Abstract
Let h be a homogeneous polynomial of degree 4 over indeterminates chi_1, ..., chi_n. Assume as usual that chi_i^2 = 1, but *don't* assume that chi_i chi_j = chi_j chi_i. Instead, assume that chi_i chi_j = -chi_j chi_i. The task is now to substitute the chi_i's with matrices (of any dimension D) that satisfy the relations so as to make h have the largest possible eigenvalue. This setup models one of the most basic tasks in quantum chemistry. Also, if the coefficients are random Gaussians, we get the "SYK model", which is used to study black holes.
In this talk I will discuss efficient algorithms for upper- and lower-bounding the optimal solution.
This is joint work in progress with Matt Hastings (Microsoft).