Perturbation Theory and the Sum of Squares

The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum chemistry it is called the reduced density matrix (RDM) method. We consider the ability of this hierarchy to reproduce weak coupling perturbation theory for three different kinds of systems: spin (or qubit) systems, bosonic systems (the anharmonic oscillator), and fermionic systems with quartic interactions. For such fermionic systems, we show that degree-$4$ SoS (called $2$-RDM in quantum chemsitry) does not reproduce second order perturbation theory but degree-$6$ SoS ($3$-RDM) does (and we conjecture that it reproduces third order perturbation theory). Indeed, we identify a fragment of degree-$6$ SoS which can do this, which may be useful for practical quantum chemical calculations as it may be possible to implement this fragment with less cost than the full degree-$6$ SoS. Remarkably, this fragment is very similar to one studied by Hastings and O'Donnell for the Sachdev-Ye-Kitaev (SYK) model.