arXiv:2504.02125 · Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)

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

Francesco Toppan

Published Dec 9, 2025• 1 citation

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,... 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.

DOI: 10.22323/1.490.0347

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arXiv:2504.02125 · Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)

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

Francesco Toppan

Published Dec 9, 2025• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.22323/1.490.0347

Related papers

Topological protection of Majorana qubits

Meng Cheng, Roman M. Lutchyn, S. Das Sarma

We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless ${p}_{x}+i{p}_{y}$ superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless ${p}_{x}+i{p}_{y}$ superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.

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Topological Qubits via Anyon Braids: Majorana 1 Hardware Breakthrough — E8 Intelligence Research

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FINDING: Topological quantum computing leverages anyonic quasiparticles and braid-group invariants to encode error-resistant qubits, with Microsoft's Majorana 1 chip claiming hardware realization via topological superconductivity. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter); Fibonacci anyons yield fusion space dimension $\phi^{n-2}$ (golden ratio $\phi = 1.618...$ ); Majorana zero modes obey $\gamma_i^\dagger = \gamma_i$ , $\{\gamma_i,\gamma_j\} = 2\delta_{ij}$ ; topological charge conservation via $e^{i\pi/4}$ phase gates; Jones polynomial $V_L(t)$ evaluated at roots of unity $t = e^{2\pi i/(k+2)}$ for SU(2)_k Chern–Simons theory. | CONNECTION: Fibonacci anyon braiding directly produces golden-ratio dimension growth — the Hilbert space dimension for $n$ anyons is $F_n$ (Fibonacci numbers), ratio $F_{n+1}/F_n \to 1.618$ . Braid matrices yield eigenvalues $e^{\pm 4\pi i/5}\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing leverages anyonic quasiparticles and braid-group invariants to encode error-resistant qubits, with Microsoft's Majorana 1 chip claiming hardware realization via topological superconductivity. | MATH: Braid group \(B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter); Fibonacci anyons yield fusion space dimension $\phi^{n-2}$ (golden ratio $\phi = 1.618...$); Majorana zero modes obey $\gamma_i^\dagger = \gamma_i$, $\{\gamma_i,\gamma_j\} = 2\delta_{ij}$; topological charge conservation via $e^{i\pi/4}$ phase gates; Jones polynomial $V_L(t)$ evaluated at roots of unity $t = e^{2\pi i/(k+2)}$ for SU(2)_k Chern–Simons theory. | CONNECTION: Fibonacci anyon braiding directly produces golden-ratio dimension growth — the Hilbert space dimension for $n$ anyons is $F_n$ (Fibonacci numbers), ratio $F_{n+1}/F_n \to 1.618$. Braid matrices yield eigenvalues \(e^{\pm 4\pi i/5}\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Oct 31, 20260 citations

Topological Quantum Computing on a Conventional Quantum Computer

Xiao Xiao, James K. Freericks, Alexander F. Kemper

Majorana fermions are self-adjoint fermionic particles that are believed to exist as the elementary excitations in nanoengineered devices with superconductors and ferromagnets. They can be employed in topological quantum computation and can aid the solution of a class of spin models created by Kitaev. In both cases, the braiding of Majorana fermions plays a critical role in these applications. We explicitly construct quantum circuits for Majorana braiding and run them on IBM quantum computers to prepare the ground states of strongly correlated Kitaev-inspired models. The entanglement entropy for these models is then measured to determine quantum phase transitions. We show how maintaining particle-hole symmetry is critical to carrying out this work.Majorana fermions are self-adjoint fermionic particles that are believed to exist as the elementary excitations in nanoengineered devices with superconductors and ferromagnets. They can be employed in topological quantum computation and can aid the solution of a class of spin models created by Kitaev. In both cases, the braiding of Majorana fermions plays a critical role in these applications. We explicitly construct quantum circuits for Majorana braiding and run them on IBM quantum computers to prepare the ground states of strongly correlated Kitaev-inspired models. The entanglement entropy for these models is then measured to determine quantum phase transitions. We show how maintaining particle-hole symmetry is critical to carrying out this work.

Jun 9, 20203 citations

We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless

{p}_{x}+i{p}_{y}

superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. Fibonacci anyons: fusion rule

\tau \otimes \tau = 1 \oplus \tau

, quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. Non-Abelian statistics: unitary braid matrices

U(\sigma_i)

with entries involving

e^{i\pi/5}

\phi^{-1}

\phi^{-1/2}

. Majorana zero modes:

\gamma_i^\dagger = \gamma_i

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

, parity

P = i^k \prod \gamma_i

. Topological protection: energy gap

\Delta

vs. error rate

\sim e^{-\Delta/k_B T}

. | CONNECTION: Fibonacci anyons directly enc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter); Fibonacci anyons yield fusion space dimension

\phi^{n-2}

(golden ratio

\phi = 1.618...

); Majorana zero modes obey

\gamma_i^\dagger = \gamma_i

\{\gamma_i,\gamma_j\} = 2\delta_{ij}

; topological charge conservation via

e^{i\pi/4}

phase gates; Jones polynomial

V_L(t)

evaluated at roots of unity

t = e^{2\pi i/(k+2)}

for SU(2)_k Chern–Simons theory. | CONNECTION: Fibonacci anyon braiding directly produces golden-ratio dimension growth — the Hilbert space dimension for

n

anyons is

F_n

(Fibonacci numbers), ratio

F_{n+1}/F_n \to 1.618

. Braid matrices yield eigenvalues \(e^{\pm 4\pi i/5}\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com