Improved Approximations for Extremal Eigenvalues of Sparse Hamiltonians

TL;DR

This paper presents improved classical approximation algorithms for estimating the maximum eigenvalues of sparse fermionic and qubit Hamiltonians with local terms, achieving better bounds than previous methods.

Contribution

It introduces new approximation ratios for sparse fermionic and qubit Hamiltonians with local terms, improving upon prior bounds and extending to more general cases.

Findings

Achieves a 1/(qk+1)-approximation for maximum eigenvalues of k-sparse fermionic Hamiltonians.

Provides a 1/(4k+1)-approximation for Hamiltonians with 2- and 4-local terms.

Develops a 1/O(qk^2)-approximation for general k-sparse, q-local fermionic Hamiltonians.

Abstract

We give a classical $1/(qk+1)$-approximation for the maximum eigenvalue of a $k$-sparse fermionic Hamiltonian with strictly $q$-local terms, as well as a $1/(4k+1)$-approximation when the Hamiltonian has both $2$-local and $4$-local terms. More generally we obtain a $1/O(qk^2)$-approximation for $k$-sparse fermionic Hamiltonians with terms of locality at most $q$. Our techniques also yield analogous approximations for $k$-sparse, $q$-local qubit Hamiltonians with small hidden constants and improved dependence on $q$.