Quantum Constraint Problems can be complete for BQP, 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, NPcomplete, MA-complete, or QMA1-complete, but this list has not been shown to be exhaustive. We present three quantum constraint problems, that are (1) BQP1-complete (also known as coRQP), (2) QCMA-complete and (3) coRP-complete. This provides the first natural complete problem for BQP1. 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.