The Power of Depth 2 Circuits over Algebras

Chandan Saha, Ramprasad Saptharishi & Nitin Saxena
We study the problem of polynomial identity testing (PIT) for depth $2$ arithmetic circuits over matrix algebra. We show that identity testing of depth $3$ ($\Sigma \Pi \Sigma$) arithmetic circuits over a field $\F$ is polynomial time equivalent to identity testing of depth $2$ ($\Pi \Sigma$) arithmetic circuits over $\mathsf{U}_2(\mathbb{F})$, the algebra of upper-triangular $2\times 2$ matrices with entries from $\F$. Such a connection is a bit surprising since we also show that, as computational...