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arXiv:2503.15405 · Physical review. B./Physical review. B

Simulated non-Abelian statistics of Majorana zero modes from a Kitaev lattice

Foster Sabatino, Matthew Brooks, Charles Tahan, Silas Hoffman

Published Jul 28, 2025• 1 citation
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Abstract

We simulate the non-Abelian exchange of Majorana zero modes (MZMs) on a quantum computer. Rather than utilizing MZMs at the boundaries of quantum Ising chains, which are typically represented as nonlocal operators on a quantum computer, using a Kitaev lattice allows us to exploit a local representation of MZMs. We detail the protocol for braiding two and four MZMs in terms of a spin Hamiltonian, i.e. physical qubit Hamiltonian. Projecting this onto a subspace of states, we extract an effective Hamiltonian which drives a non-Abelian Berry phase. Using several approximations, we construct a set of gates which mimics this accumulation of non-Abelian phase and process this construction on a quantum computer. For two and four MZMs, we realize braiding fidelities of approximately 85 and 47%, respectively.

DOI
10.1103/4qt9-5gjy

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We simulate the non-Abelian exchange of Majorana zero modes (MZMs) on a quantum computer. Rather than utilizing MZMs at the boundaries of quantum Ising chains, which are typically represented as nonlocal operators on a quantum computer, using a Kitaev lattice allows us to exploit a local representation of MZMs. We detail the protocol for braiding two and four MZMs in terms of a spin Hamiltonian, i.e. physical qubit Hamiltonian. Projecting this onto a subspace of states, we extract an effective Hamiltonian which drives a non-Abelian Berry phase. Using several approximations, we construct a set of gates which mimics this accumulation of non-Abelian phase and process this construction on a quantum computer. For two and four MZMs, we realize braiding fidelities of approximately 85 and 47%, respectively.
We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as sss-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by Z2{\mathbb{Z}}_{2}Z2​ invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by MZM hybridization. Numerically simulating the time-dependent Bogoliubov--de Gennes equation without assuming a priori existence of MZMs, we examine quantum noises on the unitary operators of non-Abelian braiding dynamics due to interactions with neighboring MZMs and other quasiparticle states. We demonstrate that after the interchange of two vortices, the lowest vortex-bound states accumulate the geometric phase $\ensuremath{\pi}/2$, and errors stemming from dynamical phases are negligibly small, irrespective of interactions of MZMs. Furthermore, we numerically simulate the braiding dynamics of four vortices in two-dimensional topological superconductors, and discuss an optimal braiding condition for realizing the high performance of non-Abelian statistics and quantum gate operations of Majorana-based qubits.
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.
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
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.
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.
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.