Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling

Abstract

We give a quantum algorithm for computing an $ϵ$-approximate Nash equilibrium of a zero-sum game in a $m\times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $\widetilde{O}(\sqrt{m + n}\cdot \epsilon^{-2.5} + \epsilon^{-3})$ and outputs a classical representation of the $ϵ$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\tilde{O}(\sqrt{m+n}\cdot {ϵ}^{-3})$ obtained by [vAG19] and the classic $\tilde{O}((m+n)\cdot {ϵ}^{-2})$ runtime due to [GK95] whenever $ϵ=\Omega ((m+n{)}^{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.