Optimizing sparse fermionic Hamiltonians

TL;DR

This paper demonstrates that sparse fermionic Hamiltonians, including certain models like the sparse SYK-4, can be efficiently approximated by Gaussian states with a constant ratio, unlike dense models.

Contribution

The paper proves that sparse fermionic Hamiltonians have a constant Gaussian approximation ratio, extending to models with quadratic and quartic terms, and clarifies why dense models like SYK-4 differ.

Findings

Sparse fermionic Hamiltonians admit constant Gaussian approximation ratios.

Efficient determination of Gaussian states for sparse Hamiltonians.

Extension of approximation ratios to SYK-q models with even q.

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.