Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling
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We give a quantum algorithm for computing an ϵ-approximate Nash equilibrium of a zero-sum game in a m × n payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time O(√(m + n)·ϵ^-2.5 + ϵ^-3) and outputs a classical representation of the ϵ-approximate Nash equilibrium. This improves upon the best prior quantum runtime of O(√(m + n)·ϵ^-3) obtained by [vAG19] and the classic O((m + n) ·ϵ^-2) runtime due to [GK95] whenever ϵ = Ω((m +n)^-1). We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.
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