  In this work we consider the ground space connectivity problem for commuting
local Hamiltonians. The ground space connectivity problem asks whether it is
possible to go from one (efficiently preparable) state to another by applying a
polynomial length sequence of 2-qubit unitaries while remaining at all times in
a state with low energy for a given Hamiltonian $H$. It was shown in [Gharibian
and Sikora, ICALP'15] that this problem is QCMA-complete for general local
Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP
verifier. Here we show that the commuting version of the problem is also
QCMA-complete. This provides one of the first examples where commuting local
Hamiltonians exhibit complexity theoretic hardness equivalent to general local
Hamiltonians.
