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

We give a quantum algorithm for computing an $\epsilon$-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 $\epsilon$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot \epsilon^{-3})$ obtained by [van Apeldoorn, Gilyen '19] and the classical $\widetilde{O}((m + n) \cdot \epsilon^{-2})$ runtime due to [Grigoradis, Khachiyan '95] whenever $\epsilon = \Omega((m +n)^{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.