The local Hamiltonian problem on a line with eight states is QMA-complete

The Local Hamiltonian problem – estimating the ground state energy of a local Hamiltonian – is a natural problem in physics, and belongs to the complexity class QMA. QMA is the quantum analogue of NP. Languages in QMA have a quantum verifier: a polynomial-time quantum algorithm that takes (poly-sized) quantum states as witnesses. In quantum mechanics, the Hamiltonian of a system is the Hermitian operator corresponding to the energy of the system: its eigenvalues are the set of energies that a system can be measured to have. It also determines the time-evolution of the system and defines the interactions between its subsystems. The least eigenvalue (ground state energy) and the corresponding eigenvector (the ground state) are key to understanding the properties of a