Local Hamiltonians and Ground States

We discuss many-body systems, where the Hamiltonian involves only few-body interactions. With the tensor product structure of the many-body Hilbert space in mind, we introduce the concept of locality. It is naturally associated with the spatial geometry of the system, where the most natural interaction between degrees of freedom are those “local” ones, for instance nearest neighbor interactions. We discuss the effect of locality on the ground-state properties. We then discuss ways of determining the ground-state energy of local Hamiltonians, and their hardness. Theories have been developed in quantum information science to show that even with the existence of a quantum computer, there is no efficient way of finding the ground-state energy for a local Hamiltonian in general. However, for practical cases, special structures may lead to simpler method, such as Hartree’s mean-field theory. We also discuss a special kind of local Hamiltonians, called the frustration-free Hamiltonians, where the ground state is also ground states of all the local interaction terms. However, to determine whether a Hamiltonian is frustration-free is in general hard.