Quantified derandomization of linear threshold circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for TC0, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for TC0. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of TC0 circuits of depth d>2. Our first main result is a quantified derandomization algorithm for TC0 circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a TC0 circuit C over n input bits with depth d and n1+exp(−d) wires, runs in almost-polynomial-time, and distinguishes between the case that C rejects at most 2n1−1/5d inputs and the case that C accepts at most 2n1−1/5d inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a TC0 circuit (i.e., C is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of TC0, and would consequently imply that NEXP⊈TC0. Specifically, if there exists a quantified derandomization algorithm that gets as input a TC0 circuit with depth d and n1+O(1/d) wires (rather than n1+exp(−d) wires), runs in time at most 2nexp(−d), and distinguishes between the case that C rejects at most 2n1−1/5d inputs and the case that C accepts at most 2n1−1/5d inputs, then there exists an algorithm with running time 2n1−Ω(1) for standard derandomization of TC0.