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

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.