The Power of Depth 2 Circuits over Algebras

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 (ΣΠΣ) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (ΠΣ) 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, ΠΣ circuits over U2(F) are strictly ‘weaker’ than ΣΠΣ circuits over F. The equivalence further shows that PIT of depth 3 arithmetic circuits reduces to PIT of width-2 planar commutative Algebraic Branching Programs(ABP). Thus, identity testing for commutative ABPs is interesting even in the case of width-2. Further, we give a deterministic polynomial time identity testing algorithm for a ΠΣ circuit over any constant dimensional commutative algebra over F. While over commutative algebras of polynomial dimension, identity testing is at least as hard as that of ΣΠΣ circuits over F.