  We study the complexity of approximating the permanent of a positive
semidefinite matrix $A\in \mathbb{C}^{n\times n}$.
  1. We design a new approximation algorithm for $\mathrm{per}(A)$ with
approximation ratio $e^{(0.9999 + \gamma)n}$, exponentially improving upon the
current best bound of $e^{(1+\gamma-o(1))n}$ [AGOS17,YP22]. Here, $\gamma
\approx 0.577$ is Euler's constant.
  2. We prove that it is NP-hard to approximate $\mathrm{per}(A)$ within a
factor $e^{(\gamma-\epsilon)n}$ for any $\epsilon>0$. This is the first
exponential hardness of approximation for this problem. Along the way, we prove
optimal hardness of approximation results for the $\|\cdot\|_{2\to q}$ ``norm''
problem of a matrix for all $-1 < q < 2$.
