Entanglement as a Structural Complexity Axis: A PAC-Bayesian View of Generalization in Quantum Policies and Value Functions

Parameterized quantum circuits (PQCs) are increasingly used as policies and value functions in quantum reinforcement learning, yet it remains unclear when and why quantum policies generalize. We give a PAC-Bayesian account in which generalization is governed not by the raw number of circuit parameters, but by the effective dimension of the Fisher geometry induced by the circuit. This quantity is inflated by entanglement, making entangling connectivity an independent axis of complexity.In controlled experiments that fix the number of trainable rotations and vary only entanglement, we find that circuits with larger Fisher effective dimension exhibit larger train-test gaps, while parameter count is a weak predictor. The resulting bound acts primarily as a ranking certificate: it correctly orders circuits with identical parameter count, which parameter-counting bounds cannot do. We validate this mechanism across supervised classification, quantum contextual bandits, and value-function generalization, where entangled circuits consistently generalize worse than non-entangled circuits of equal parameter count, with gaps shrinking as sample size increases.Our strongest evidence comes from low-variance decision models, including single-observable classifiers, value heads, and one-step policies. In end-to-end multi-step policy learning, entanglement effects remain statistically significant but high return variance leaves the full ordering only partially resolved. Partial-correlation analysis shows that Fisher effective dimension screens off entangling pattern, and controls for training accuracy, readout, and optimizer rule out major optimization confounders. The effect also persists on an IBM Heron quantum processor under real noise. Overall, our results reframe quantum policy design around an entanglement--generalization trade-off rather than expressivity alone.

Staged Hybridisation for Visual Quantum Reinforcement Learning via Knowledge Distillation

Visual environments are a demanding setting for quantum reinforcement learning (QRL): high-dimensional observations, unstable RL optimisation, and constrained variational quantum circuits (VQCs) are difficult to train jointly. This paper studies knowledge distillation (KD) as a staged hybridisation strategy for visual QRL. Instead of training a hybrid visual agent end-to-end from pixels, we first train a classical visual teacher, freeze its encoder as a feature interface, and distil the teacher's policy behaviour into compact downstream heads. These heads can be classical or VQC-based, enabling small quantum-compatible students to be evaluated under the same frozen representation as compact classical controls. We evaluate the pipeline on CartPole Pixels and Acrobot Pixels. The results show that staged KD enables shallow VQC heads to acquire non-trivial visual-control behaviour in settings where direct pixel-based training would be substantially more difficult. Angle-encoded VQC heads retain near-teacher performance, while amplitude-encoded heads push compactness to an extreme regime, at the cost of greater fragility, stronger budget sensitivity, and higher simulation time. Overall, staged KD reframes visual QRL as a compact-head learning problem, opening a practical route for training small quantum-compatible policies outside the standard end-to-end RL loop.

QnRL: Quantum-Native Reinforcement Learning

Quantum reinforcement learning (QRL) is a promising approach to learn effective decision strategies across several applications with stochastic environments. Instead of directly modeling the random variables that govern these environments, existing QRL architectures indirectly approximate environment behavior by estimating expected outcomes, which limits their expressive power and adaptive potential. Overcoming such challenges requires a novel QRL approach that exploits the distributional nature of quantum computers to directly model environment random variables as quantum state distributions. Hence, in this paper, a novel framework dubbed quantum-native reinforcement learning (QnRL) is proposed. QnRL is a distributional RL framework that learns conditional distributions naturally in Hilbert space via superimposed and entangled quantum states. Thus, QnRL can directly model the behavior of stochastic learning environments via the natural properties of quantum systems. QnRL accomplishes this via a novel, proposed quantum amplitude kickback (QuAK) algorithm that enables comparing the $n$-th power of the $m$-th moment of multiple superimposed distributions. It is theoretically proven that a conditional action policy distribution is distilled from the moments of a quantum generative model entirely within Hilbert space via QuAK, and optimized via QnRL. This complex distribution composition is also shown to provide extra dimensions for expressing environment correlations that are unknown to purely classical and classically-sampled quantum distributional models. Experimental results across diverse environments show that QnRL achieves up to $82.9\%$ higher evaluation scores, with up to $94.3\%$ fewer parameters on average, more accurately estimates the expected return for unseen observations, and better adapts to varying stochastic conditions compared to the baseline.