  The k-local Hamiltonian problem is a natural complete problem for the
complexity class QMA, the quantum analog of NP. It is similar in spirit to
MAX-k-SAT, which is NP-complete for k<=2. It was known that the problem is
QMA-complete for any k <= 3. On the other hand 1-local Hamiltonian is in P, and
hence not believed to be QMA-complete. The complexity of the 2-local
Hamiltonian problem has long been outstanding. Here we settle the question and
show that it is QMA-complete. We provide two independent proofs; our first
proof uses only elementary linear algebra. Our second proof uses a powerful
technique for analyzing the sum of two Hamiltonians; this technique is based on
perturbation theory and we believe that it might prove useful elsewhere. Using
our techniques we also show that adiabatic computation with two-local
interactions on qubits is equivalent to standard quantum computation.
