Optimizing sparse fermionic Hamiltonians

• URL: http://arxiv.org/abs/2211.16518v2
• Date: Thu, 3 Aug 2023 17:21:59 GMT
• Title: Optimizing sparse fermionic Hamiltonians
• Authors: Yaroslav Herasymenko, Maarten Stroeks, Jonas Helsen, Barbara Terhal
• Abstract summary: We consider the problem of approximating the ground state energy of a fermionic Hamiltonian using a Gaussian state. We prove that strictly $q$-local $rm textit sparse$ fermionic Hamiltonians have a constant Gaussian approximation ratio.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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, 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 $q>4$, 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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