Quantum computation with hybrid parafermion-spin qubits

We propose a universal set of single- and two-qubit quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit to a $\mathbb{Z}_{2m}$ parafermion qubit with arbitrary integer $m$. The special case $m=1$ reproduces the results previously derived for Majorana qubits. Our formalism utilizes Fock parafermions, facilitating a transparent treatment of hybrid parafermion-spin systems. Furthermore, we highlight the previously overlooked importance of particle-hole symmetry in these systems. We give concrete examples how the hybrid qubit system could be realized experimentally for $\mathbb{Z}_4$ and $\mathbb{Z}_6$ parafermions. In addition, we discuss a simple readout scheme for the fractional parafermion charge via the measurement of the spin qubit resonant frequency.