Fault-tolerant and Continuous Holonomic Gates for Topologically Protected Qubits.

We present the design of the topologically protected qubit that allows fault tolerant discrete operations together with the continuous holonomic exp(iθσ_z)-gates. Topological protection of qubits is meaningless without ability to perform fault tolerant operations. These transformations should involve at least the generators of the discrete fault tolerant Clifford group and a rotation at an arbitrary angle such as π/3. We propose a superconducting circuit that satisfies these criteria. The proposed qubit allows protected π/2-rotations in charge (Q)- and flux (Φ)-channels. Furthermore, by moving the state of the qubit along the closed loop in the {Q,Φ}-space we can controllably and continuously gain a finite Berry phase difference between 0 and 1 states that is weakly sensitive to the errors in charge and flux biases.
The qubit is based on superconducting 0-π elements which effectively act as π-periodic Josephson junctions. The discrete gate operations are characterized by the exponential suppression of the flux and charge noise. The continuous holonomic operation maintains exponential suppression of the flux noise and has the quadratic suppression of the charge noise.