Recent advances in quantum computing and in particular, the introduction of quantum GANs, have led to increased interest in quantum zero-sum game theory, extending the scope of learning algorithms for classical games into the quantum realm. In this paper, we focus on learning in quantum zero-sum games under Matrix Multiplicative Weights Update (a generalization of the multiplicative weights update method) and its continuous analogue, Quantum Replicator Dynamics. When each player selects their state according to quantum replicator dynamics, we show that the system exhibits conservation laws in a quantum-information theoretic sense. Moreover, we show that the system exhibits Poincare recurrence, meaning that almost all orbits return arbitrarily close to their initial conditions infinitely often. Our analysis generalizes previous results in the case of classical games.

翻译：近期量子计算领域的进展，特别是量子生成对抗网络的引入，激发了人们对量子零和博弈理论的兴趣，并将经典博弈学习算法的适用范围拓展至量子领域。本文聚焦于矩阵乘性权重更新方法（乘性权重更新法的泛化形式）及其连续类比——量子复制动力学框架下的量子零和博弈学习问题。当每位参与者依据量子复制动力学选择其状态时，我们证明该系统在量子信息论意义上展现出守恒律。此外，我们证明该系统具有庞加莱回复性，即几乎所有轨道都会无限频繁地任意接近其初始状态。本文的分析将经典情形下的已有结论进行了推广。