  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 [vAG19] and the classic $\widetilde{O}((m + n) \cdot \epsilon^{-2})$ runtime
due to [GK95] 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.
