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Free energy-based reinforcement learning (FERL) with clamped quantumBoltzmann machines (QBM) was shown to significantly improve the learningefficiency compared to classical Q-learning with the restriction, however, todiscrete state-action space environments. In this paper, the FERL approach isextended to multi-dimensional continuous state-action space environments toopen the doors for a broader range of real-world applications. First, freeenergy-based Q-learning is studied for discrete action spaces, but continuousstate spaces and the impact of experience replay on sample efficiency isassessed. In a second step, a hybrid actor-critic scheme for continuousstate-action spaces is developed based on the Deep Deterministic PolicyGradient algorithm combining a classical actor network with a QBM-based critic.The results obtained with quantum annealing, both simulated and with D-Wavequantum annealing hardware, are discussed, and the performance is compared toclassical reinforcement learning methods. The environments used throughoutrepresent existing particle accelerator beam lines at the European Organisationfor Nuclear Research (CERN). Among others, the hybrid actor-critic agent isevaluated on the actual electron beam line of the Advanced Plasma WakefieldExperiment (AWAKE).

Hybrid actor-critic algorithm for quantum reinforcement learning at CERN beam lines