Quantum Bayesian Optimization
This work addresses the limitation of existing quantum bandits to linear or multi-armed settings, enabling more sophisticated real-world optimization problems, though it is incremental in extending quantum methods to Bayesian optimization.
The paper tackles the problem of optimizing black-box functions with non-linear rewards by introducing the Q-GP-UCB algorithm, achieving a regret upper bound of O(polylog T) compared to the classical lower bound of Omega(sqrt(T)). Simulations and a real quantum computer experiment verify the theoretical quantum speedup.
Kernelized bandits, also known as Bayesian optimization (BO), has been a prevalent method for optimizing complicated black-box reward functions. Various BO algorithms have been theoretically shown to enjoy upper bounds on their cumulative regret which are sub-linear in the number T of iterations, and a regret lower bound of Omega(sqrt(T)) has been derived which represents the unavoidable regrets for any classical BO algorithm. Recent works on quantum bandits have shown that with the aid of quantum computing, it is possible to achieve tighter regret upper bounds better than their corresponding classical lower bounds. However, these works are restricted to either multi-armed or linear bandits, and are hence not able to solve sophisticated real-world problems with non-linear reward functions. To this end, we introduce the quantum-Gaussian process-upper confidence bound (Q-GP-UCB) algorithm. To the best of our knowledge, our Q-GP-UCB is the first BO algorithm able to achieve a regret upper bound of O(polylog T), which is significantly smaller than its regret lower bound of Omega(sqrt(T)) in the classical setting. Moreover, thanks to our novel analysis of the confidence ellipsoid, our Q-GP-UCB with the linear kernel achieves a smaller regret than the quantum linear UCB algorithm from the previous work. We use simulations, as well as an experiment using a real quantum computer, to verify that the theoretical quantum speedup achieved by our Q-GP-UCB is also potentially relevant in practice.