A Polylogarithmic PRG for Degree $2$ Threshold Functions in the Gaussian Setting

We construct and analyze a new pseudorandom generator for degree 2 polynomial threshold functions with respect to the Gaussian measure. In particular, we obtain one whose seed length is polylogarithmic in both the dimension and the desired error, a substantial improvement over existing constructions. Our generator is obtained as an appropriate weighted average of pseudorandom generators against read once branching programs. The analysis requires a number of ideas including a hybrid argument and a structural result that allows us to treat our degree 2 threshold function as a function of a number of linear polynomials and one approximately linear polynomial.