Observing and braiding topological Majorana modes on programmable quantum simulators

The advent of coherent, controllable many-body quantum systems with dozens to hundreds of qubits has enabled digital quantum simulators that may outperform classical computation for quantum many-body problems, including topological phases of matter. Majorana fermions—non-Abelian excitations that are their own antiparticles—offer a route to fault-tolerant, topological quantum computation, but solid-state realizations face challenges from disorder, limited control, and confusion with trivial zero-energy states. Quantum simulators, with precise parameter control, provide an alternative platform to realize and probe such phases, including Floquet-driven topological phases. Prior signatures of topological modes have been observed in photonics and programmable processors, but detailed, qualitative probing of the excitations—especially the robust identification of edge-localized Majorana modes and their braiding—has remained difficult. The present work sets out to quantitatively simulate a 1D topological superconductor on noisy superconducting quantum hardware, to detect and characterize Majorana modes via Fourier analysis of multi-qubit observables, to distinguish them from trivial localized modes using two-point correlations, and to demonstrate a practical non-adiabatic braiding protocol compatible with current noise levels.

Early proposals for Majorana-based quantum memories in solid-state platforms encountered difficulties due to disorder, lack of control, and the presence of trivial zero-energy states masquerading as Majoranas. Photonic experiments and digital quantum processors have reported signatures of topological modes in non-equilibrium settings, though often limited by system size and not directly probing full topological phases. Floquet engineering has emerged as a powerful approach to realize robust topological phases with periodic drives and to mitigate resource depth on digital hardware. Prior studies often used methods tailored to free-fermion models and provided indirect signatures; comprehensive detection, structural characterization of Majorana edge modes, and demonstration of their braiding statistics in a larger, programmable device remained open challenges. This work builds on these developments by combining Floquet dynamics, Jordan–Wigner mappings, and tailored measurement protocols to directly reconstruct Majorana mode wavefunctions, differentiate them from trivial localized modes, and implement a non-adiabatic braiding operation.

The target model is a periodically driven (Floquet) spin system equivalent to a 1D topological superconductor under Jordan–Wigner transformation. The time-periodic Hamiltonian H(t)=∑{j=1}^N (J(t) X_j X{j+1} + λ(t) Z_j Z_{j+1}) + h(t) ∑_{j=1}^N Z_j is implemented in three sequential segments per period T: (i) on-site field h during [0,T1]; (ii) XX coupling J during [T1,T2]; (iii) ZZ coupling λ during [T2,T]. This yields a Floquet unitary U_F that factorizes into layers of on-site Z rotations, nearest-neighbor XX evolutions, and nearest-neighbor ZZ evolutions with angles θ=h T1, ϕ=J (T2−T1), φ=λ (T−T2). The circuit uses local single- and two-qubit gates in constant depth per Floquet cycle and maps efficiently to native gates on IBM devices (CNOT plus single-qubit rotations for special angles such as π/4, otherwise two CNOTs for general angles).

• Verified detection of Majorana edge modes via Fourier analysis: Measuring boundary-localized observables and computing F_j(ω) across Floquet cycles reveals clear peaks at ω=0 (MZM) and ω=π (MPM). Parameter sweeps show transitions between MZM, trivial, and MPM phases.
• Reconstruction of Majorana wavefunctions: Using F_j(ω) over sites and representations, the spatial profiles of MZMs and MPMs were recovered on 10-qubit chains. Non-interacting cases match theory closely; interacting cases (added ZZ term) show increased noise yet retain localized structure.
• Distinguishing topological from trivial modes: A two-point correlation function T_{μν} over random product initial states differentiates unpaired Majorana modes localized at opposite ends (yielding near-zero T_{12}) from trivial localized modes near the same boundary (yielding positive T_{12}). An engineered trivial scenario (decoupled end qubits) was contrasted against a topological setting to validate the diagnostic.
• Non-adiabatic braiding (FAS): Introduced and implemented a fast approximate swap protocol that exchanges left and right Majorana modes in 1D without deep adiabatic evolution. Experiments on 5-qubit chains measured wavefunctions before and after FAS, observing the expected exchange and relative sign change consistent with non-Abelian braiding statistics.
• Hardware and parameters: Experiments used IBM 27-qubit devices (ibm_hanoi, ibm_montreal, ibm_mumbai, ibm_toronto); tomography typically on 10 qubits; frequency-resolved boundary oscillations on 21 qubits; braiding on 5 qubits due to depth sensitivity. Floquet cycles D=11 and D=21 were employed. Each data point averaged over 8192 shots; decay-compensated rescaling with Γ extracted from data (e.g., Γ≈0.0328 for MZM, 0.0376 for MPM, 0.120 with added ZZ) highlighted long-lived Majorana signals over many cycles.
• Robustness and lifetime: In interacting regimes, Majorana observables decay with a characteristic lifetime τ; nevertheless, persistent boundary signals over dozens of cycles were observed, enabling reliable mode extraction and validation on noisy hardware.

The study addresses the central challenge of conclusively identifying and manipulating topological Majorana modes on noisy, programmable quantum hardware. By leveraging Floquet engineering and boundary-focused measurements, the authors obtain frequency-resolved signatures and reconstruct spatial mode profiles, confirming edge localization and distinguishing topological from trivial excitations. The two-point correlation diagnostic provides a practical tool to detect zero-frequency modes and classify their nature based on spatial correlations. The fast approximate swap (FAS) demonstrates non-adiabatic braiding compatible with current device depths, yielding exchange statistics consistent with Majorana non-Abelian behavior. The framework generalizes to continuous-time static Hamiltonians by replacing discrete with continuous Fourier analysis and points toward scalable simulations of topological matter, including nanowire models and higher-dimensional phases, as hardware connectivity and coherence improve.

This work demonstrates end-to-end detection, verification, and braiding of topological Majorana modes on superconducting quantum processors via Floquet simulation. Key contributions include: (i) a constant-depth-per-cycle protocol realizing a 1D topological superconductor; (ii) Fourier-based reconstruction of Majorana wavefunctions and phase identification; (iii) a correlation-based method to distinguish topological from trivial zero modes; and (iv) a non-adiabatic FAS braiding implementation validating exchange statistics on current hardware. Future directions include optimizing parameters to enhance Majorana lifetimes (e.g., MBL-assisted stabilization without inducing trivial phases), extending to static Hamiltonian simulations and nanowire models with spinful ladders, exploring higher-connectivity devices for 2D topological phases, and applying the wavefunction-extraction techniques to local integrals of motion in many-body localized systems.

• Finite lifetimes due to Floquet heating limit the number of usable cycles; interactions and bulk dispersion reduce τ.
• Noise (decoherence, gate and measurement errors) breaks fermionic parity symmetry, undermining topological protection and contributing to signal decay.
• Disorder-assisted stabilization (MBL) may require disorder strengths that risk driving the system into trivial phases, necessitating careful parameter choices.
• Circuit depth constraints restrict system size and the feasibility of fully adiabatic braiding; FAS mitigates but remains approximate.
• Experiments focus on boundary phenomena in 1D Floquet settings; generalization to larger systems and 2D phases awaits higher connectivity and coherence.