Abstract
Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proofofprinciple sublineartime algorithm for solving SDPs with lowrank 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(m⋅poly(log n,r,1/ε)) given access to a samplingbased lowoverhead data structure for the constraint matrices, where ε 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) samplingbased 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 A₁,…,A_τ, we propose a procedure that gives a good approximation of A = A₁+⋯+A_τ.
 Symmetric approximation: we propose a sampling procedure that gives the spectral decomposition of a lowrank Hermitian matrix A. To the best of our knowledge, this is the first samplingbased algorithm for spectral decomposition, as previous works only give singular values and vectors.
BibTeX  Entry
@InProceedings{chia_et_al:LIPIcs:2020:12691,
author = {NaiHui Chia and Tongyang Li and HanHsuan Lin and Chunhao Wang},
title = {{QuantumInspired Sublinear Algorithm for Solving LowRank Semidefinite Programming}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {23:123:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771597},
ISSN = {18688969},
year = {2020},
volume = {170},
editor = {Javier Esparza and Daniel Kr{\'a}ľ},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12691},
URN = {urn:nbn:de:0030drops126919},
doi = {10.4230/LIPIcs.MFCS.2020.23},
annote = {Keywords: Spectral decomposition, Semidefinite programming, Quantuminspired algorithm, Sublinear algorithm}
}
Keywords: 

Spectral decomposition, Semidefinite programming, Quantuminspired algorithm, Sublinear algorithm 
Collection: 

45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) 
Issue Date: 

2020 
Date of publication: 

18.08.2020 