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

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

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Spectral decomposition
  • Semi-definite programming
  • Quantum-inspired algorithm
  • Sublinear algorithm

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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.

Cite As Get BibTex

Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.23

Author Details

Nai-Hui Chia
  • Department of Computer Science, University of Texas at Austin, TX, USA
Tongyang Li
  • Department of Computer Science, Institute for Advanced Computer Studies, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
Han-Hsuan Lin
  • Department of Computer Science, University of Texas at Austin, TX, USA
Chunhao Wang
  • Department of Computer Science, University of Texas at Austin, TX, USA

Funding

NHC, HHL, and CW were supported by Scott Aaronson’s Vannevar Bush Faculty Fellowship.
  • Li, Tongyang: TL was supported by IBM PhD Fellowship, QISE-NET Triplet Award (NSF DMR-1747426), and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program.

Acknowledgements

We thank Scott Aaronson, Andr{á}s Gilyén, Ewin Tang, Ronald de Wolf, and anonymous reviewers for their detailed feedback on preliminary versions of this paper.

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