Quantum Bandits Research Establishes New Lower Bounds and Algorithms

The study delves into the theoretical foundations of quantum multi-armed bandits (QMAB) and quantum linear bandits (QLB), building upon a model where a learner interacts with each arm or action via a quantum reward oracle. Previous work had established algorithms with regret bounds of O(K log T) for QMAB and O(d^2 polylog T) for QLB, leaving open questions regarding the tightness of these bounds and potential improvements. This paper makes significant progress by proving the first minimax lower bounds for these quantum bandit problems. Specifically, it establishes bounds of Ω(K log(T/K)) for QMAB and Ω(d log(T/d)) for finite-action QLB. These findings directly address and resolve the question of whether regret independent of the time horizon T is achievable in these quantum settings. The core of the argument relies on a high-confidence single-arm quantum testing lower bound, derived using the polynomial method and a Remez-type inequality. Complementing these lower bounds, the researchers also present a new design-based elimination algorithm for finite-action QLB. When the action set size is polynomial in d, this algorithm achieves regret linear in d, which is a notable improvement over the prior d^2 dependence and matches the newly established lower bound up to polylogarithmic factors. This algorithm effectively combines a low-bias, low-variance quantum mean estimator with a small-support G-optimal design, optimizing query allocation.