by extrapolating the properties of quantum systems. The method is based on Gaussian process regression with a combination of kernels chosen through an iterative procedure maximizing the predicting power of the kernels. The method is capable of extrapolating across the transition lines. The calculations within a given phase can be used to predict not only the closest sharp transition but also a transition removed from the available data by a separate phase. This makes the present method particularly valuable for searching phase transitions in the parts of the parameter space that cannot be probed experimentally or theoretically.

arXiv:1803.08195•Physical Review Letters•Dec 17, 2018•67 citations

Benchmarking of quantum fidelity kernels for Gaussian process regression

Xuyang Guo, Jun Feng Dai, Roman V. Krems

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.

arXiv:2407.15961•Machine Learning Science and Technology•Sep 1, 2024•1 citation

A Bayesian inference framework for compression and prediction of quantum states

Yannic Rath, Aldo Glielmo, George H. Booth

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.

arXiv:2008.03595•The Journal of Chemical Physics•Sep 24, 2020•11 citations

Quantum Gaussian processes for prediction of channel observations

Jonas Jäger, Yaroslav Khmelnitskiy, Paolo Braccia +4 more

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.

quant-pharXiv:2608.19306•Aug 21, 2026•0 citations
For applications in chemistry and physics, machine learning (ML) is generally used to solve one of three problems: interpolation, classification or clustering. These problems use information about physical systems in a certain range of parameters or variables in order to make predictions at unknown values of these variables within the same range. The present work illustrates the application of ML to prediction of physical properties outside the range of the training parameters. We define `physical extrapolation' to refer to accurate predictions y(x∗)y(\mathbf{x^\ast})y(x∗) of a given physical property at a point x∗=[x1∗,...,xD∗]\mathbf{x^\ast} = [ x^\ast_1, ..., x^\ast_{\cal D} ]x∗=[x1∗​,...,xD∗​] in the D\cal DD-dimensional space, if, at least, one of the variables xi∗∈[x1∗,...,xD∗]x^\ast_i \in \left [ x^\ast_1, ..., x^\ast_{\cal D} \right ]xi∗​∈[x1∗​,...,xD∗​] is {\it outside} of the range covering the training data. We show that Gaussian processes (GPs) can be used to build ML models capable of physical extrapolation of quantum properties of complex systems across quantum phase transitions. The approach is based on training GP models of variable complexity by the evolution of the physical functions. We show that, as the complexity of the models increases, they become capable of predicting new transitions. We also show that, where the evolution of the physical functions is analytic and relatively simple, GP models with simple kernels already yield accurate generalization results, allowing for accurate predictions of quantum properties in a different quantum phase. For more complex problems, it is necessary to build models with complex kernels. The complexity of the kernels is increased using the Bayesian Information Criterion (BIC). We illustrate the importance of the BIC by comparing the results with random kernels of various complexity and illustrate a method to obtain meaningful extrapolation results without direct validation in the extrapolated region.
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.
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.
We present a machine-learning method for predicting sharp transitions in a Hamiltonian phase diagram by extrapolating the properties of quantum systems. The method is based on Gaussian process regression with a combination of kernels chosen through an iterative procedure maximizing the predicting power of the kernels. The method is capable of extrapolating across the transition lines. The calculations within a given phase can be used to predict not only the closest sharp transition but also a transition removed from the available data by a separate phase. This makes the present method particularly valuable for searching phase transitions in the parts of the parameter space that cannot be probed experimentally or theoretically.
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.
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.
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.