  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.
