  We study the design of polylogarithmic depth algorithms for approximately
solving packing and covering semidefinite programs (or positive SDPs for
short). This is a natural SDP generalization of the well-studied positive LP
problem.
  Although positive LPs can be solved in polylogarithmic depth while using only
$\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best known
positive SDP solvers due to Jain and Yao require $O(\log^{14} n
/\varepsilon^{13})$ parallelizable iterations. Several alternative solvers have
been proposed to reduce the exponents in the number of iterations. However, the
correctness of the convergence analyses in these works has been called into
question, as they both rely on algebraic monotonicity properties that do not
generalize to matrix algebra.
  In this paper, we propose a very simple algorithm based on the optimization
framework proposed for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n
/ \varepsilon^2)$ iterations, matching that of the best LP solver. To surmount
the obstacles encountered by previous approaches, our analysis requires a new
matrix inequality that extends Lieb-Thirring's inequality, and a
sign-consistent, randomized variant of the gradient truncation technique
proposed in.
