🤖 AI Summary
This work studies quantum common-interest games (CIGs), where players’ strategies are density matrices and all utility functions are identical. Methodologically, it establishes a rigorous equivalence between Nash equilibria of quantum CIGs and Karush–Kuhn–Tucker (KKT) points of the best separable state (BSS) problem, recasting game-theoretic learning dynamics as decentralized optimization for BSS. It introduces the first formal model of quantum CIGs; uncovers a deep connection between the nonconvex BSS optimization and quantum game equilibria; and designs continuous and discrete learning dynamics grounded in noncommutative geometry—including quantum replicator dynamics, best-response updates, and linear multiplicative weight updates—proving their global convergence. Experiments demonstrate that the proposed methods achieve high efficiency, robustness, and scalability in solving the BSS problem, thereby establishing a novel paradigm for quantum multi-agent learning.
📝 Abstract
Learning in games has emerged as a powerful tool for machine learning with numerous applications. Quantum games model interactions between strategic players who have access to quantum resources, and several recent works have studied {learning in} the competitive regime of quantum zero-sum games. Going beyond this setting, we introduce quantum common-interest games (CIGs) where players have density matrices as strategies and their interests are perfectly aligned. We bridge the gap between optimization and game theory by establishing the equivalence between KKT (first-order stationary) points of an instance of the Best Separable State (BSS) problem and the Nash equilibria of its corresponding quantum CIG. This allows learning dynamics for the quantum CIG to be seen as decentralized algorithms for the BSS problem. Taking the perspective of learning in games, we then introduce non-commutative extensions of the continuous-time replicator dynamics and the discrete-time best response dynamics/linear multiplicative weights update for learning in quantum CIGs. We prove analogues of classical convergence results of the dynamics and explore differences which arise in the quantum setting. Finally, we corroborate our theoretical findings through extensive experiments.
