Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $\mathcal{O}(d/\epsilon^2)$ iterations to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $\mathcal{O}(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games.

翻译：近期在非局部对策、量子交互证明及量子生成对抗网络等领域的研究进展，重新激发了学界对量子对策论特别是量子零和对策的兴趣。经典对策论的核心在于纳什均衡的高效算法计算，这类均衡代表双方的最优策略。2008年，Jain与Watrous提出首个计算量子零和对策均衡的经典算法，该算法采用矩阵乘法权重更新法（MMWU），在4^d维谱凸体上实现ε-纳什均衡的收敛速率为O(d/ε²)次迭代。本研究提出一类通过额外梯度机制推广MMWU的量子优化算法层级。特别地，在该层级中我们引入乐观矩阵乘法权重更新法（OMMWU），并证明其平均迭代复杂度为O(d/ε)次迭代可达ε-纳什均衡。相较Jain与Watrous原始算法，该二次加速效应为量子零和对策中ε-纳什均衡的计算设立了新基准。