Quantum Robust Inner Minimization for Reinforcement Learning with Quadratic Speed-Up in Query Complexity

Robust reinforcement learning (RRL) aims to tackle unexpected environmental changes by optimizing policies against the worst case. However, RRL remains impractical due to the cost of the Max-Min optimization, where it suffers from the exhaustive query complexity for finding the worst-case (dubbed 'Min') within the environmental uncertainty set $\mathcal{U}$, i.e., $\mathcal{O}(|\mathcal{U}|)$. By viewing this via a lens of quantum perspective, we raise a pivotal question: *If we can query from the environment with quantum superpositions, is it possible to accelerate the Max-Min optimization of RRL?* Our answer is 'Yes'. Our method, called quantum robust inner minimization (QRIM), encodes the uncertainty set with quantum superposition and amplifies low-return cases, thus enabling RL for solving the robust (i.e., worst-case) Bellman equation. Importantly, QRIM achieves a quadratic speed-up in query complexity without altering the outer RL pipeline, i.e., $\mathcal{O}(\sqrt{|\mathcal{U}|})$. Validated through classical simulations to real quantum hardware execution, QRIM learns more robust policies with quadratically reduced queries than classical RL.