License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.23
URN: urn:nbn:de:0030-drops-126919
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12691/
Go to the corresponding LIPIcs Volume Portal


Chia, Nai-Hui ; Li, Tongyang ; Lin, Han-Hsuan ; Wang, Chunhao

Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming

pdf-format:
LIPIcs-MFCS-2020-23.pdf (0.6 MB)


Abstract

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(m⋅poly(log n,r,1/ε)) given access to a sampling-based low-overhead 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) 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 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 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.

BibTeX - Entry

@InProceedings{chia_et_al:LIPIcs:2020:12691,
  author =	{Nai-Hui Chia and Tongyang Li and Han-Hsuan Lin and Chunhao Wang},
  title =	{{Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12691},
  URN =		{urn:nbn:de:0030-drops-126919},
  doi =		{10.4230/LIPIcs.MFCS.2020.23},
  annote =	{Keywords: Spectral decomposition, Semi-definite programming, Quantum-inspired algorithm, Sublinear algorithm}
}

Keywords: Spectral decomposition, Semi-definite programming, Quantum-inspired algorithm, Sublinear algorithm
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI