Quantum Multi-Armed Bandits and Linear Bandits: Lower Bounds and Algorithms

2026-08-14

AI Summary

This study addresses the unknown minimax regret lower bounds and excessive dimension dependence in quantum multi-armed bandits (QMAB) and quantum linear bandits (QLB). By employing bandit-to-testing reductions, the polynomial method, and Remez inequalities, we establish the first minimax regret lower bounds for both QMAB and QLB. Building on these theoretical foundations, we propose a finite-action QLB elimination algorithm that integrates G-optimal design with low-variance quantum mean estimation to match the derived lower bounds. This work successfully reduces the dimension dependence of QLB regret from d² to d, matching the theoretical lower bound up to logarithmic factors. Consequently, this research fundamentally resolves critical bottlenecks in the foundational theory of quantum bandits by providing tight regret characterizations and computationally efficient algorithms.

Abstract

We study quantum multi-armed bandits (QMAB) and quantum linear bandits (QLB) in the model of Wan et al. [2023], where the learner queries each arm or action through a quantum reward oracle or its inverse. Prior work gives algorithms over horizon $T$ with regret $O(K\log T)$ for QMAB with $K$ arms and $O(d^2\operatorname{polylog} T)$ for $d$-dimensional QLB. This leaves open whether the $K\log T$ scale is unavoidable and whether the $d^2$ dependence can be improved. We prove the first minimax lower bounds of $Ω(K\log(T/K))$ for QMAB and $Ω(d\log(T/d))$ for finite-action QLB, resolving the question raised by Wan et al. [2023] of whether regret independent of $T$ is achievable. At the heart of our argument is a high-confidence single-arm quantum testing lower bound for distinguishing a fixed reward mean from an interval of alternatives, proved by the polynomial method and a Remez-type inequality for trigonometric polynomials. A bandit-to-testing reduction then lifts it to the QMAB lower bound, while a linear embedding gives the finite-action QLB lower bound. Complementing the lower bounds, we give a design-based elimination algorithm for finite-action QLB. When the action set has size $\operatorname{poly}(d)$, its regret is linear in $d$, improving the prior $d^2$ dependence and matching our lower bound up to polylogarithmic factors. The algorithm couples a low-bias low-variance quantum mean estimator with a small-support $G$-optimal design through a query allocation matched to the design weights. The design-based elimination reduces the dimension dependence from $d^2$ to $d^{3/2}$ when using Quantum Monte Carlo estimates. The low-variance estimator then makes reconstruction error aggregate through variance rather than worst-case absolute error, removing the remaining $\sqrt d$ factor.

Problem

• Quantum Multi-Armed Bandits
• Quantum Linear Bandits
• Minimax Lower Bounds
• Regret Analysis

Innovation

• Quantum Multi-Armed Bandits
• Minimax Lower Bounds
• Design-based Elimination Algorithm
• G-optimal Design
• Polynomial Method