  For $\cos(2\theta)$ qubits based on voltage-controlled semiconductor nanowire
Josephson junctions we introduce a single-qubit $Z$ gate inspired by the
noise-bias preserving gate of the Kerr-cat qubit. This scheme relies on a $\pi$
rotation in phase space via a beamsplitter-like transformation between a qubit
and ancilla qubit. The rotation is implemented by adiabatically changing the
potential energies of the two qubits such as to preserve a double-well
potential at all times. This gate constrains the dynamics in the subspace of a
$\cos(2\theta)$ qubit at all times, therefore yielding high-fidelity operation
while preserving the qubit's coherence. We introduce a circuit to realize this
gate and support our findings with numerical simulations.
