Quantum Constraint Problems can be complete for $\mathsf{BQP}$, $\mathsf{QCMA}$, and more

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. These provide some of the first natural problems in these classes. This suggests significantly more diversity than the case of classical constraint problems, which are known to only ever be P or NP-complete. The constructions also offer a quantum view of a new BPP-complete problem.