| Summary: | We consider the problem of approximating the ground state energy of a fermionic Hamiltonian using a Gaussian state. In sharp contrast to the dense case [1, 2], we prove that strictly $q$-local $\rm {\textit {sparse}}$ fermionic Hamiltonians have a constant Gaussian approximation ratio; the result holds for any connectivity and interaction strengths. Sparsity means that each fermion participates in a bounded number of interactions, and strictly $q$-local means that each term involves exactly $q$ fermionic (Majorana) operators. We extend our proof to give a constant Gaussian approximation ratio for sparse fermionic Hamiltonians with both quartic and quadratic terms. With additional work, we also prove a constant Gaussian approximation ratio for the so-called sparse SYK model with strictly $4$-local interactions (sparse SYK-$4$ model). In each setting we show that the Gaussian state can be efficiently determined. Finally, we prove that the $O(n^{-1/2})$ Gaussian approximation ratio for the normal (dense) SYK-$4$ model extends to SYK-$q$ for even $q\gt4$, with an approximation ratio of $O(n^{1/2 – q/4})$. Our results identify non-sparseness as the prime reason that the SYK-$4$ model can fail to have a constant approximation ratio [1, 2].
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