  We analyze whether circuit-QED Hamiltonians are stoquastic focusing on
systems of coupled flux qubits: we show that scalable sign-problem free path
integral Monte Carlo simulations can typically be performed for such systems.
Despite this, we corroborate the recent finding [arXiv:1903.06139] that an
effective, non-stoquastic qubit Hamiltonian can emerge in a system of
capacitively coupled flux qubits. We find that if the capacitive coupling is
sufficiently small, this non-stoquasticity of the effective qubit Hamiltonian
can be avoided if we perform a canonical transformation prior to projecting
onto an effective qubit Hamiltonian. Our results shed light on the power of
circuit-QED Hamiltonians for the use of quantum adiabatic computation and the
subtlety of finding a representation which cures the sign problem in these
systems
