Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

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 $κ$ in quantum space $O(\log n + \log κ)$. 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 $κ$ can be approximated to inverse polynomial relative error with classical circuits of depth $\tilde O(\log n \cdot \log κ)$, 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].