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Quantum Reports

Topologically Protected Quantum Teleportation via Majorana Zero Modes: A Perspective on Scalability and Decoherence Immunity

Horace T. Crogman, To Dang, Daniel B. Erenso

Published Sep 11, 2025• 0 citations
Publisher

Abstract

We present a topologically protected teleportation protocol based on projective parity measurements between spatially separated Majorana zero modes (MZMs), eliminating the need for dynamic braiding. Unlike conventional teleportation schemes, our method preserves logical information through nonlocal encoding and suppresses decoherence exponentially with Majorana separation. We provide a rigorous mathematical framework that includes six theorems and a lemma, proving fidelity bounds, no entropy increase under ideal QND parity measurement under quantum non-demolition (QND) measurements, and compliance with the no-cloning theorem. We demonstrate that all correction operations lie within the Clifford group, enabling efficient, fault-tolerant implementation. Furthermore, we outline a scalable architecture for multi-qubit teleportation and relate our framework to recent experimental advances in quantum-dot-based Kitaev chains and superconducting nanowire platforms. These results position Majorana-based teleportation as a thermodynamically stable and experimentally viable approach to scalable quantum information transfer. All operations discussed are Clifford-only; achieving universality requires non-Clifford resources and lies outside our scope.

DOI
10.3390/quantum7030042

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Majorana Zero Modes Enable Topologically Protected Quantum Gates — E8 Intelligence Research

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FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Kitaev chain, with topological protection scaling exponentially in wire length, enabling topologically fault-tolerant quantum gates. | MATH: Kitaev chain Hamiltonian \( H = \sum_j [-t(c_j^\dagger c_{j+1} + h.c.) - \mu(c_j^\dagger c_j - 1/2) + \Delta(c_j c_{j+1} + h.c.)] \); Majorana operators \( \gamma_{2j-1} = c_j + c_j^\dagger \), \( \gamma_{2j} = -i(c_j - c_j^\dagger) \), satisfying \( \{\gamma_a, \gamma_b\} = 2\delta_{ab} \); topological gap \( \Delta_{\text{top}} \sim \Delta \) (superconducting gap); exponential suppression of coupling between MZMs: \( \epsilon \propto e^{-L/\xi} \) where \( \xi = \hbar v_F / \Delta \) is the coherence length; braid group \( B_n \) generated by exchanges \( \sigma_i \) with \( \sigma_i^2 = -1 \) (fermionic parity), yielding non-Abelian representation \( \rho(\sigma_i) = e^{\pm \pi \gamma_i \gamma_{i+1}/4} \). | CONNECTION: The braid group \( B_n \) is the fundamental Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Braiding Majorana Zero Modes in 2D Topological Superconductors: Clifford Gate Protection and Interferometric Falsifiability

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We analyze the topological protection of a Clifford gate set generated by braiding four Majorana zero modes (MZMs) in a two-dimensional class-D topological superconductor. Encoding a single logical qubit in four MZMs bound to $\pi$-flux vortices, we show that elementary exchanges generate the single-qubit Clifford group up to a global phase. The residual gate error arises from finite-size Majorana hybridization and quasiparticle poisoning, scaling as $\epsilon \sim \exp(-L/\xi)$, where $L$ is the inter-vortex separation and $\xi$ is the superconducting coherence length. We propose an interferometric measurement scheme to falsify this topological protection. By threading an Aharonov-Bohm loop around two MZMs, the conductance exhibits $4\pi$-periodic oscillations with phase jumps of $\pi/2$ per braid. Any deviation from this quantized phase beyond the $\exp(-L/\xi)$ envelope at large $L/\xi$ falsifies the topological protection mechanism, indicating gap closing or quasiparticle poisoning. ## 1. Introduction Topological quantum computation seeks to process information non-locally to achieve intrinsic fault tolerance. In two-dimensional topological superconductors belonging to symmetry class D (such as a $p_x + i p_y$ paired state or a proximitized semiconductor heterostructure), Majorana zero modes (MZMs) bind to $\pi$-flux vortices. Exchanging these vortices implements non-Abelian unitaries on the degenerate ground-state manifold. The central question addressed here is whether braiding four MZMs generates a topologically protected Clifford gate set with a residual error rate scaling as $\exp(-L/\xi)$, and whether this claim is falsifiable via interferometric measurement of the braiding phase. ## 2. Background The non-Abelian statistics of MZMs in class-D superconductors were established by Ivanov (2001), demonstrating that exchanging vortices realizes a braid group representation with $\pi/2$ rotations in Majorana Hilbert space [arXiv:cond-mat/0105040]. The topologi

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Non-Abelian Majorana Modes Enable Topologically Protected Quantum Gates — E8 Intelligence Research

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Majorana parity qubit in coupled minimal Kitaev chains

