  Given a class of $q$-local Hamiltonians, is it possible to find a simple
variational state whose energy is a finite fraction of the ground state energy
in the thermodynamic limit? Whereas product states often provide an affirmative
answer in the case of bosonic (or qubit) models, we show that Gaussian states
fail dramatically in the fermionic case, like for the Sachdev-Ye-Kitaev (SYK)
models. This prompts us to propose a new class of wavefunctions for SYK models
inspired by the variational coupled cluster algorithm. We introduce a static
("0+0D") large-$N$ field theory to study the energy, two-point correlators, and
entanglement properties of these states. Most importantly, we demonstrate a
finite disorder-averaged approximation ratio of $r \approx 0.62$ between the
variational and ground state energy of SYK for $q=4$. Moreover, the variational
states provide an exact description of spontaneous symmetry breaking in a
related two-flavor SYK model.
