Quantum Speedups in Linear Programming via Sublinear Multi-Gibbs Sampling
Supplementary Material: pdf
Primary Area: optimization
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Keywords: Linear programming, zero-sum games, quantum algorithms, quantum Gibbs sampling
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TL;DR: We propose a state-of-the-art quantum algorithm for linear programming and zero-sum games
Abstract: As a basic optimization technique, linear programming has found wide applications in many areas. In this paper, we propose an improved quantum algorithm for solving a linear programming problem with $m$ constraints and $n$ variables in time $\widetilde{O} (\sqrt{m+n}\gamma^{2.25})$, where $\gamma = Rr/\varepsilon$ is the additive error $\varepsilon$ scaled down with bounds $R$ and $r$ on the size of the primal and dual optimal solutions, improving the prior best $\widetilde O(\sqrt{m+n}\gamma^{2.5})$ by Bouland, Getachew, Jin, Sidford, and Tian (ICML 2023) and Gao, Ji, Li, and Wang (NeurIPS 2023).
Our algorithm solves linear programming via a zero-sum game formulation, under the framework of the sample-based optimistic multiplicative weight update. At the heart of our construction, is an improved quantum multi-Gibbs sampler for diagonal Hamiltonians with time complexity \emph{sublinear} in inverse temperature $\beta$, breaking the general $O(\beta)$-barrier.
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Submission Number: 7683
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