Current State

Research into topological superconductors (TSCs) and Majorana fermions between 2024 and 2026 spans both theoretical foundations and experimental/device-level proposals. While fully conclusive experimental demonstrations of Majorana-based quantum computation remain elusive, substantial progress has been made in understanding the critical phenomena, braiding schemes, and computational capabilities of these platforms.

Key Approaches

Vortex-bound Majorana modes and braiding control. A central experimental challenge is the controlled manipulation of Majorana zero modes (MZMs) hosted in vortices of 2D TSCs. One detailed proposal addresses this directly by introducing a programmable spin-transfer torque (STT) matrix integrated with a TSC layer [AG-2024.04-2314]. Using finite element analysis combined with time-dependent Ginzburg-Landau and Bogoliubov-de Gennes equations, the authors numerically demonstrate deterministic vortex driving, braiding, and fusion, quantifying how vortex acceleration and finite MZM coupling affect braiding fidelity. This STT-matrix approach is argued to be scalable toward fault-tolerant topological quantum computation [AG-2024.04-2314].

Boundary criticality and Majorana surface modes. On the theoretical side, 2D correlated time-reversal-invariant TSCs have been investigated at quantum phase transitions using sign-problem-free determinant quantum Monte Carlo simulations [AG-2025.10-751]. This work maps the full boundary phase diagram—ordinary, special, and extraordinary transitions—and reveals that the boundary Majorana fermion at the special transition generates a new Gross-Neveu-Yukawa fixed point, with critical exponents computed to two-loop order. Iron-based superconductors are discussed as candidate experimental platforms [AG-2025.10-751].

Quantum Hall systems and Majorana conformal field theory. Fuzzy sphere regularization has been applied to resolve the long-standing question of the nature of the transition between the Halperin state and the Moore–Read Pfaffian in quantum Hall bilayers [AG-2025.09-817]. The critical theory is identified with the 3D gauged Majorana conformal field theory, with the transition driven by the closing of the neutral fermion gap, and operator content extracted in both integer and half-integer spin sectors—the first realization of a fermionic theory on the fuzzy sphere [AG-2025.09-817].

Beyond Majorana: parafermionic and non-Abelian platforms. A significant theoretical advance proposes a charge-4e TSC supporting parafermion zero modes rather than Majorana modes [AG-2026.02-669]. This platform, obtainable by proliferating vortex-antivortex pairs in a stack of two 2e p+ip TSCs or by melting a ν=2/3 quantum Hall state, traps ℤ₃ parafermion zero modes in vortex cores that encode qutrits. Braiding parafermion defects alone generates the full many-qutrit Clifford group, and a single-probe interferometric measurement enables universal magic-state preparation [AG-2026.02-669]. Complementary work predicts that tuning bandwidth in fractional Chern insulators at ν=2/3 can drive transitions to intrinsically non-Abelian superconductors supporting Majorana zero modes, with a rich landscape of exotic superconductors at more general fillings ν=p/(2p+1) [AG-2025.12-1132].

Superconducting circuit implementations. Microwave spectroscopy of a three-Josephson-junction circuit has demonstrated degeneracies indicative of intrinsic nontrivial topology, with three topological invariants identified—one linked to a hidden quantum mechanical supersymmetry—and phase diagrams shown to be robust to junction imperfections [AG-2024.01-1770]. Separately, superconducting quantum processors have been used to observe Thouless pumping and its interplay with disorder at the 41-qubit scale [AG-2024.01-1296] and second-order topological pumps with corner-localized states on 4×4 arrays [AG-2024.02-2023], establishing these processors as versatile platforms for topological physics.

Open Problems

Precise experimental braiding of vortex-bound MZMs in solid-state systems remains undemonstrated; the STT-matrix scheme [AG-2024.04-2314] is still a numerical proposal. The realization of parafermion zero modes in charge-4e TSCs [AG-2026.02-669] and non-Abelian TSCs emerging from fractional Chern insulators [AG-2025.12-1132] await experimental confirmation. The boundary critical phenomena predicted for iron-based superconductors [AG-2025.10-751] have not yet been directly measured.

Where to Read Next

For vortex braiding schemes, see [AG-2024.04-2314]; for parafermion-based universal computation, [AG-2026.02-669]; for boundary criticality and Majorana fixed points, [AG-2025.10-751]; for quantum Hall Majorana criticality, [AG-2025.09-817]; and for non-Abelian TSCs from fractional Chern insulators, [AG-2025.12-1132].

Caveat

The archive preprints are predominantly theoretical or quantum-simulation-based. Direct experimental reports of Majorana signatures in semiconductor nanowires, magnetic adatom chains, or iron-based superconductors—the most actively debated experimental systems in 2024–2026—are not represented in this shortlist, so the review reflects theoretical and device-proposal progress more than direct condensed-matter experimental measurements.