We propose the first online quantum algorithm for zero-sum games with $\tilde O(1)$ regret under the game setting. Moreover, our quantum algorithm computes an $\varepsilon$-approximate Nash equilibrium of an $m \times n$ matrix zero-sum game in quantum time $\tilde O(\sqrt{m+n}/\varepsilon^{2.5})$, yielding a quadratic improvement over classical algorithms in terms of $m, n$. Our algorithm uses standard quantum inputs and generates classical outputs with succinct descriptions, facilitating end-to-end applications. As an application, we obtain a fast quantum linear programming solver. Technically, our online quantum algorithm "quantizes" classical algorithms based on the optimistic multiplicative weight update method. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem, which may be of independent interest.

翻译：我们提出了首个在线量子算法，用于在博弈设定下实现$\tilde O(1)$遗憾的零和博弈。此外，我们的量子算法能在量子时间$\tilde O(\sqrt{m+n}/\varepsilon^{2.5})$内计算一个$m \times n$矩阵零和博弈的$\varepsilon$-近似纳什均衡，相比经典算法在$m, n$维度上实现了二次加速。该算法使用标准量子输入，并生成具有简洁描述的经典输出，便于端到端应用。作为应用，我们获得了一个快速量子线性规划求解器。技术上，我们的在线量子算法将基于乐观乘法权重更新法的经典算法"量子化"。算法核心是一个用于吉布斯采样问题的快速量子多重采样过程，该过程可能具有独立的研究价值。