Parametrised quantum circuits offer expressive and data-efficient representations for machine learning. Due to quantum states residing in a high-dimensional Hilbert space, parametrised quantum circuits have a natural interpretation in terms of kernel methods. The representation of quantum circuits in terms of quantum kernels has been studied widely in quantum supervised learning, but has been overlooked in the context of quantum RL. This paper proposes the use of kernel policies and quantum policy gradient algorithms for quantum-accessible environments. After discussing the properties of such policies and a demonstration of classical policy gradient on a coherent policy in a quantum environment, we propose parametric and non-parametric policy gradient and actor-critic algorithms with quantum kernel policies in quantum environments. This approach, implemented with both numerical and analytical quantum policy gradient techniques, allows exploiting the many advantages of kernel methods, including data-driven forms for functions (and their gradients) as well as tunable expressiveness. The proposed approach is suitable for vector-valued action spaces and each of the formulations demonstrates a quadratic reduction in query complexity compared to their classical counterparts. We propose actor-critic algorithms based on stochastic policy gradient, deterministic policy gradient, and natural policy gradient, and demonstrate additional query complexity reductions compared to quantum policy gradient algorithms under favourable conditions.
Parametrised quantum circuits offer expressive and data-efficient representations for machine learning. Due to quantum states residing in a high-dimensional Hilbert space, parametrised quantum circuits have a natural interpretation in terms of kernel methods. The representation of quantum circuits in terms of quantum kernels has been studied widely in quantum supervised learning, but has been overlooked in the context of quantum RL. This paper proposes the use of kernel policies and quantum policy gradient algorithms for quantum-accessible environments. After discussing the properties of such policies and a demonstration of classical policy gradient on a coherent policy in a quantum environment, we propose parametric and non-parametric policy gradient and actor-critic algorithms with quantum kernel policies in quantum environments. This approach, implemented with both numerical and analytical quantum policy gradient techniques, allows exploiting the many advantages of kernel methods, including data-driven forms for functions (and their gradients) as well as tunable expressiveness. The proposed approach is suitable for vector-valued action spaces and each of the formulations demonstrates a quadratic reduction in query complexity compared to their classical counterparts. We propose actor-critic algorithms based on stochastic policy gradient, deterministic policy gradient, and natural policy gradient, and demonstrate additional query complexity reductions compared to quantum policy gradient algorithms under favourable conditions.
Understanding the power and limitations of quantum access to data in machine learning tasks is primordial to assess the potential of quantum computing in artificial intelligence. Previous works have already shown that speed-ups in learning are possible when given quantum access to reinforcement learning environments. Yet, the applicability of quantum algorithms in this setting remains very limited, notably in environments with large state and action spaces. In this work, we design quantum algorithms to train state-of-the-art reinforcement learning policies by exploiting quantum interactions with an environment. However, these algorithms only offer full quadratic speed-ups in sample complexity over their classical analogs when the trained policies satisfy some regularity conditions. Interestingly, we find that reinforcement learning policies derived from parametrized quantum circuits are well-behaved with respect to these conditions, which showcases the benefit of a fully-quantum reinforcement learning framework.
Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.
Variational quantum circuits have recently gained popularity as quantum machine learning models. While considerable effort has been invested to train them in supervised and unsupervised learning settings, relatively little attention has been given to their potential use in reinforcement learning. In this work, we leverage the understanding of quantum policy gradient algorithms in a number of ways. First, we investigate how to construct and train reinforcement learning policies based on variational quantum circuits. We propose several designs for quantum policies, provide their learning algorithms, and test their performance on classical benchmarking environments. Second, we show the existence of task environments with a provable separation in performance between quantum learning agents and any polynomial-time classical learner, conditioned on the widely-believed classical hardness of the discrete logarithm problem. We also consider more natural settings, in which we show an empirical quantum advantage of our quantum policies over standard neural-network policies. Our results constitute a first step towards establishing a practical near-term quantum advantage in a reinforcement learning setting. Additionally, we believe that some of our design choices for variational quantum policies may also be beneficial to other models based on variational quantum circuits, such as quantum classifiers and quantum regression models.
Quantum computing has a superior advantage in tackling specific problems, such as integer factorization and Simon's problem. For more general tasks in machine learning, by applying variational quantum circuits, more and more quantum algorithms have been proposed recently, especially in supervised learning and unsupervised learning. However, little work has been done in reinforcement learning, arguably more important and challenging. Previous work in quantum reinforcement learning mainly focuses on discrete control tasks where the action space is discrete. In this work, we develop a quantum reinforcement learning algorithm based on soft actor-critic -- one of the state-of-the-art methods for continuous control. Specifically, we use a hybrid quantum-classical policy network consisting of a variational quantum circuit and a classical artificial neural network. Tested in a standard reinforcement learning benchmark, we show that this quantum version of soft actor-critic is comparable with the original soft actor-critic, using much less adjustable parameters. Furthermore, we analyze the effect of different hyper-parameters and policy network architectures, pointing out the importance of architecture design for quantum reinforcement learning.
Reading out a parametrized quantum circuit through finite-shot measurements introduces a discrete stochastic component into hybrid quantum-classical neural networks, making gradient-based training inefficient. In this work, we introduce a surrogate-assisted actor-critic method for training hybrid quantum policies that avoids differentiating the quantum circuit during backpropagation. A classical neural surrogate trained on the circuit's input-output pairs from a replay buffer provides a differentiable substitute for actor updates and critic-target computation, while forward interaction and evaluation retain the quantum circuit's measurements. Our experiments on standard MuJoCo continuous-control tasks demonstrate promising policy learning. Notably, on Humanoid, more runs reach high returns than with learned classical baselines. Moreover, the best average return in our ablation comes from a smaller circuit / surrogate, suggesting that strong performance does not depend on a large network. We also derive conditional upstream gradient-error bounds that separate surrogate fitting from the mean-feature relaxation. Together, the results support surrogate-assisted training as an effective and efficient approach to learning hybrid quantum policies.
Quantum kernels are among the most promising methods in quantum machine learning, providing a natural way to embed data into high-dimensional Hilbert spaces via quantum circuits and capturing non-linear relationships in data. Their learning behaviour is governed by the function class they represent, which is formally characterized by the structure of the associated Reproducing Kernel Hilbert Space (RKHS). However, a theoretical characterization of this function class, and of how it depends on the choice of feature map and its hyperparameters, is still lacking. In this work, we focus on the Instantaneous Quantum Polynomial (IQP) feature map and derive a Fourier representation of the associated RKHS generators as finite sums of trigonometric functions evaluated on polynomially transformed input variables, with a finite set of accessible frequencies determined by the circuit construction and scaled by the bandwidth hyperparameter. This representation identifies three bandwidth regimes with qualitatively distinct expressive properties from localised, through multimodal, to periodic. Numerical experiments on synthetic datasets with controlled periodic and multimodal structure, across data dimensions and feature-map configurations, reproduce the three regimes in both supervised classification and unsupervised anomaly detection, benchmarked against a classical Gaussian kernel. These results make the relationship between expressivity and data structure explicit for IQP kernels: expressive power translates into a practical benefit only when the accessible frequencies match with the structure of the data, which provides a criterion for selecting the bandwidth when the periodic or multimodal structure of data is known.