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arXiv:2301.03763

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

Adam Bouland, Yosheb Getachew, Yujia Jin, Aaron Sidford, Kevin Tian

Published Jan 10, 2023• 3 citations
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Abstract

We give a quantum algorithm for computing an $ε$-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 $\widetilde{O}(\sqrt{m + n}\cdot ε^{-2.5} + ε^{-3})$ and outputs a classical representation of the $ε$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot ε^{-3})$ obtained by [vAG19] and the classic $\widetilde{O}((m + n) \cdot ε^{-2})$ 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.

DOI
10.48550/arxiv.2301.03763

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We give a quantum algorithm for computing an εεε-approximate Nash equilibrium of a zero-sum game in a m×nm \times nm×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)\widetilde{O}(\sqrt{m + n}\cdot ε^{-2.5} + ε^{-3})O(m+n​⋅ε−2.5+ε−3) and outputs a classical representation of the εεε-approximate Nash equilibrium. This improves upon the best prior quantum runtime of O~(m+n⋅ε−3)\widetilde{O}(\sqrt{m + n} \cdot ε^{-3})O(m+n​⋅ε−3) obtained by [vAG19] and the classic O~((m+n)⋅ε−2)\widetilde{O}((m + n) \cdot ε^{-2})O((m+n)⋅ε−2) runtime due to [GK95] whenever ε=Ω((m+n)−1)ε= Ω((m +n)^{-1})ε=Ω((m+n)−1). We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.
We derive sublinear-time quantum algorithms for computing the Nash equilibrium of two-player zero-sum games, based on efficient Gibbs sampling methods. We are able to achieve speed-ups for both dense and sparse payoff matrices at the cost of a mildly increased dependence on the additive error compared to classical algorithms. In particular we can find ε\varepsilonε-approximate Nash equilibrium strategies in complexity O~(n+m/ε3)\tilde{O}(\sqrt{n+m}/\varepsilon^3)O~(n+m​/ε3) and O~(s/ε3.5)\tilde{O}(\sqrt{s}/\varepsilon^{3.5})O~(s​/ε3.5) respectively, where n×mn\times mn×m is the size of the matrix describing the game and sss is its sparsity. Our algorithms use the LP formulation of the problem and apply techniques developed in recent works on quantum SDP-solvers. We also show how to reduce general LP-solving to zero-sum games, resulting in quantum LP-solvers that have complexities O~(n+mγ3)\tilde{O}(\sqrt{n+m}γ^3)O~(n+m​γ3) and O~(sγ3.5)\tilde{O}(\sqrt{s}γ^{3.5})O~(s​γ3.5) for the dense and sparse access models respectively, where γγγ is the relevant "scale-invariant" precision parameter
We propose the first online quantum algorithm for solving zero-sum games with O~(1)\widetilde O(1)O(1) regret under the game setting. Moreover, our quantum algorithm computes an ε\varepsilonε-approximate Nash equilibrium of an m×nm \times nm×n matrix zero-sum game in quantum time O~(m+n/ε2.5)\widetilde O(\sqrt{m+n}/\varepsilon^{2.5})O(m+n​/ε2.5). Our algorithm uses standard quantum inputs and generates classical outputs with succinct descriptions, facilitating end-to-end applications. Technically, our online quantum algorithm "quantizes" classical algorithms based on the optimistic multiplicative weight update method. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem, which may be of independent interest.
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 O(d/ϵ2) iterations to ϵ -Nash equilibria in the 4d -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 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.
Computing Nash equilibria of zero-sum games in classical and quantum settings is extensively studied. For general-sum games, computing Nash equilibria is PPAD-hard and the computing of a more general concept called correlated equilibria has been widely explored in game theory. In this paper, we initiate the study of quantum algorithms for computing ε\varepsilonε-approximate correlated equilibria (CE) and coarse correlated equilibria (CCE) in multi-player normal-form games. Our approach utilizes quantum improvements to the multi-scale Multiplicative Weight Update (MWU) method for CE calculations, achieving a query complexity of O~(mn)\tilde{O}(m\sqrt{n})O~(mn​) for fixed ε\varepsilonε. For CCE, we extend techniques from quantum algorithms for zero-sum games to multi-player settings, achieving query complexity O~(mn/ε2.5)\tilde{O}(m\sqrt{n}/\varepsilon^{2.5})O~(mn​/ε2.5). Both algorithms demonstrate a near-optimal scaling in the number of players mmm and actions nnn, as confirmed by our quantum query lower bounds.
The Nash equilibrium plays a crucial role in game theory. Most of results are based on classical resources. Our goal in this paper is to explore multipartite zero-sum game with quantum settings. We find that in two different settings there is no strategy for a tripartite classical game being fair. Interestingly, this is resolved by providing dynamic zero-sum quantum games using single quantum state. Moreover, the gains of some players may be changed dynamically in terms of the committed state. Both quantum games are robust against the preparation noise and measurement errors.
We propose a projected variational quantum extragradient (VQEG) framework for computing approximate Nash equilibria in two-player zero-sum matrix games. Mixed strategies are parameterized as Born distributions of parameterized quantum circuits (PQCs), transforming the classical bilinear saddle point problem into a smooth but generally minmax optimization in circuit-parameter space. The expected payoff is expressed as the expectation of a diagonal observable, enabling gradient evaluation via the parameter shift rule and compatibility with shot based quantum hardware. To support arbitrary game sizes, we introduce a dominated embedding that maps (m,n) games to qubit-compatible power-of-two dimensions while preserving equilibrium structure. We then develop a projected extragradient method using stochastic gradient estimates derived from finite measurement shots, and establish variance bounds scaling as O(1/S) with respect to the number of measurement shots S, along with convergence to approximate first-order stationarity under standard assumptions. Since stationarity does not guarantee equilibrium optimality, we evaluate performance using the game-space Nash gap. Numerical results demonstrate high-precision solutions on structured instances up to 32x32, while highlighting challenges in unstructured settings.