Quantum zero-sum games provide a framework for non-local games, quantum interactive proofs, and quantum machine learning, where players optimize a bilinear payoff over quantum states. In contrast to classical bilinear games over polyhedral domains, for which gradient methods achieve linear last-iterate convergence, comparable guarantees over spectraplexes have remained open. Recent work achieved only an $O(1/\varepsilon)$ average-iterate rate and suggested that semidefinite geometry may preclude classical-style linear rates. We refute this obstruction. We prove that quantum zero-sum games admit algorithms with $O(\log(1/\varepsilon))$ last-iterate convergence to Nash equilibrium. In particular, matrix variants of Nesterov's iterative smoothing and Optimistic Gradient Descent--Ascent match the asymptotic rate of the classical polyhedral case. The key technical ingredient is a new error-bound theory for semidefinite games, establishing metric subregularity of the relevant monotone operator over spectrahedra despite the absence of polyhedral structure. We also give a geometric characterization of Nash equilibria via slack operators, classifying strategic directions as essential, neutral, or non-essential. Under strict complementarity or nondegeneracy, this reduces to a sharp classical-style dichotomy. Finally, we revisit Optimistic Matrix Multiplicative Weights Update. By extending the Quantal Response Equilibrium framework to spectraplex games, we prove an $\widetilde O(1/\varepsilon)$ last-iterate guarantee, while showing that any $O(\log(1/\varepsilon))$ speedup for this method must depend on a natural, dimension-dependent condition number. Experiments support the theoretical picture, with Optimistic Gradient Descent--Ascent outperforming Optimistic Matrix Multiplicative Weights Update in the regimes studied.

翻译：零和量子博弈为非局域博弈、量子交互证明及量子机器学习提供了框架，其中博弈双方通过量子态优化双线性收益。与多面体域上经典双线性博弈（梯度方法可实现线性末步收敛）不同，谱多面体上的可比拟保证一直是开放问题。近期工作仅达到$O(1/\varepsilon)$平均步长收敛速率，并暗示半定几何结构可能阻碍经典线性速率的实现。我们反驳了这一障碍。我们证明零和量子博弈存在算法，其纳什均衡的末步收敛速率为$O(\log(1/\varepsilon))$。具体而言，Nesterov迭代平滑及乐观梯度下降-上升的矩阵变体与经典多面体情形下的渐近速率相匹配。关键技术突破是建立半定博弈的新误差界理论，证明在谱面体上相关单调算子满足度量次正则性，尽管缺乏多面体结构。我们还通过松弛算子给出纳什均衡的几何刻画，将战略方向分为本质、中性或非本质三类。在严格互补性或非退化条件下，这简化为经典的尖锐二分。最后，我们重新审视乐观矩阵乘性权重更新方法。通过将量子响应均衡框架推广至谱多面体博弈，我们证明该方法具有$\widetilde O(1/\varepsilon)$末步保证，同时表明其任何$O(\log(1/\varepsilon))$加速必然依赖于一个自然的、与维度相关的条件数。实验支持理论结论，在所研究场景中乐观梯度下降-上升的表现优于乐观矩阵乘性权重更新。