  The QMA-completeness of the local Hamiltonian problem is a landmark result of
the field of Hamiltonian complexity that studies the computational complexity
of problems in quantum many-body physics. Since its proposal, substantial
effort has been invested in better understanding the problem for physically
motivated important families of Hamiltonians. In particular, the
QMA-completeness of approximating the ground state energy of local Hamiltonians
has been extended to the case where the Hamiltonians are geometrically local in
one and two spatial dimensions. Among those physically motivated Hamiltonians,
stoquastic Hamiltonians play a particularly crucial role, as they constitute
the manifestly sign-free Hamiltonians in Monte Carlo approaches. Interestingly,
for such Hamiltonians, the problem at hand becomes more ''classical'', being
hard for the class MA (the randomized version of NP) and its complexity has
tight connections with derandomization. In this work, we prove that both the
two- and one-dimensional geometrically local analogues remain MA-hard with high
enough qudit dimension. Moreover, we show that related problems are
StoqMA-complete.