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Majorana zero modes provide a route to fault-tolerant qubits by encoding information non-locally in fermion parity. Their sensitivity to noise is expected to decrease exponentially with increasing separation between the Majoranas, a suppression known as topological protection. Kitaev chains engineered in quantum dot-superconductor arrays provide a tunable platform in which separated Majorana zero modes can emerge at the ends of the chain, even in two-site chains. These minimal-chain modes are known as poor man's Majoranas and retain characteristic Majorana properties, including near-zero energy and equal electron-hole character, but have only limited protection. A key outstanding challenge is to move beyond identifying such modes in electrical transport measurements and achieve coherent qubit control in the time domain. Here, we demonstrate a Majorana parity qubit by realizing coherent coupling between two-site Kitaev chains. Since total fermion parity is conserved, the system separates into global even and odd parity manifolds. We observe coherent parity oscillations in both manifolds with equal oscillation frequencies at the Majorana sweet spot, as predicted for isolated Majorana zero modes. We further show that the oscillation frequency and coherence depend systematically on inter-chain coupling and quantum-dot detunings, in close agreement with our model for short, partially protected chains. Our results establish the first coherent control of a Majorana qubit, encoded in the fermion parity of Majorana zero modes in minimal Kitaev chains.

cond-mat.mes-hallCross-listedarXiv:2607.09511•Jul 13, 2026•0 citations
We present a topologically protected teleportation protocol based on projective parity measurements between spatially separated Majorana zero modes (MZMs), eliminating the need for dynamic braiding. Unlike conventional teleportation schemes, our method preserves logical information through nonlocal encoding and suppresses decoherence exponentially with Majorana separation. We provide a rigorous mathematical framework that includes six theorems and a lemma, proving fidelity bounds, no entropy increase under ideal QND parity measurement under quantum non-demolition (QND) measurements, and compliance with the no-cloning theorem. We demonstrate that all correction operations lie within the Clifford group, enabling efficient, fault-tolerant implementation. Furthermore, we outline a scalable architecture for multi-qubit teleportation and relate our framework to recent experimental advances in quantum-dot-based Kitaev chains and superconducting nanowire platforms. These results position Majorana-based teleportation as a thermodynamically stable and experimentally viable approach to scalable quantum information transfer. All operations discussed are Clifford-only; achieving universality requires non-Clifford resources and lies outside our scope.
Quantum error correction is widely considered to be an essential ingredient for overcoming decoherence and achieving large-scale quantum computation. Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we demonstrate in a superconducting quantum processor that the spin-mapped version of the Majorana zero modes can be used to perform quantum teleportation. The teleportation transfers the quantum state encoded on two-qubit Majorana zero mode states between two Kitaev chains, using only braiding operations. The Majorana encoding is a quantum-error-detecting code for phase flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from $70.76 \pm 0.35 \% $ to 84.60±0.11%84.60 \pm 0.11 \%84.60±0.11%, well beyond the classical bound in either case.
Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we perform a quantum simulation of topological quantum computing, by teleporting a qubit encoded in the Majorana zero modes of a Kitaev chain. The quantum simulation is performed by mapping the Kitaev chain to its equivalent spin version and realizing the ground states in a superconducting quantum processor. The teleportation transfers the quantum state encoded in the spin-mapped version of the Majorana zero mode states between two Kitaev chains. The teleportation circuit is realized using only braiding operations and can be achieved despite being restricted to Clifford gates for the Ising anyons. The Majorana encoding is a quantum error detecting code for phase-flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from 70.76±0.35% to 84.60±0.11%, well beyond the classical bound in either case.
FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Kitaev chain, with topological protection scaling exponentially in wire length, enabling topologically fault-tolerant quantum gates. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)−μ(cj†cj−1/2)+Δ(cjcj+1+h.c.)] H = \sum_j [-t(c_j^\dagger c_{j+1} + h.c.) - \mu(c_j^\dagger c_j - 1/2) + \Delta(c_j c_{j+1} + h.c.)] H=∑j​[−t(cj†​cj+1​+h.c.)