Optimality of high-degree energy and free-energy approximation constants

Determine whether the constants in the high-degree product-state approximations for the ground-state energy and finite-temperature free energy of quadratic fermionic Hamiltonians on finite regular graphs are optimal.

Background

The paper establishes product-state approximation bounds for quadratic fermionic Hamiltonians on finite simple regular graphs of degree D, with errors scaling as O(D{-1/2}) for both ground-state energy and finite-temperature free energy. The proofs use covariance monogamy together with edge averaging, the Cauchy–Schwarz inequality, and only the operator-norm normalization of the interaction terms.

The authors explicitly identify the optimality of the resulting constants as unresolved. Improvements could potentially arise from exploiting additional structure, such as bipartite graph geometry, translation invariance, or non-uniform mode counts across regions, but the paper does not resolve whether the stated constants are sharp.

References

It remains open, however, whether the constants in the resulting high-degree energy and free-energy estimates are optimal, since their derivation also uses edge averaging, Cauchy--Schwarz, and only the operator-norm normalization of the interactions.

— Approximation theorems for fermionic Gaussian states  (2610.01860 - Negari et al., 1 Oct 2026) in Section 6, Discussion and outlook