  Multi-arm bandit (MAB) and stochastic linear bandit (SLB) are important
models in reinforcement learning, and it is well-known that classical
algorithms for bandits with time horizon $T$ suffer $\Omega(\sqrt{T})$ regret.
In this paper, we study MAB and SLB with quantum reward oracles and propose
quantum algorithms for both models with $O(\mbox{poly}(\log T))$ regrets,
exponentially improving the dependence in terms of $T$. To the best of our
knowledge, this is the first provable quantum speedup for regrets of bandit
problems and in general exploitation in reinforcement learning. Compared to
previous literature on quantum exploration algorithms for MAB and reinforcement
learning, our quantum input model is simpler and only assumes quantum oracles
for each individual arm.
