Protected phase gate for the 0-π qubit using its internal modes

Protected superconducting qubits such as the 0-π qubit promise to substantially reduce physical error rates through a multi-mode encoding. This protection comes at the cost of controllability, as standard techniques for quantum gates are ineffective. We propose a protected phase gate for the 0-π qubit that utilises an internal mode of the circuit as an ancilla. The gate is achieved by varying the qubit-ancilla coupling via a tunable Josephson element. Our scheme is a modified version of a protected gate proposed by Brooks, Kitaev and Preskill that uses an external oscillator as an ancilla. We find that our scheme is compatible with the protected regime of the 0-π qubit, and does not suffer from spurious coupling to additional modes of the 0-π circuit. Through numerical simulations, we study how the gate error scales with the circuit parameters of the 0-π qubit and the tunable Josephson element that enacts the gate.