  We show that the mixed discriminant of $n$ positive semidefinite $n \times n$
real symmetric matrices can be approximated within a relative error $\epsilon
>0$ in quasi-polynomial $n^{O(\ln n -\ln \epsilon)}$ time, provided the
distance of each matrix to the identity matrix in the operator norm does not
exceed some absolute constant $\gamma_0 >0$. We deduce a similar result for the
mixed discriminant of doubly stochastic $n$-tuples of matrices from the Marcus
- Spielman - Srivastava bound on the roots of the mixed characteristic
polynomial. Finally, we construct a quasi-polynomial algorithm for
approximating the sum of $m$-th powers of principal minors of a matrix,
provided the operator norm of the matrix is strictly less than 1. As is shown
by Gurvits, for $m=2$ the problem is $\#P$-hard and covers the problem of
computing the mixed discriminant of positive semidefinite matrices of rank 2.
