Quantum theory of three-junction flux qubit with non-negligible loop inductance: Towards scalability

The three-junction flux qubit (quantum bit) consists of three Josephson junctions connected in series on a superconducting loop. We present a numerical treatment of this device for the general case in which the ratio beta(Q) of the geometrical inductance of the loop to the kinetic inductance of the Josephson junctions is not necessarily negligible. Relatively large geometric inductances allow the flux through each qubit to be controlled independently with on-chip bias lines, an essential consideration for scalability. We derive the three-dimensional potential in terms of the macroscopic degrees of freedom, and include the possible effects of asymmetry among the junctions and of stray capacitance associated with them. To find solutions of the Hamiltonian, we use basis functions consisting of the product of two plane wave states and a harmonic oscillator eigenfunction to compute the energy levels and eigenfunctions of the qubit numerically. We present calculated energy levels for the relevant range of beta(Q). As beta(Q) is increased beyond 0.5, the tunnel splitting between the ground and first excited states decreases rapidly, and the device becomes progressively less useful as a qubit.