  Is it possible to prove that the properties of Hamiltonians, such as the
ground state energy, results of dynamical evolution, or thermal state
expectation values, can be efficiently calculated when the Hamiltonians have
physically motivated constraints such as translational or rotational
invariance? We report that rotational invariance does not reduce the difficulty
of finding the ground state energy of the system. Crucially, the construction
it preserves the translational invariance of a Hamiltonian. The failure of the
construction for the properties of thermal states at finite temperatures is
discussed.
