A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games

TL;DR

The paper addresses the challenge of efficiently computing ε-Nash equilibria in quantum zero-sum games, where prior work (MMWU) incurred a convergence rate of O(d/ε^2). By developing an Optimistic Matrix Mirror Prox (OMMP) framework and instantiating it with a von Neumann entropy regularizer to yield the Optimistic Matrix Multiplicative Weights Update (OMMWU), the authors achieve a linear-in-ε convergence rate of O(d/ε) for average iterates. The methodology leverages a gradient-based view of quantum game feedback, establishing monotonicity and Lipschitz continuity of the game’s gradient operator, and combines extra-gradient ideas with iterate-averaging to obtain the speedup, while maintaining a single gradient evaluation per iteration. The results are supported by convergence analysis and experiments showing improved average-case performance and numerical stability, suggesting practical advantages for quantum game-theoretic problems in quantum information, QGANs, and related areas. Overall, the work provides a principled, scalable route to faster ε-Nash computation in the spectraplex setting with clear theoretical guarantees and empirical validation.

Abstract

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 $\mathcal{O}(d/ε^2)$ iterations to $ε$-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 $\mathcal{O}(d/ε)$ iterations to $ε$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $ε$-Nash equilibria in quantum zero-sum games.

Figures (10)

• Figure 1: Design of Classical Zero-Sum Game Algorithms. This diagram provides the update rules for and relationships between different learning algorithms for classical zero-sum games. The left-hand side presents the Bregman generalized methods, parameterized by distance-generating/regularization function $h$. The right-hand side presents instantiations based on the $\ell_2$ norm and entropy function, inducing an orthogonal projection $\textrm{Orth}\Pi$ and logit map $\Lambda$, respectively. Note that moving from $h=\ell_2$ to $h=$entropy results in a logarithmic improvement in the total number of rounds $T$, by reducing the dependence on the simplex dimension $d$. Furthermore, moving from the mirror map $\textnormal{Mir}\Pi$ of the Dual Averaging method (a "lazy" variant of the classic Mirror Descent Ascent algorithm) to the proximal maps $\textnormal{Prox}\Pi$ of the Mirror Prox method results in a quadratic improvement in convergence, achieving the desired $O(1/\epsilon)$ rate. Furthermore, by reusing the "past gradient," the Single-Call Mirror Prox method retains the rate of Mirror Prox, but reduces the total number of gradient calls per iteration from two to one.
• Figure 2: Design of Quantum Zero-Sum Game Algorithms. This diagram provides the update rules for and relationships between the learning algorithms for quantum zero-sum games as proposed in this work. The left-hand side presents the Bregman generalized methods, parameterized by distance-generating/regularization function $h$. The right-hand side presents instantiations based on the von Neumann entropy function and Frobenius ($\ell_2$) norm, inducing an orthogonal projection $\textrm{Orth}\Pi$ and logit map $\Lambda$, respectively. Note that moving from $\ell_2$ to an entropy function as the regularizer results in a logarithmic improvement in the total number of rounds $T$, by reducing the dependence on the spectraplex dimension $D=4^d$. Furthermore, moving from the mirror map $\textnormal{Mir}\Pi$ of the Matrix Dual Averaging method (Jain and Watrous' MMWU proposal jain2009parallel) to the proximal maps $\textnormal{Prox}\Pi$ of the Matrix Mirror Prox method results in a quadratic improvement in convergence, achieving the desired $O(1/\epsilon)$ rate. Furthermore, by reusing the "past gradient," the Single-Call Matrix Mirror Prox method retains the rate of Matrix Mirror Prox, but reduces the total number of gradient calls per iteration from two to one.
• Figure 3: An illustration of a single round of the quantum zero-sum game.
• Figure 4: An illustration of the game feedback in a single round of the quantum zero-sum game (gradient-based view).
• Figure 5: Intuitive illustrations of the strong and weak solutions $\Psi^*$ of the variational inequalities given in \ref{['eqn:vi']} and \ref{['eqn:vi_weak']}, respectively. For strong solutions, the inner product between $\Psi-\Psi^*$ and $\mathcal{F}(\Psi^*)$ must be non-positive for all $\Psi$ in the joint spectraplex $\mathcal{C}$. Meanwhile, for weak solutions, the inner product between $\Psi^*-\Psi$ and $\mathcal{F}(\Psi)$ must be non-negative for all $\Psi\in\mathcal{C}$.

Theorems & Definitions (45)

• Corollary 1: OMMWU Iteration Complexity
• Theorem 1: Main Result
• Conjecture 1
• Lemma 1
• Lemma 2
• Definition 1: Convex Function
• Definition 2: $\mu$-Strongly Convex Function
• Definition 3: $\beta$-Smooth Function
• Definition 4: Dual Norm
• Proposition 1: $\beta$-Lipschitz