Quantum Bayesian Optimization

TL;DR

This paper introduces a quantum Bayesian optimization algorithm that achieves a regret upper bound of O(polylog T), significantly outperforming classical bounds, and demonstrates its effectiveness through simulations and real quantum computer experiments.

Contribution

The paper presents the first quantum Bayesian optimization algorithm with a regret upper bound of O(polylog T), surpassing classical lower bounds and applicable to complex non-linear problems.

Findings

Q-GP-UCB achieves regret of O(polylog T)

Simulation and quantum computer experiments confirm quantum speedup

Smaller regret than previous quantum linear UCB algorithms

Abstract

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.