Optimized preparation of magic state for parafermionic qudits via non-adiabatic braiding*

Parafermionic zero modes (PZMs) offer some benefits compared to Majorana zero modes (MZMs), exhibiting resilience against dissipation, realizing a full many-qudit Clifford group, and allowing more efficient magic state distillation. Notwithstanding, neither PZMs nor MZMs support topological non-Clifford gates, and thus, per Gottesman--Knill theorem, can be classically simulated. However, adiabatic manipulations yielding geometric phases can realize high-quality magic gates in MZMs, enabling universal computation.

We show that such an approach is not applicable to parafermions: any adiabatic manipulation that preserves the ground state degeneracy yields Clifford gates. We suggest an alternative protocol, using non-adiabatic effects to construct the magic gate for qudits realized by parafermions while preserving a fixed number of zero modes and thus avoiding dynamical phase accumulation.

We show that protocol parameters, such as speed and coupling strength, can be optimized, resulting in a protocol robust against generic low-frequency noise with a specified yield of high-fidelity magic states.