Permanent does not have succinct polynomial size arithmetic circuits of constant depth

Abstract

We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φn} using division by constants,3 where Φn has size at most p(n) and depth O(1), such that Φn computes the n×n permanent. A circuit family {Φn} is succinct if there exists a nonuniform Boolean circuit family {Cn} with O(logn) many inputs and size no(1) such that Cn can correctly answer direct connection language queries about Φn – succinctness is a relaxation of uniformity.To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this, we obtain the lower bound by giving an explicit construction, computable in the polynomial hierarchy, of a hitting set for arithmetic circuits.