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

Abstract: 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 $$\widetilde{O}(\sqrt{m + n}\cdot \epsilon^{-2.5} + \epsilon<sup>{-3})$$ and outputs a classical representation of the $\epsilon$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $$\widetilde{O}(\sqrt{m + n} \cdot \epsilon<sup>{-3})$$ obtained by [vAG19] and the classic $\widetilde{O}((m + n) \cdot \epsilon<sup>{-2})$ runtime due to [GK95] whenever $\epsilon = \Omega((m +n)<sup>{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.