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

• URL: http://arxiv.org/abs/2311.10859v1
• Date: Fri, 17 Nov 2023 20:38:38 GMT
• Title: A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games
• Authors: Francisca Vasconcelos, Emmanouil-Vasileios Vlatakis-Gkaragkounis, Panayotis Mertikopoulos, Georgios Piliouras, Michael I. Jordan
• Abstract summary: We introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $mathcalO(d/epsilon)$ to $epsilon$-Nash equilibria. This quadratic speed-up sets a new benchmark for computing $epsilon$-Nash equilibria in quantum zero-sum games.
• Score: 102.46640028830441
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathcal{O}(d/\epsilon^2)$ 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.

Related papers

• Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices [7.174256268278207]
  We propose a quantum algorithm inspired by the classical multi-row iteration method. Our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems.
  arXiv  Detail & Related papers  (2024-09-06T03:32:02Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Payoff-based learning with matrix multiplicative weights in quantum games [35.111876815522116]
  We study the problem of learning in quantum games - and other classes of semidefinite games - with scalar, payoff-based feedback. We introduce a suite of minimal-information matrix multiplicative weights (3MW) methods tailored to different information frameworks. We show that the 3MW method with deterministic payoff feedback retains the convergence rate of the vanilla, full information MMW algorithm in quantum min-max games.
  arXiv  Detail & Related papers  (2023-11-04T14:56:17Z)
• Alleviating the quantum Big-$M$ problem [0.237499051649312]
  Classically known as the "Big-$M$" problem, it affects the physical energy scale. We take a systematic, encompassing look at the quantum big-$M$ problem, revealing NP-hardness in finding the optimal $M$. We propose a practical translation algorithm, based on SDP relaxation, that outperforms previous methods in numerical benchmarks.
  arXiv  Detail & Related papers  (2023-07-19T18:00:05Z)
• Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum Games [10.79781442303645]
  We propose the first online quantum algorithm for solving zero-sum games. Our algorithm generates classical outputs with succinct descriptions. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem.
  arXiv  Detail & Related papers  (2023-04-27T14:02:54Z)
• Quantum Clustering with k-Means: a Hybrid Approach [117.4705494502186]
  We design, implement, and evaluate three hybrid quantum k-Means algorithms. We exploit quantum phenomena to speed up the computation of distances. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version.
  arXiv  Detail & Related papers  (2022-12-13T16:04:16Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Policy Optimization for Markov Games: Unified Framework and Faster Convergence [81.3266426402464]
  We show that the state-wise average policy of this algorithm converges to an approximate Nash equilibrium (NE) of the game. We extend this algorithm to multi-player general Markov Games and show a $mathcalwidetildeO(T-1/2)$ convergence rate to Correlated Equilibria (CCE)
  arXiv  Detail & Related papers  (2022-06-06T14:23:13Z)
• Improving Quantum Simulation Efficiency of Final State Radiation with Dynamic Quantum Circuits [1.3375143521862154]
  We make use of a new quantum hardware capability called dynamical quantum computing to improve the scaling of the QPS algorithm. We modify the quantum parton shower circuit to incorporate mid-circuit qubit measurements, resets, and quantum operations conditioned on classical information.
  arXiv  Detail & Related papers  (2022-03-18T15:31:19Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.