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We give a quantum algorithm for computing an $\epsilon$-approximate Nashequilibrium of a zero-sum game in a $m \times n$ payoff matrix with boundedentries. Given a standard quantum oracle for accessing the payoff matrix ouralgorithm 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 priorquantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot \epsilon^{-3})$ obtainedby [vAG19] and the classic $\widetilde{O}((m + n) \cdot \epsilon^{-2})$ runtimedue to [GK95] whenever $\epsilon = \Omega((m +n)^{-1})$. We obtain this resultby designing new quantum data structures for efficiently sampling from aslowly-changing Gibbs distribution.

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