upper bound. However, they usually assume that the objective function sits within the reproducing kernel Hilbert space and their algorithms suffer from the curse of dimensionality. In this paper, we propose the new Q-NLB-UCB algorithm which enjoys an \emph{input dimension-free}

O(\mathrm{poly}\log T)

upper bound, making it applicable for high-dimensional tasks. At the heart of our algorithm design are quantum Monte Carlo mean estimator, parametric function approximation technique, and a new quantum non-linear regression oracle, which can be of independent interests in more quantum machine learning problems. Our algorithm is also validated for its efficiency compared with other quantum algorithms on both high-dimensional synthetic and real-world tasks.

DOI: 10.48550/arxiv.2503.03023

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Maoli Liu, Zhuohua Li, John C. S. Lui

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Yuqi Huang, Vincent Y. F Tan, Sharu Theresa Jose

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Ω(\sqrt{T})

regret lower bound and achieve the new

O(\mathrm{poly}\log T)

O(\mathrm{poly}\log T)

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 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(K\log(1+T/K))

for QMAB and the first lower bound of

Ω(d\log(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

\operatorname{poly}(d)

, its regret is nearly linear in

d

and matches our lower bound up to polylogarithmic factors.

\tilde O(T^{(d_z+1)/(d_z+2)})

over time horizon

T

. Motivated by recent advancements in quantum computing and the demonstrated success of quantum Monte Carlo in simpler bandit settings, we introduce the first quantum Lipschitz bandit algorithms to address the challenges of continuous action spaces and non-linear reward functions. Specifically, we first leverage the elimination-based framework to propose an efficient quantum Lipschitz bandit algorithm named Q-LAE. Next, we present novel modifications to the classical Zooming algorithm, which results in a simple quantum Lipschitz bandit method, Q-Zooming. Both algorithms exploit the computational power of quantum methods to achieve an improved regret bound of

\tilde O(T^{d_z/(d_z+1)})

. Comprehensive experiments further validate our improved theoretical findings, demonstrating superior empirical performance compared to existing Lipschitz bandit methods.

T

and

K

actions, our theoretical analysis reveals a significantly improved parameter scaling of

Ω((TK)^3)

for QNTK-UCB, a substantial reduction compared to

Ω((TK)^8)

required by classical NeuralUCB algorithms for similar regret guarantees. Empirical evaluations on non-linear synthetic benchmarks and quantum-native variational quantum eigensolver tasks demonstrate QNTK-UCB's superior sample efficiency in low-data regimes. This work highlights how the inherent properties of QNTK provide implicit regularization and a sharper spectral decay, paving the way for achieving ``quantum advantage'' in online learning.