  Recent developments in domains such as non-local games, quantum interactive
proofs, and quantum generative adversarial networks have renewed interest in
quantum game theory and, specifically, quantum zero-sum games. Central to
classical game theory is the efficient algorithmic computation of Nash
equilibria, which represent optimal strategies for both players. In 2008, Jain
and Watrous proposed the first classical algorithm for computing equilibria in
quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU)
method to achieve a convergence rate of $\mathcal{O}(d/\epsilon^2)$ iterations
to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this
work, we propose a hierarchy of quantum optimization algorithms that generalize
MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy,
we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU)
algorithm and establish its average-iterate convergence complexity as
$\mathcal{O}(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This
quadratic speed-up relative to Jain and Watrous' original algorithm sets a new
benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games.
