  The determinant can be computed by classical circuits of depth $O(\log^2 n)$,
and therefore it can also be computed in classical space $O(\log^2 n)$. Recent
progress by Ta-Shma [Ta13] implies a method to approximate the determinant of
Hermitian matrices with condition number $\kappa$ in quantum space $O(\log n +
\log \kappa)$. However, it is not known how to perform the task in less than
$O(\log^2 n)$ space using classical resources only. In this work, we show that
the condition number of a matrix implies an upper bound on the depth complexity
(and therefore also on the space complexity) for this task: the determinant of
Hermitian matrices with condition number $\kappa$ can be approximated to
inverse polynomial relative error with classical circuits of depth $\tilde
O(\log n \cdot \log \kappa)$, and in particular one can approximate the
determinant for sufficiently well-conditioned matrices in depth $\tilde{O}(\log
n)$. Our algorithm combines Barvinok's recent complex-analytic approach for
approximating combinatorial counting problems [Bar16] with the
Valiant-Berkowitz-Skyum-Rackoff depth-reduction theorem for low-degree
arithmetic circuits [Val83].
