Gates for protected superconducting qubits via their internal modes

Following the original proposal for a 0-π qubit [1], Brooks, Kitaev, and Preskill (BKP) presented a method for performing a protected gate on a π-periodic superconducting qubit [2]. This protected gate can is achieved by adiabatically coupling a 0-π qubit to a high-inductance LC oscillator. This analysis was agnostic with regards to the actual circuit implementation of the 0-π qubit. Further analyses [3, 4] showed that coupling to the LC oscillator also leads to unwanted interactions between the internal modes of the 0-π qubit. This interaction persists even in the presence of zero circuit disorder seemingly ruining the possibility of performing a BKP-style protected gate. In this note we analyse whether the internal degrees of freedom of the 0-π qubit can be used as the requisite ancilla mode for the protected gate proposed by BKP.

[1] A Kitaev. Protected qubit based on a superconducting current mirror, 2006.

[2] P Brooks, A Kitaev, and J Preskill. Protected gates for superconducting qubits. Physical Review A, 87(5), 2013.

[3] P Groszkowski, A Di Paolo, A L Grimsmo, A Blais, D I Schuster, A A Houck, and J Koch. Coherence properties of the 0-π qubit. New Journal of Physics, 20(4), 2018.

[4] A Di Paolo, A L Grimsmo, P Groszkowski, J Koch, and A Blais. Control and coherence time enhancement of the 0-π qubit. New Journal of Physics, 21(4), 2019.