On the limits of depth reduction at depth 3 over small finite fields

Abstract

In a surprising recent result, Gupta–Kamath–Kayal–Saptharishi have proved that over Q any nO(1)-variate and n-degree polynomial in VP can also be computed by a depth three ΣΠΣ circuit of size 2O(nlog3/2⁡n).2 Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ circuit that computes the determinant (or the permanent) polynomial of a n×n matrix must be of size 2Ω(n). In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ΣΠΣ circuit computing it must be of size 2Ω(nlog⁡n). 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 nO(1)-variate and n-degree polynomials in VP by depth 3 circuits of size 2o(nlog⁡n). The result of Grigoriev and Karpinski can only rule out such a possibility for ΣΠΣ circuits of size 2o(n).We also give an example of an explicit polynomial (NWn,ϵ(X)) in VNP (which is not known to be in VP), for which any ΣΠΣ circuit computing it (over fixed-size fields) must be of size 2Ω(nlog⁡n). The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson, and is closely related to the polynomials considered in many recent papers (by Kayal–Saha–Saptharishi, Kayal–Limaye–Saha–Srinivasan, and Kumar–Saraf), where strong depth 4 circuit size lower bounds are shown.