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

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.

Mitigating Bias in Low-SNR Financial Reinforcement Learning via Quantum Representations

The financial market is a typical low signal-to-noise ratio (SNR) setting, which often destabilizes off-policy maximum-entropy methods like Soft Actor-Critic (SAC). Specifically, noisy state representations may produce unreliable Q-value estimates, and bootstrapping amplifies these errors, forming a failure mode we call the "Financial Entropy Trap". In this paper, we propose FPQC-SAC, an efficient and plug-and-play SAC variant that places a compact and bounded Parameterized Quantum Circuit (PQC) before the actor and critic networks to constrain feature propagation at the representation level, rather than filtering raw inputs or regularizing Q-values after bootstrapping. Notably, FPQC-SAC reduces the impact of extreme market fluctuations on Bellman target estimation, while trainable quantum entanglement preserves flexible cross-asset interactions. Empirical evaluations on real-world portfolio management tasks demonstrate that FPQC-SAC substantially enhances out-of-sample stability and cumulative returns by achieving a 66.89% relative gain in cumulative return over standard unconstrained SAC and outperforms the best continuous-control deep reinforcement learning baseline by approximately 27%. Open-source code is available at https://github.com/ZeyuLIU-UST/FPQC-SAC-main.

QnRL: Quantum-Native Reinforcement Learning

Quantum reinforcement learning (QRL) is a promising approach to learn effective decision strategies across several applications with stochastic environments. Instead of directly modeling the random variables that govern these environments, existing QRL architectures indirectly approximate environment behavior by estimating expected outcomes, which limits their expressive power and adaptive potential. Overcoming such challenges requires a novel QRL approach that exploits the distributional nature of quantum computers to directly model environment random variables as quantum state distributions. Hence, in this paper, a novel framework dubbed quantum-native reinforcement learning (QnRL) is proposed. QnRL is a distributional RL framework that learns conditional distributions naturally in Hilbert space via superimposed and entangled quantum states. Thus, QnRL can directly model the behavior of stochastic learning environments via the natural properties of quantum systems. QnRL accomplishes this via a novel, proposed quantum amplitude kickback (QuAK) algorithm that enables comparing the $n$-th power of the $m$-th moment of multiple superimposed distributions. It is theoretically proven that a conditional action policy distribution is distilled from the moments of a quantum generative model entirely within Hilbert space via QuAK, and optimized via QnRL. This complex distribution composition is also shown to provide extra dimensions for expressing environment correlations that are unknown to purely classical and classically-sampled quantum distributional models. Experimental results across diverse environments show that QnRL achieves up to $82.9\%$ higher evaluation scores, with up to $94.3\%$ fewer parameters on average, more accurately estimates the expected return for unseen observations, and better adapts to varying stochastic conditions compared to the baseline.