Best arm identification (BAI) is a key problem in stochastic multi-armed bandits, where K arms each has an associated reward distribution, and the objective is to minimize the number of queries needed to identify the best arm with high confidence. In this paper, we explore BAI using quantum oracles. For the case where each query probes only one arm (m=1), we devise a quantum algorithm with a query complexity upper bound of O((K/Delta)log(1/delta)), where delta is the confidence parameter and Delta is the reward gap between best and second best arms. This improves on the classical bound by a factor of 1/Delta. For the general case where a single query can probe m arms (1
We investigate the top-
marm identification problem in a quantum multi-armed bandit (MAB) setting. Given
Karms whose rewards follow Bernoulli distributions with unknown means, the objective is to return the
marms with the largest means with confidence at least
1−δ. Many practical tasks-e.g., selecting the best
mrouting paths in a quantum network or the best
mquantum circuits for a given computation-fit naturally into this framework. We introduce Q-GapE, an elimination-based algorithm that queries each arm through a weak quantum oracle.
Q- GapE attains the query complexity guarantee
O~(∑k=1K(1/Δk(m))log(1/δ)), where
Δk(m)denotes the gap between arm
k's mean and the
m-th largest mean. Compared with the best classical-oracle bound, this result offers a quadratic quantum speed-up. Extensive simulations confirm that
Q−GapEconsistently outperforms leading classical MAB algorithms in both query complexity and running time. These findings highlight the potential of quantum bandit algorithms to deliver substantial efficiency gains over their classical counterparts.
We study quantum multi-armed bandits (QMAB) and quantum linear bandits (QLB), 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(KlogT) for QMAB with
K arms and
O(d2polylogT) for
d-dimensional QLB. This leaves open the optimal dependence on
K and
T and whether the dependence on
d can be further improved. In this work, we prove the first tight minimax regret bound of
Θ(Klog(1+T/K)) for QMAB and the first lower bound of
Ω(dlog(1+T/d)) for finite-action QLB, ruling out regret independent of
T. Our lower bounds rely on a high-confidence single-arm quantum testing lower bound for distinguishing a fixed reward mean from an interval of alternatives. A bandit-to-testing reduction then lifts it to the QMAB lower bound, while a linear embedding gives the finite-action QLB lower bound. The matching QMAB upper bound is obtained using a tail bound for the Quantum Monte Carlo (QMC) estimator. For finite-action QLB, we propose a phased elimination algorithm that combines a low-bias low-variance quantum mean estimator with a small-support
G-optimal design through a query allocation matched to the design weights. When the action set has size
poly(d), its regret is nearly linear in
d and matches our lower bound up to polylogarithmic factors.
Identifying the best arm of a multi-armed bandit is a central problem in bandit optimization. We study a quantum computational version of this problem with coherent oracle access to states encoding the reward probabilities of each arm as quantum amplitudes. Specifically, we provide an algorithm to find the best arm with fixed confidence based on variable-time amplitude amplification and estimation. This algorithm gives a quadratic speedup compared to the best possible classical result in terms of query complexity. We also prove a matching quantum lower bound (up to poly-logarithmic factors).
In this paper, we study multi-armed bandits (MAB) and stochastic linear bandits (SLB) with heavy-tailed rewards and quantum reward oracle. Unlike the previous work on quantum bandits that assumes bounded/sub-Gaussian distributions for rewards, here we investigate the quantum bandits problem under a weaker assumption that the distributions of rewards only have bounded
(1+v)-th moment for some
v∈(0,1]. In order to achieve regret improvements for heavy-tailed bandits, we first propose a new quantum mean estimator for heavy-tailed distributions, which is based on the Quantum Monte Carlo Mean Estimator and achieves a quadratic improvement of estimation error compared to the classical one. Based on our quantum mean estimator, we focus on quantum heavy-tailed MAB and SLB and propose quantum algorithms based on the Upper Confidence Bound (UCB) framework for both problems with $\Tilde{O}(T^{\frac{1-v}{1+v}})$ regrets, polynomially improving the dependence in terms of
T as compared to classical (near) optimal regrets of $\Tilde{O}(T^{\frac{1}{1+v}})$, where
T is the number of rounds. Finally, experiments also support our theoretical results and show the effectiveness of our proposed methods.
Multi-armed bandits are one of the theoretical pillars of reinforcement learning. Recently, the investigation of quantum algorithms for multi-armed bandit problems was started, and it was found that a quadratic speed-up (in query complexity) is possible when the arms and the randomness of the rewards of the arms can be queried in superposition. Here we introduce further bandit models where we only have limited access to the randomness of the rewards, but we can still query the arms in superposition. We show that then the query complexity is the same as for classical algorithms. This generalizes the prior result that no speed-up is possible for unstructured search when the oracle has positive failure probability.
Quantum reinforcement learning has emerged as a framework combining quantum computation with sequential decision-making, and applications to the multi-armed bandit (MAB) problem have been reported. The graph bandit problem extends the MAB setting by introducing spatial constraints, where the accessibility of arms is restricted by graph connectivity, yet quantum approaches to this setting remain limited. In this paper, we formulate best-arm identification in graph bandits and propose a quantum algorithmic framework, termed quantum spatial best-arm identification, which is applicable to general graph structures. This framework uses quantum walks to encode superpositions over graph-constrained actions, thereby extending amplitude amplification and generalizing the quantum BAI algorithm via Szegedy’s walk framework. We focus our theoretical analysis on complete and bipartite graphs, deriving the maximal success probability of identifying the best arm and the time step at which it is achieved. Our results clarify how quantum walk-based search can be adapted to structurally constrained decision problems and provide a foundation for quantum best-arm identification in graph-structured environments.