文章基本信息

标题：Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
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作者：Nai-Hui Chia ; Tongyang Li ; Han-Hsuan Lin 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：170
页码：23:1-23:15
DOI：10.4230/LIPIcs.MFCS.2020.23
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with m constraint matrices, each of dimension n and rank r, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time O(mâ<.poly(log n,r,1/Îµ)) given access to a sampling-based low-overhead data structure for the constraint matrices, where Îµ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: - Weighted sampling: assuming sampling access to each individual constraint matrix Aâ,,â¦,A_Ï", we propose a procedure that gives a good approximation of A = Aâ,+â<¯+A_Ï". - Symmetric approximation: we propose a sampling procedure that gives the spectral decomposition of a low-rank Hermitian matrix A. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.
关键词：Spectral decomposition; Semi-definite programming; Quantum-inspired algorithm; Sublinear algorithm