  Estimation of the minimum eigenvalue of a quantum Hamiltonian can be
formalised as the Local Hamiltonian problem. We study the natural special case
of the Local Hamiltonian problem where the same 2-local interaction, with
differing weights, is applied across each pair of qubits. First we consider
antiferromagnetic/ferromagnetic interactions, where the weights of the terms in
the Hamiltonian are restricted to all be of the same sign. We show that for
symmetric 2-local interactions with no 1-local part, the problem is either
QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and
antiferromagnetic XY interactions are shown to be QMA-complete. We also prove
StoqMA-completeness of the antiferromagnetic transverse field Ising model.
Second, we study the Local Hamiltonian problem under the restriction that the
interaction terms can only be chosen to lie on a particular graph. We prove
that nearly all of the QMA-complete 2-local interactions remain QMA-complete
when restricted to a 2D square lattice. Finally we consider both restrictions
at the same time and discover that, with the exception of the antiferromagnetic
Heisenberg interaction, all of the interactions which are QMA-complete with
positive coefficients remain QMA-complete when restricted to a 2D triangular
lattice.
