A Quantum Interior Point Method for LPs and SDPs
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We present a quantum interior point method with worst case running time O(n^2.5/ξ^2μκ^3 (1/ϵ)) for SDPs and O(n^1.5/ξ^2μκ^3 (1/ϵ)) for LPs, where the output of our algorithm is a pair of matrices (S,Y) that are ϵ-optimal ξ-approximate SDP solutions. The factor μ is at most √(2)n for SDPs and √(2n) for LP's, and κ is an upper bound on the condition number of the intermediate solution matrices. For the case where the intermediate matrices for the interior point method are well conditioned, our method provides a polynomial speedup over the best known classical SDP solvers and interior point based LP solvers, which have a worst case running time of O(n^6) and O(n^3.5) respectively. Our results build upon recently developed techniques for quantum linear algebra and pave the way for the development of quantum algorithms for a variety of applications in optimization and machine learning.
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