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arXiv:2603.03645

Resource-Efficient Emulation of Majorana Zero Mode Braiding on a Superconducting Trijunction

Rahul Singh, Weixin Lu, Kaelyn J Ferris, Javad Shabani

quant-phSubmitted Mar 4, 2026• Announced Mar 5, 2026• 0 citations
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Abstract

Topological superconductivity could host quasiparticles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana Zero Modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.

Comments
11 pages, 7 figures

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Topological superconductivity could host quasiparticles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana Zero Modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.
Topological superconductivity could host quasi-particles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana zero modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.
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.
The realization and manipulation of Majorana zero modes have drawn significant attention for their crucial role in enabling topological quantum computation. Conventional approaches to the braiding of Majorana zero modes rely on adiabatic processes. In this work, using a composite 2-Kitaev-chain system accommodating Majorana zero modes as a working example, we propose a nonadiabatic and non-Abelian geometry phase-based protocol to execute operations on these Majorana zero modes. This is possible by locally coupling the edge sites of both quantum chains with an embedded lattice defect, successfully simulating the braiding operation of two Majorana modes in a highly nonadiabatic fashion. To further enhance the robustness against control imperfections, we apply a multiple-pulse composite strategy to our quantum chain setting for second-order protection of the braiding operations. Our proposal can also support the fast and robust realization of the $\ensuremath{\pi}/8$ gate, an essential ingredient for universal quantum computation. This work hence offers a potential pathway toward the nonadiabatic and fault-tolerant control of Majorana zero modes.
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
Majorana zero modes have emerged as one of the most promising platforms for topological quantum computation, since their non-Abelian braiding statistics allow quantum information to be encoded nonlocally and manipulated through braiding operations that are, in principle, protected against local perturbations. In practice, however, a braid is only as robust as its physical implementation: finite-time operation, residual couplings, and environmental noise can all convert local excitations into logical errors during the exchange process. Here, we address this question through a microscopic comparison of two representative braiding architectures, a nanowire trijunction and a quantum-dot-assisted setup, simulating the full time-dependent Bogoliubov--de Gennes dynamics under both noiseless and noisy conditions. We show that the dot-assisted architecture consistently achieves a lower error over a shorter timescale than the trijunction, owing to its more localized exchange mechanism. This advantage persists in the presence of noise, and a spatially resolved analysis further reveals that, in the dot-assisted geometry, fast noise localized on the dot produces a smaller error than equivalent noise on the wires, whereas slow, quasi-static noise on the dot becomes the dominant limitation. Taken together, these findings link the different error contributions directly to device geometry, pointing to concrete design principles for noise-resilient Majorana-based quantum gates.
Majorana-based topological qubits are expected to exploit the nonabelian braiding statistics of Majorana modes in topological superconductors to realize fault-tolerant topological quantum computation.Scalable qubit designs require several Majorana modes localized on quantum wires networks, with braiding operations relying on the presence of the groundstate degeneracy of the topologically nontrivial superconducting phase.However, this degeneracy is lifted due to the hybridization between Majorana modes localized at a finite distance.Here, we describe a braiding protocol in a trijunction where each branch consists of a lattice of Majorana modes overlapping at a finite distance.We find that the energy splitting between the groundstate and the lowest-energy state decreases exponentially with the number of Majorana modes if the system is in its topologically nontrivial regime.This result does not rely on the specific braiding geometry and on the details of the braiding scheme but is a consequence of the supersymmetry and nontrivial topology of the effective low-energy Hamiltonian describing the Majorana lattice.