In this paper, we introduce a class of learning dynamics for general quantum games, that we call "follow the quantum regularized leader" (FTQL), in reference to the classical "follow the regularized leader" (FTRL) template for learning in finite games. We show that the induced quantum state dynamics decompose into (i) a classical, commutative component which governs the dynamics of the system's eigenvalues in a way analogous to the evolution of mixed strategies under FTRL; and (ii) a non-commutative component for the system's eigenvectors which has no classical counterpart. Despite the complications that this non-classical component entails, we find that the FTQL dynamics incur no more than constant regret in all quantum games. Moreover, adjusting classical notions of stability to account for the nonlinear geometry of the state space of quantum games, we show that only pure quantum equilibria can be stable and attracting under FTQL while, as a partial converse, pure equilibria that satisfy a certain "variational stability" condition are always attracting. Finally, we show that the FTQL dynamics are Poincar\'e recurrent in quantum min-max games, extending in this way a very recent result for the quantum replicator dynamics.

翻译：本文针对一般量子博弈引入了一类学习动力学，称之为“跟随量子正则化领导者”（FTQL），这一术语借鉴了有限博弈中经典“跟随正则化领导者”（FTRL）学习框架。我们证明，由此诱导的量子状态动力学可分解为：（i）经典交换分量，该分量以类似FTRL下混合策略演变的方式控制系统特征值的演化；（ii）非交换分量，负责系统特征向量的演化，该分量无经典对应物。尽管这一非经典分量带来诸多复杂性，但我们发现FTQL动力学在所有量子博弈中的遗憾值均不超过常数。此外，通过调整经典稳定性概念以适配量子博弈状态空间的非线性几何结构，我们证明在FTQL下只有纯量子均衡才可能是稳定且吸引的；作为部分逆命题，满足特定“变分稳定性”条件的纯均衡始终具有吸引性。最后，我们证明FTQL动力学在量子最小-最大博弈中具有庞加莱回归性，从而将量子复制动力学的最新结果进行了推广。