Arithmetic Circuits: A Chasm at Depth 3

Abstract

We show that, over ${\mathbb Q}$, if an $n$-variate polynomial of degree $d = n^{O(1)}$ is computable by an arithmetic circuit of size $s$ (respectively, by an arithmetic branching program of size $s$), then it can also be computed by a depth-3 circuit (i.e., a $\Sigma \Pi \Sigma$ circuit) of size $\exp(O(\sqrt{d \log n \log d\log s}))$ (respectively, of size $\exp(O(\sqrt{d \log n \log s}))$. In particular this yields a $\Sigma \Pi \Sigma$ circuit of size $\exp(O(\sqrt{d} \cdot \log d))$ computing the $d \times d$ determinant $\mathsf{Det}_d$. It also means that if we can prove a lower bound of $\exp(\omega(\sqrt{d} \cdot \log d))$ on the size of any $\Sigma \Pi \Sigma$ circuit computing the $d \times d$ permanent $\mathsf{Perm}_d$, then we get superpolynomial lower bounds for the size of any arithmetic branching program computing $\mathsf{Perm}_d$. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The $\Sigma \Pi \Sigma $ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than $d$. Indeed such a counterintuitive construction is unavoidable---it is known that in any $\Sigma \Pi \Sigma$ circuit $C$ computing either $\mathsf{Det}_d$ or $\mathsf{Perm}_d$, if every multiplication gate has fanin at most $d$ (or any constant multiple thereof), then $C$ must have size at least $\exp(\Omega(d))$.