The Power of Depth 2 Circuits over Algebras

Abstract

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 (SPS) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (PS) arithmetic circuits over U2(F), the algebra of upper-triangular 2 2 matrices with entries from F. Such a connection is a bit surprising since we also show that, as computational models, PS circuits over U2(F) are strictly ‘weaker’ than SPS circuits over F. The equivalence further implies that PIT of SPS circuits reduces to PIT of width-2 commutative Algebraic Branching Programs(ABP). Further, we give a deterministic polynomial time identity testing algorithm for a PS circuit of size s over commutative algebras of dimension O(log s/ log log s) over F. Over commutative algebras of dimension poly(s), we show that identity testing of PS circuits is at least as hard as that of SPS circuits over F.