The Sum-of-Squares for Fermionic Systems, and the SYK Model

Perhaps the most important problem in physics or quantum chemistry is to determine properties of the ground state of an interacting system of fermions.  As a quantum mechanical problem, there may be no efficient classical witness to the ground state energy, or even to an approximation of that energy.  A commonly considered witness is a so-called “Gaussian state”, or free fermion wavefunction.  As a prominent example , the Sachdev-Ye-Kitaev (SYK) model has no Gaussian state which achieves a good approximation to the energy; this model is sometimes considered as one of the “most entangled” or “most strongly interacting” models possible.  I will discuss applications of the sum-of-squares method to this model.  Sum-of-squares is a semidefinite programming relaxation.  I will show that this method can give classically efficient constant-factor lower bounds on the energy, and it inspires a quantum algorithm which gives constant-factor upper bounds.  Joint work with R. O’Donnell.