A new construction for a QMA complete 3-local Hamiltonian
  We present a new way of encoding a quantum computation into a 3-local Hamiltonian. Our construction is novel in that it does not include any terms that induce legal-illegal clock transitions. Therefore, the weights of the terms in the Hamiltonian do not scale with the size of the problem as in previous constructions. This improves the construction by Kempe and Regev, who were the first to prove that 3-local Hamiltonian is complete for the complexity class QMA, the quantum analogue of NP.   Quantum k-SAT, a restricted version of the local Hamiltonian problem using only projector terms, was introduced by Bravyi as an analogue of the classical k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA with one-sided error (QMA_1) and that quantum 2-SAT is in P. We give an encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only 3-local terms. As an intermediate step to this 3-local construction, we show that quantum 3-SAT for particles with dimensions 3x2x2 (a qutrit and two qubits) is QMA_1 complete. The complexity of quantum 3-SAT with qubits remains an open question. 
Daniel Nagaj, Shay Mozes