  Computing the ground-state energy of interacting electron (fermion) problems
has recently been shown to be hard for QMA, a quantum analogue of the
complexity class NP. Fermionic problems are usually hard, a phenomenon widely
attributed to the so-called sign problem occurring in Quantum Monte Carlo
simulations. The corresponding bosonic problems are, according to conventional
wisdom, tractable. Here, we discuss the complexity of interacting boson
problems and show that they are also QMA-hard. In addition, we show that the
bosonic version of the so-called N-representability problem is QMA-complete, as
hard as its fermionic version. As a consequence, these problems are unlikely to
have efficient quantum algorithms.
