Optimizing sparse fermionic Hamiltonians

We consider the problem of approximating ground state energy a fermionic Hamiltonian using Gaussian state. In sharp contrast to dense case [1, 2], we prove that strictly q-local xmlns:mml="http://www.w3.org/1998/Math/MathML">sparse Hamiltonians have constant approximation ratio; result holds for any connectivity and interaction strengths. Sparsity means each fermion participates in bounded number interactions, term involves exactly xmlns:mml="http://www.w3.org/1998/Math/MathML">q (Majorana) operators. extend our proof give ratio sparse with both quartic quadratic terms. With additional work, also so-called SYK model xmlns:mml="http://www.w3.org/1998/Math/MathML">4-local interactions (sparse SYK-4 model). setting show can be efficiently determined. Finally, xmlns:mml="http://www.w3.org/1998/Math/MathML">O(n−1/2) normal (dense) extends even xmlns:mml="http://www.w3.org/1998/Math/MathML">q>4, an class="MJX-TeXAtom-ORD">1/2–q/4). Our results identify non-sparseness as prime reason fail 2].