Gaussian-only approximation for quartic fermionic Hamiltonians

Determine whether an Omega(1/\sqrt{n})-approximation for homogeneous quartic fermionic Hamiltonians can be achieved using only fermionic Gaussian states.

Background

The paper establishes an \Omega(1/\sqrt{n}) approximation for homogeneous quartic fermionic Hamiltonians by combining a Gaussian-state candidate with a simple non-Gaussian sparse Fock-basis state. It also recalls that Gaussian states alone achieve an optimal \Theta(1/\sqrt{n}) ratio on certain SYK instances, but the general Gaussian-only question remains unresolved in the discussion.

References

Is such an approximation possible using only Gaussian states?

— Improved classical optimization of fermionic Hamiltonians beyond Gaussian states  (2610.12844 - Parekh, 9 Oct 2026) in Section Discussion

If so, is this possible in polynomial time?

— Improved classical optimization of fermionic Hamiltonians beyond Gaussian states  (2610.12844 - Parekh, 9 Oct 2026) in Section Discussion