Optimizing sparse fermionic Hamiltonians
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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 (Hastings O'Donnell, 2022), we prove that strictly q-local 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>4, 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.
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