Universal quantum computing with parafermions assisted by a half-fluxon

Braiding of anyons such as Majoranas or parafermions provides only Clifford gates which do not form a universal set of quantum gates. We propose a robust and resource-efficient scheme to perform non-Clifford gates on logical qudits encoded in parafermionic zero modes via the Aharonov-Casher effect. This gate can be implemented by moving a half-flux quantum around the pair of parafermionic zero modes. The parafermion modes can be realized in a two-dimensional setup via existing proposals, and a half-fluxon can be created as a part of half-fluxon/anti-half-fluxon pair in a spin-triplet Josephson junction. We provide evidence that the half-fluxon can be braided robustly around the parafermions and hence this is a reliable proposal for the implementation of the non-Clifford gate without magic state distillation. Supplementing this gate with the braiding of parafermions provides the avenue for universal quantum computing with parafermions.

Figure 1

(a) Fractional Chern insulator based setup for parafermionic defects shown by black patches at the superconductor-insulator interface. (b) Spin-triplet Josephson junction with a localized dipole current that facilitates LHF-FAHF pair creation. The shaded ring-shaped regions present the superconductors, while the region between them is the insulator. The bias current is shown as $I_b$, dipole current as $I$, and the distance between the injection and collection leads is marked as D. (c) Josephson junction setup combined with the FCI-based parafermionic setup [10] such that the free AHF's flux can go around the pair of parafermionic defects to implement $\sqrt{Z_N}$ gate. Josephson junction has been placed on top of the setup with parafermionic zero modes. Here the current injection leads as shown in panel (b) are not shown for simplicity.

Figure 2

Processes that contribute to the parafermionic zero mode operator. Blue and orange arrows correspond to up and down spin, while black and white circles correspond to quasiparticles and quasiholes, respectively. $l_1$ is the length of the domain wall between the pairing and insulating region. (a) Right-moving quasiparticle from pairing to insulating region. The insulator with spin-orbit coupling (SOC) or ferromagnet reflects the quasiparticle back with down spin in (b). (c) Andreev reflection from the pairing region sends back a quasihole with spin reversed back to up spin. (d) Quasihole gets reflected back from the insulating region to the pairing region with down spin.

Figure 3

Gate sequence composed of HF braidings in opposite directions (at time steps $t_1$ and $t_3$) with a PF braiding operation (at time step $t_2$ and shown in green) implements a non-Clifford operation. Red circle denotes the HF, and blue circles denote the PFs. The first HF braiding induces a change of the $s$-wave superconductor's order parameter phase from ${\mathrm{\Phi }}_s$ to ${\mathrm{\Phi }}_s+\pi$. The overall operation restores the order parameter phase back to ${\mathrm{\Phi }}_s$ and hence acts on the original ground-state manifold.

Figure 4

Energy of possible vacuum configurations as a function of defect strength $\varepsilon$. The energy for the half-fluxon solution stays between the quadrupole and the fluxon solution, which are the two stable vacuum configurations. A defect strength can be chosen such that one of these configurations is the vacuum configuration.