On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

Abstract

In a surprising recent result, Gupta et al. [GKKS13b] have proved that over ℚ any n ^O(1)-variate and n-degree polynomial in VP can also be computed by a depth three ∑ ∏ ∑ circuit of size $2^{O(\sqrt{n} \log^{3/2}n)}$ . Over fixed-size finite fields, Grigoriev and Karpinski proved that any ∑ ∏ ∑ circuit that computes the determinant (or the permanent) polynomial of a n×n matrix must be of size 2^Ω(n). In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ∑ ∏ ∑ circuit computing it must be of size 2^Ω(nlogn). The explicit polynomial that we consider is the iterated matrix multiplication polynomial of n generic matrices of size n×n. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the n ^O(1)-variate and n-degree polynomials in VP by depth 3 circuits of size 2^o(nlogn). The result of [GK98] can only rule out such a possibility for ∑ ∏ ∑ circuits of size 2^o(n).

We also give an example of an explicit polynomial (NW _n,ε (X)) in VNP (which is not known to be in VP), for which any ∑ ∏ ∑ circuit computing it (over fixed-size fields) must be of size 2^Ω(nlogn). The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson [NW94], and is closely related to the polynomials considered in many recent papers where strong depth 4 circuit size lower bounds were shown [KSS13,KLSS14,KS13b,KS14].