Braiding of Majorana fermions, topological quantum computation and the nature of Clifford algebras

Louis Hirsch Kauffman

Louis Hirsch Kauffman, Samuel J. Lomonaco

Ali Hamed Safwan, Raditya Weda Bomantara

Abstract Majorana fermions and their generalizations to Zn parafermions are considered promising building blocks of fault-tolerant quantum computers for their ability to encode quantum information nonlocally. In such topological quantum computers, highly robust quantum gates are obtained by braiding pairs of these quasi-particles. However, it is well-known that braiding Majorana fermions or parafermions only leads to a Clifford gate, hindering quantum universality. This paper establishes an exact mapping between Majorana fermions to Z4 parafermions in systems under total parity nonconserving and total parity conserving setting. It is revealed that braiding of Majorana fermions may lead to non-Clifford quantum gates in the 4-dimensional qudit representation spanned by Z4 parafermions, whilst braiding of Z4 parafermions may similarly yield non-Clifford quantum gates in the qubit representation spanned by Majorana fermions. This finding suggests that topologically protected universal quantum computing may be possible with Majorana fermions (Z4 parafermions) by supplementing the usual braiding operations with the braiding of Z4 parafermions (Majorana fermions) that could be formed out of Majorana fermions (Z4 parafermions) via the mapping prescribed here. Finally, the paper discusses how braiding of Majorana fermions or Z4 parafermions could be obtained via a series of parity measurements.

Andrew Stewart Caldin

FINDING: Six Majorana zero modes (MZMs) form three qubit pairs whose braiding generates a finite Clifford-group representation — the minimal non-Abelian platform for universal-adjacent topological quantum computation. | MATH: 2n MZMs → n qubits; 6 MZMs → 3 qubits → Hilbert space dim 2³ = 8. Braid group B₆ (or its image) maps to Clifford group C₃ ⊂ U(8). Ising anyon σ braiding: γᵢ → γⱼ, γⱼ → −γᵢ (i≠j), γₖ → γₖ. Majorana algebra {γᵢ,γⱼ} = 2δᵢⱼ. Fusion: σ×σ = 1 + ψ. Braid generator eigenvalues e^{±iπ/8} (topological phase, related to ν=5/2). | CONNECTION: Clifford group order for n qubits = 2^{n²+2n}·∏(4ᵏ−1); for n=3: 2^{15}·(4−1)(16−1)(64−1) = 32768·3·15·63 = 92,897,280. Braid group B₆ has 6 generators σ₁…σ₅ with relations σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁ (Yang–Baxter / braid relation) — a cubic algebraic relation, echoing the 3-fold symmetry of A₂ root system and the 2π/3 rotation in SU(3). The 6 MZMs sit at vertices of an octahedron (6 vertices, 8 faces = 2³ Hilbert states); braiding = SO(6) rota Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Debanksh Guha, Meghansh Govil, Suraj Kumar Saw +2 more

Motohiko Ezawa

Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits based on the observation that the circuit Laplacian can be made identical to the Hamiltonian. A set of AC voltages along the chain plays the role of the wave function. We generate an arbitral number of topological segments in a Kitaev chain. A pair of topological edge states emerge at the edges of a topological segment. Its wave function is observable by the position and the phase of a peak in impedance measurement. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. We also present an operator formalism by regarding a topological edge state as a topological soliton intertwining the trivial segment and the topological segment. Our analysis shows that the electric-circuit approach can simulate the Majorana-fermion approach to topological quantum computations.