Pinned QMA: The power of fixing a few qubits in proofs

What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this can greatly increase the complexity of static questions -- ground state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the Pinned commuting and Pinned Stoquastic Local Hamiltonian problems are QMA complete. Second, we show that pinning a single qubit via often repeated measurements also results in universal quantum computation already with commuting and stoquastic Hamiltonians. Finally, we discuss variants of the Ground State Connectivity (GSCON) problem in light of pinning, and show that Stoquastic GSCON is QCMA complete. We hence identify a comprehensive picture of the computational power of pinning, reminiscent of the power of the one clean qubit model.