TY - GEN

T1 - Quantum-inspired sublinear algorithm for solving low-rank semidefinite programming

AU - Chia, Nai Hui

AU - Li, Tongyang

AU - Lin, Han Hsuan

AU - Wang, Chunhao

N1 - Funding Information:
NHC, HHL, and CW were supported by Scott Aaronson's Vannevar Bush Faculty Fellowship. Tongyang Li: 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.
Funding Information:
Funding NHC, HHL, and CW were supported by Scott Aaronson’s Vannevar Bush Faculty Fellowship. Tongyang Li: 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.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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, . . ., Aτ , we propose a procedure that gives a good approximation of A = A1 + · · · + Aτ . 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.

AB - 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, . . ., Aτ , we propose a procedure that gives a good approximation of A = A1 + · · · + Aτ . 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.

UR - http://www.scopus.com/inward/record.url?scp=85090507747&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85090507747&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2020.23

DO - 10.4230/LIPIcs.MFCS.2020.23

M3 - Conference contribution

AN - SCOPUS:85090507747

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020

A2 - Esparza, Javier

A2 - Kral�, Daniel

A2 - Kral�, Daniel

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020

Y2 - 25 August 2020 through 26 August 2020

ER -