🤖 AI Summary
This work addresses the low computational efficiency of Nash equilibrium computation in quantum zero-sum games. We propose a class of Matrix Multiplicative Weights Update (MMWU) algorithms based on the extragradient mechanism. Notably, we introduce Optimistic MMWU (OMMWU) to quantum game equilibrium computation for the first time, achieving an $varepsilon$-Nash equilibrium on the $4^d$-dimensional spectral simplex in $O(d/varepsilon)$ iterations—yielding a quadratic speedup over the classical MMWU’s $O(d/varepsilon^2)$. Our method integrates quantum optimization principles, extragradient correction, and optimistic prediction, substantially accelerating convergence. Both theoretical analysis and empirical experiments establish a new benchmark for equilibrium computation in quantum zero-sum games, providing a foundational algorithmic framework for the intersection of quantum game theory and quantum optimization.
📝 Abstract
Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $mathcal{O}(d/epsilon^2)$ iterations to $epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $mathcal{O}(d/epsilon)$ iterations to $epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $epsilon$-Nash equilibria in quantum zero-sum games.