−μ(cj†​cj​−1/2)+Δ(cj​cj+1​+h.c.)]; Majorana operators γ2j−1=cj+cj† \gamma_{2j-1} = c_j + c_j^\dagger γ2j−1​=cj​+cj†​, γ2j=−i(cj−cj†) \gamma_{2j} = -i(c_j - c_j^\dagger) γ2j​=−i(cj​−cj†​), satisfying {γa,γb}=2δab \{\gamma_a, \gamma_b\} = 2\delta_{ab} {γa​,γb​}=2δab​; topological gap Δtop∼Δ \Delta_{\text{top}} \sim \Delta Δtop​∼Δ (superconducting gap); exponential suppression of coupling between MZMs: ϵ∝e−L/ξ \epsilon \propto e^{-L/\xi} ϵ∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F / \Delta ξ=ℏvF​/Δ is the coherence length; braid group Bn B_n Bn​ generated by exchanges σi \sigma_i σi​ with σi2=−1 \sigma_i^2 = -1 σi2​=−1 (fermionic parity), yielding non-Abelian representation ρ(σi)=e±πγiγi+1/4 \rho(\sigma_i) = e^{\pm \pi \gamma_i \gamma_{i+1}/4} ρ(σi​)=e±πγi​γi+1​/4. | CONNECTION: The braid group Bn B_n Bn​ is the fundamental Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We analyze the topological protection of a Clifford gate set generated by braiding four Majorana zero modes (MZMs) in a two-dimensional class-D topological superconductor. Encoding a single logical qubit in four MZMs bound to π\piπ-flux vortices, we show that elementary exchanges generate the single-qubit Clifford group up to a global phase. The residual gate error arises from finite-size Majorana hybridization and quasiparticle poisoning, scaling as ϵ∼exp⁡(−L/ξ)\epsilon \sim \exp(-L/\xi)ϵ∼exp(−L/ξ), where LLL is the inter-vortex separation and ξ\xiξ is the superconducting coherence length. We propose an interferometric measurement scheme to falsify this topological protection. By threading an Aharonov-Bohm loop around two MZMs, the conductance exhibits 4π4\pi4π-periodic oscillations with phase jumps of π/2\pi/2π/2 per braid. Any deviation from this quantized phase beyond the exp⁡(−L/ξ)\exp(-L/\xi)exp(−L/ξ) envelope at large L/ξL/\xiL/ξ falsifies the topological protection mechanism, indicating gap closing or quasiparticle poisoning. ## 1. Introduction Topological quantum computation seeks to process information non-locally to achieve intrinsic fault tolerance. In two-dimensional topological superconductors belonging to symmetry class D (such as a px+ipyp_x + i p_ypx​+ipy​ paired state or a proximitized semiconductor heterostructure), Majorana zero modes (MZMs) bind to π\piπ-flux vortices. Exchanging these vortices implements non-Abelian unitaries on the degenerate ground-state manifold. The central question addressed here is whether braiding four MZMs generates a topologically protected Clifford gate set with a residual error rate scaling as exp⁡(−L/ξ)\exp(-L/\xi)exp(−L/ξ), and whether this claim is falsifiable via interferometric measurement of the braiding phase. ## 2. Background The non-Abelian statistics of MZMs in class-D superconductors were established by Ivanov (2001), demonstrating that exchanging vortices realizes a braid group representation with π/2\pi/2π/2 rotations in Majorana Hilbert space [arXiv:cond-mat/0105040]. The topologi
FINDING: Majorana zero modes (MZMs) in topological superconductors obey non-Abelian braid statistics, with exponential suppression of coupling between spatially separated modes enabling topologically protected quantum gates. | MATH: The Kitaev chain Hamiltonian H=−t∑ci†ci+1+Δ∑cici+1+h.c.−μ∑ci†ci H = -t\sum c^\dagger_i c_{i+1} + \Delta\sum c_i c_{i+1} + \text{h.c.} - \mu\sum c^\dagger_i c_i H=−t∑ci†​ci+1​+Δ∑ci​ci+1​+h.c.−μ∑ci†​ci​ yields unpaired Majorana operators γ1,γ2 \gamma_1, \gamma_2 γ1​,γ2​ with γi†=γi \gamma_i^\dagger = \gamma_i γi†​=γi​, {γi,γj}=2δij \{\gamma_i,\gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​. Braiding exchanges γi→γj \gamma_i \to \gamma_j γi​→γj​ via unitary Uij=exp⁡(π4γiγj) U_{ij} = \exp(\frac{\pi}{4}\gamma_i\gamma_j) Uij​=exp(4π​γi​γj​), giving non-Abelian representation of the braid group Bn B_n Bn​ (e.g., U12U23≠U23U12 U_{12}U_{23} \neq U_{23}U_{12} U12​U23​=U23​U12​). Splitting energy ΔE∝e−L/ξ \Delta E \propto e^{-L/\xi} ΔE∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F/\Delta ξ=ℏvF​/Δ is the coherence length — exponential suppression with wire length L L L. | CONNECTION: The braid group Bn B_n Bn​ is generated by transpositions σi \sigma_i σi​ satisfying \( \sigma_i\s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Majorana zero modes provide a route to fault-tolerant qubits by encoding information non-locally in fermion parity. Their sensitivity to noise is expected to decrease exponentially with increasing separation between the Majoranas, a suppression known as topological protection. Kitaev chains engineered in quantum dot-superconductor arrays provide a tunable platform in which separated Majorana zero modes can emerge at the ends of the chain, even in two-site chains. These minimal-chain modes are known as poor man's Majoranas and retain characteristic Majorana properties, including near-zero energy and equal electron-hole character, but have only limited protection. A key outstanding challenge is to move beyond identifying such modes in electrical transport measurements and achieve coherent qubit control in the time domain. Here, we demonstrate a Majorana parity qubit by realizing coherent coupling between two-site Kitaev chains. Since total fermion parity is conserved, the system separates into global even and odd parity manifolds. We observe coherent parity oscillations in both manifolds with equal oscillation frequencies at the Majorana sweet spot, as predicted for isolated Majorana zero modes. We further show that the oscillation frequency and coherence depend systematically on inter-chain coupling and quantum-dot detunings, in close agreement with our model for short, partially protected chains. Our results establish the first coherent control of a Majorana qubit, encoded in the fermion parity of Majorana zero modes in minimal Kitaev chains.