One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for T C 0 , 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 T C 0 . In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of T C 0 circuits of depth 2"> d 2 .

Our first main result is a quantified derandomization algorithm for T C 0 circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a T C 0 circuit C over n input bits with depth d and n 1+ exp ( − d ) wires, runs in almost-polynomial-time, and distinguishes between the case that C rejects at most 2 n 1 − 1 5 d inputs and the case that C accepts at most 2 n 1 − 1 5 d inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a T C 0 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 T C 0 , and would consequently imply that NEX P T C 0 . Specifically, if there exists a quantified derandomization algorithm that gets as input a T C 0 circuit with depth d and n 1+ O (1 d ) wires (rather than n 1+ exp ( − d ) wires), runs in time at most 2 n exp ( − d ) , and distinguishes between the case that C rejects at most 2 n 1 − 1 5 d inputs and the case that C accepts at most 2 n 1 − 1 5 d inputs, then there exists an algorithm with running time 2 n 1 − (1) for standard derandomization of T C 0 .