Draft:Parafermions (condensed matter)

Not to be confused with Parastatistics.

Parafermions is a type of anyons appearing in condensed matter system, specifically fractional quantum Hall effect system coupled to superconductors. While opening a superconducting gap in 1D semiconductor results in Majorana zero modes, opening a superconducting gap in a 1D wire with fractional excitations results in parafermionic zero modes.

The $\mathbb {Z} _{p}$ parafermionic operators $\chi$ satisfy the following algebra:

$${\begin{aligned}\chi _{j}\chi _{k}&=e^{{\frac {2i\pi }{p}}\mathrm {sgn} (k-j)}\chi _{k}\chi _{j}\\\chi _{j}^{p}&=1\\\chi _{j}^{-1}&=\chi ^{\dagger }.\end{aligned}}$$

For $p=2$, this algebra describes regular fermions.

Generalized Jordan–Wigner transformation

Similar to how fermions can be mapped to a two-level (qubit) system via Jordan–Wigner transformation, parafermions can be mapped to a p-level system (qudit) using generalized Jordan–Wigner transformation.

Consider parafermionic chain $\chi _{i}$ and define

$${\begin{aligned}\chi _{2i-1}&=\left(\prod _{j<i}Z_{j}\right)X_{i}^{\dagger }\\\chi _{2i}&=\omega ^{\frac {p+1}{2}}\left(\prod _{j\leqslant i}Z_{j}\right)X_{i}^{\dagger }.\end{aligned}}$$

The $X_{i}$ and $Z_{i}$ are qudit analog of Pauli operators.

While braiding of parafermions is not universal, for odd $p$ generates full Clifford group, making them more suitable for quantum computing than Majorana zero modes.