  This paper introduces a new interior point method algorithm that solves
semidefinite programming (SDP) with variable size $n \times n$ and $m$
constraints in the (current) matrix multiplication time $m^{\omega}$ when $m
\geq \Omega(n^2)$. Our algorithm is optimal because even finding a feasible
matrix that satisfies all the constraints requires solving an linear system in
$m^{\omega}$ time. Our work improves the state-of-the-art SDP solver [Jiang,
Kathuria, Lee, Padmanabhan and Song, FOCS 2020], and it is the first result
that SDP can be solved in the optimal running time.
  Our algorithm is based on two novel techniques:
  $\bullet$ Maintaining the inverse of a Kronecker product using lazy updates.
  $\bullet$ A general amortization scheme for positive semidefinite matrices.
