Progress Towards Quantum Annealer v2.0 II: Theory

Superconducting flux qubits are typically modeled as lumped-element circuits. While this is usually a good approximation, several transmission line-based flux qubit architectures have macroscopic spatial dimensions. Thus an accurate description of these circuits could require accounting for the spatial variation of the fields along the transmission line, beyond the usual procedure of defining an effective low-frequency L and C. In this talk we describe such a model for the case of the CPW-based fluxmon qubit. We show that the standard, lumped-element circuit Hamiltonian should be augmented by a capacitive coupling to a set of normal modes describing the field oscillations along the CPW. For a circuit with CPW length 2 mm and otherwise typical parameters, we find that these modes have much higher frequencies than the qubit operating frequency. Thus they can be eliminated from the Hamiltonian using the Born-Oppenheimer approximation, which gives a non-negligible renormalization of the lumped-element Hamiltonian.