Extremal eigenvalues of local Hamiltonians

TL;DR

This paper uses classical algorithms to efficiently approximate bounds on the extremal eigenvalues of local Hamiltonians, providing practical bounds for operator norms and ground-state energies.

Contribution

It introduces a method to find bounds on extremal eigenvalues of local Hamiltonians using classical algorithms, applicable to spin systems with bounded participation.

Findings

Bounds on operator norm and ground-state energy are achievable by product states.

Classical algorithms can efficiently find these bounds.

Applicable to spin Hamiltonians with limited local interactions.

Abstract

We apply classical algorithms for approximately solving constraint satisfaction problems to find bounds on extremal eigenvalues of local Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on the number of terms in which each spin participates, and find extensive bounds for the operator norm and ground-state energy of such Hamiltonians under this constraint. In each case the bound is achieved by a product state which can be found efficiently using a classical algorithm.