Braided Majorana qubits as a minimal setting for Topological Quantum Computation?

TL;DR

This work argues that braided Majorana qubits constitute a minimal, braided-quantum model suitable for Topological Quantum Computation by encoding braid statistics and topological protection within a compact framework. It advances three equivalent mathematical descriptions of braiding: (i) a graded Hopf algebra with braided tensors, (ii) truncated reps of the quantum superalgebra $U_q(osp(1|2))$ at roots of unity, and (iii) a Volichenko-type metasymmetry realized through mixed-bracket Heisenberg-Lie algebras. A central result is that braided Majorana qubits realize Gentile-type parastatistics with level $s$ (finite for root-of-unity values) and a maximal excitation count of $s-1$ in a multiparticle sector, with ordinary fermions recovered at $s=2$ and untruncated spectra at generic $t$. The paper outlines a program to test their viability for implementing logical operations and hardware in TQC, while highlighting rich mathematical structures and future directions such as knot logic integration and concrete physical realizations.

Abstract

I point out that a possible minimal setting to realize Kitaev's proposal of a Topological Quantum Computation which offers topological protection from decoherence could in principle be realized by braided Majorana qubits. Majorana qubits and their braiding were introduced in Nucl. Phys. B 980, 115834 (2022) and further analyzed in J. Phys. A: Math. Theor. 57, 435203 (2024). Braided Majorana qubits implement a Gentile-type parastatistics with at most $s-1$ excited states accommodated in a multiparticle sector (the integer $s=2,3,4,\ldots$ labels quantum group reps at roots of unity). It is argued that braided Majorana qubits could play, for topological quantum computers, the same role as standard bits for ordinary computers and as qubits for "ordinary" quantum computers.