  We show that lower bounds on the border rank of matrix multiplication can be
used to non-trivially derandomize polynomial identity testing for small
algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n
\times n \times n$ matrix multiplication, we construct a hitting set generator
with seed length $O(\sqrt{n} \cdot \underline{R}^{-1}(s))$ that hits
$n$-variate circuits of multiplicative complexity $s$. If the matrix
multiplication exponent $\omega$ is not 2, our generator has seed length
$O(n^{1 - \varepsilon})$ and hits circuits of size $O(n^{1 + \delta})$ for
sufficiently small $\varepsilon, \delta > 0$. Surprisingly, the fact that
$\underline{R}(n) \ge n^2$ already yields new, non-trivial hitting set
generators for circuits of sublinear multiplicative complexity.
