The Complexity of the Local Hamiltonian Problem

Abstract

The k-LOCAL Hamiltonian problem is a natural complete problem for the complexity class QMA, the quantum analogue 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 2-local interactions on qubits is equivalent to standard quantum computation.

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