积酉流形上的多人量子博弈黎曼优化
Riemannian Optimization for Multi-Player Quantum Games on Product Unitary Manifolds
- Ruhr University Bochum（波鸿鲁尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出酉策略矩阵指数算法（USMEA），一种用于扩展EWL多人量子博弈混合策略的黎曼优化方法，实现联合学习酉动作与混合概率，并验证收敛性。
AI中文摘要：
量子博弈论是经典博弈论的扩展，它将量子原理应用于博弈论中。Eisert-Wilkens-Lewenstein（EWL）量子博弈是将两人经典囚徒困境转化为量子囚徒困境的早期例子。在EWL博弈中，玩家选择由酉矩阵表示的纯量子策略。这种扩展可以通过实现具有更高收益的合作均衡来解决经典困境。在本文中，我们首先讨论用于具有混合策略的多人量子博弈的扩展EWL（EEWL）。在EEWL中，每个玩家控制一组酉算子作为量子动作，并在这些动作上使用经典混合策略。收益被定义为作用于共享量子态上的厄米奖励算子的期望值，该量子态根据EEWL协议生成和测量。然后，我们提出酉策略矩阵指数算法（USMEA），一种用于EEWL混合策略设置的几何感知顺序算法，其中每个玩家联合学习一组可训练的局部酉动作及相关的经典混合概率。因此，它充当多智能体量子决策系统的学习与控制层。我们在标准光滑性和步长条件下分析USMEA的收敛性质，并通过数值实验验证该理论。这些结果表明经典优化方法如何系统地整合到工程化量子战略互动的设计与分析中。
英文摘要：
Quantum game theory is an extension of classical game theory that uses quantum principles in game theory. The Eisert-Wilkens-Lewenstein (EWL) quantum game is an early example of the two-player classical Prisoner's Dilemma transformed into a quantum Prisoner's Dilemma. In the EWL game, the players choose pure quantum strategies represented by unitary matrices. This extension can resolve the classical dilemma by enabling cooperative equilibrium with higher payoff. In this paper, we first discuss the Extended EWL (EEWL) for multiplayer quantum games with mixed strategies. In EEWL, each player controls a set of unitary operators as quantum actions and uses a classical mixed strategy over these actions. The payoffs are defined as expectation values of Hermitian reward operators acting on a shared quantum state, which is generated and measured according to the EEWL protocol. We then propose the Unitary Strategy Matrix Exponential Algorithm (USMEA), a geometry-aware sequential algorithm for the EEWL mixed-strategy setting, in which each player jointly learns a trainable set of local unitary actions and the associated classical mixing probabilities. Thereby it acts as a learning-and-control layer for multi-agent quantum decision systems. We analyze the convergence properties of USMEA under standard smoothness and step-size conditions and validate the theory with numerical experiments. These results show how classical optimization methods can be systematically integrated into the design and analysis of engineered quantum strategic interactions.
