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arXiv:1904.03180

Quantum algorithms for zero-sum games

Joran van Apeldoorn, András Gilyén

Published Apr 5, 2019• 18 citations
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Abstract

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 $\varepsilon$-approximate Nash equilibrium strategies in complexity $\tilde{O}(\sqrt{n+m}/\varepsilon^3)$ and $\tilde{O}(\sqrt{s}/\varepsilon^{3.5})$ respectively, where $n\times m$ is the size of the matrix describing the game and $s$ 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 $\tilde{O}(\sqrt{n+m}γ^3)$ and $\tilde{O}(\sqrt{s}γ^{3.5})$ for the dense and sparse access models respectively, where $γ$ is the relevant "scale-invariant" precision parameter

DOI
10.48550/arxiv.1904.03180

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•Jan 1, 2019•60 citations
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 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 give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved O~(n+m)\widetilde{O}(\sqrt{n}+\sqrt{m})O(n​+m​) dependence on matrix dimension nnn and constraint number mmm. Compared with the O~(mn)\widetilde{O}(mn)O(mn) runtime of existing classical methods, this suggests a quartic quantum speedup when m≈nm \approx nm≈n. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs. Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let sss denote the the input matrix sparsity and γ:=Rr/εγ:=Rr/\varepsilonγ:=Rr/ε capture dependence on the primal (R)(R)(R) and dual (r)(r)(r) radii as well as target accuracy (ε)(\varepsilon)(ε). When γ2≤min⁡{m,n/s}γ^2\leq\min\{m,n/s\}γ2≤min{m,n/s}, our solver runs in time O~(nsγ4.5+msγ2)\widetilde{O}\left(nsγ^{4.5}+msγ^2\right)O(nsγ4.5+msγ2). For γ=O(1)γ=O(1)γ=O(1), this is O~((n+m)s)\widetilde{O}\left((n+m)s\right)O((n+m)s) and sublinear in the O(mns)O(mns)O(mns) input size. Similar to the quantum algorithms, this matches known lower bounds with respect to mmm and nnn, up to logarithmic factors. This implies that, with respect to dimensions mmm and nnn, there is no super-quadratic quantum advantage for generic sparse SDP solving.
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.
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.