  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 $O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))$
given access to a sampling-based low-overhead data structure for the constraint
matrices, where $\varepsilon$ 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:
  $\bullet$ Weighted sampling: assuming sampling access to each individual
constraint matrix $A_{1},\ldots,A_{\tau}$, we propose a procedure that gives a
good approximation of $A=A_{1}+\cdots+A_{\tau}$.
  $\bullet$ Symmetric approximation: we propose a sampling procedure that gives
the \emph{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.
