Topological qubits encode information nonlocally using the global properties of a quantum system, promising intrinsic resilience to local noise. Here we survey key theoretical proposals and experimental advances, focusing on Majorana zero modes in nanowires, planar heterostructures, and progress toward braiding operations.

Theoretical Foundations

  • Kitaev’s 2001 wire model: Alexei Kitaev showed that a 1D p-wave superconducting chain hosts unpaired Majorana zero modes (MZMs) at its ends, forming a nonlocal qubit basis [1].
  • Fu–Kane proposal (2008): Liang Fu and C.L. Kane demonstrated that interfacing an s-wave superconductor with a 3D topological insulator surface yields localized MZMs at vortices [2].
  • Non-Abelian anyon formalism: Freedman, Kitaev, Larsen & Wang (2002) and Nayak et al. (2008) formalized how braiding MZMs implements fault‑tolerant gates in a topological quantum computer [3, 4].

Experimental Signatures of Majorana Modes

  • Hybrid nanowires (2012): Mourik et al. observed zero‑bias conductance peaks in InSb–Al nanowires under magnetic fields, consistent with MZMs [5].
  • Full‑shell nanowires (2020): Vaitiekėnas et al. demonstrated flux‑induced topological phase transitions in Al‑InAs core–shell wires, showing robust zero‑bias peaks over multiple flux lobes [6].
  • 2D planar heterostructures (2023): Aghaee et al. reported InAs–Al devices passing the “topological gap” protocol, mapping closing and reopening of induced superconducting gaps and localizing MZMs at wire ends [7].

Toward Braiding and Fusion

  • Cavity‑controlled braiding (2024 preprint): Quiroga et al. propose using microwave cavities to split chains and observe fusion rules and braiding of MZMs via photon parity measurements [8].

Beyond Majoranas: Parafermions and Fibonacci Anyons

  • Fractional quantum Hall platforms: Proposals by Lindner et al. (2012) and Clarke et al. (2013) outline realizing parafermionic modes in ν=2/3 quantum Hall edges coupled to superconductors, enabling richer anyon statistics and universal topological gates [9].

Challenges and Outlook

Despite clear zero‑bias peaks and gap signatures, definitive braiding remains to be demonstrated experimentally. Key hurdles include precise control over quasiparticle poisoning, disorder management in heterostructures, and scalable networks of nanowire junctions. Ongoing efforts target rapid, fault‑tolerant braid operations in multiterminal devices and integration with superconducting circuit readout.

References

[1] Kitaev, A. Y. (2001). Unpaired Majorana fermions in quantum wires. Physics-Uspekhi, 44(10S), 131-136.

[2] Fu, L., & Kane, C. L. (2008). Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator. Physical Review Letters, 100(9), 096407.

[3] Freedman, M., Kitaev, A., Larsen, M., & Wang, Z. (2002). Topological quantum computation. Bulletin of the American Mathematical Society, 40(1), 31-38.

[4] Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Das Sarma, S. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics, 80(3), 1083-1159.

[5] Mourik, V., et al. (2012). Signatures of Majorana Fermions in Hybrid Superconductor–Semiconductor Nanowire Devices. Science, 336(6084), 1003-1007.

[6] Vaitiekėnas, S., et al. (2020). Flux-induced topological superconductivity in full-shell nanowires. Science, 367(6485), eaav3392.

[7] Aghaee, M., et al. (2023). InAs–Al hybrid devices passing the topological gap protocol. Physical Review B, 107(23), 235414.

[8] Quiroga, L., et al. (2024). Cavity Control of Topological Qubits: Fusion Rule, Anyon Braiding and Majorana‑Schrödinger Cat States. arXiv:2409.04515.

[9] Lindner, N. H., et al. (2012). Fractionalizing Majorana Fermions: Non-Abelian Statistics on the Edges of Abelian Quantum Hall States. Physical Review X, 2(4), 041002.