  A quantum constraint problem is a frustration-free Hamiltonian problem: given
a collection of local operators, is there a state that is in the ground state
of each operator simultaneously? It has previously been shown that these
problems can be in P, NP-complete, MA-complete, or QMA_1-complete, but this
list has not been shown to be exhaustive. We present three quantum constraint
problems, that are (1) BQP_1-complete (also known as coRQP), (2)
QCMA_1-complete and (3) coRP-complete. This provides the first natural complete
problem for BQP_1. We also show that all quantum constraint problems can be
realized on qubits, a trait not shared with classical constraint problems.
These results suggest a significant diversity of complexity classes present in
quantum constraint problems.
