  We show that the two-dimensional (2D) local Hamiltonian problem with the
constraint that the ground state obeys area laws is QMA-complete. We also prove
similar results in 2D translation-invariant systems and for the 3D Heisenberg
and Hubbard models with local magnetic fields. Consequently, unless MA = QMA,
not all ground states of 2D local Hamiltonians with area laws have efficient
classical representations that support efficient computation of local
expectation values. In the future, even if area laws are proved for ground
states of 2D gapped systems, the computational complexity of these systems
remains unclear.
