Researchers from the Microsoft Azure Quantum team have built a quantum computing chip using topological qubits. Such machines are much more robust against local noise and decoherence and can therefore be more easily scaled up to larger systems. Their Nature article describes the experiment with two parallel topological nanowires, bridged by a trivial superconducting wire (forming an H-shaped device). A Majorana quasiparticle (also called a Majorana Zero Mode, or MZM for short) is formed at each end of the topological nanowires.
Microsoft has published intermediate results, but this is not yet definitive proof of topological qubits. If this breakthrough holds up, it could enable researchers to build larger chips with more qubits much faster than expected—possibly within years rather than decades.
In this article, I want to give some background information for everyone to understand the basics of Majorana, and to be able to follow the articles written about it and give you some handles that can help you to stay skeptical and have an open mind to what people are saying on this topic. I start with giving some basics to understand what Majoranas actually are and how they differ from the particles we already know. After that, I will go more in-depth on the topological conductor and why it is made with nanowires. The Microsoft team also published a roadmap of what the future might bring on topological qubit arrays
1. What are Majoranas?
In 1937, Italian physicist Ettore Majorana introduced a relativistic wave equation, which is now known as the Majorana equation. This equation was designed to describe fermions that are their own antiparticles, a concept that led to the identification of Majorana particles. While these particles were initially thought of in the context of relativistic particles, the term has since been expanded to include any fermion that is its own antiparticle, meaning it is electrically neutral.
In addition to particle physics, the concept of Majorana particles has found an exciting application in condensed matter physics, where Majorana modes have been theorized as emergent quantum states in systems such as superconductors. These modes arise from collective electron interactions and are distinct from typical fermions. Unlike ordinary fermions, Majorana modes are chargeless, spinless, zero-energy quasi-particles that exist as topological states in the material. Their unique properties make them especially resilient to local noise, as they are protected by topological symmetries. This means that the Majorana states are separated energetically from the bulk states, making it impossible to shift them from their zero-energy state, thus providing stability against disturbances.
The development of Majorana modes in condensed matter systems has generated considerable interest due to their potential applications in topological quantum computing, where the resilience of Majorana states could allow for more stable qubits. While experimental confirmation of these modes remains a challenge, their theoretical and practical implications continue to drive much research in both particle physics and condensed matter theory.
1.1 The Self-Antiparticle: Majorana Modes
\[ [\gamma_i, \gamma_j] = 2\delta_{ij}\]
where γi,j is simultaneously the Majorana creation and annihilation operator in second quantization.This makes Majoranas different from bosons and fermions, as they uniquely behave as particles that are their own antiparticles. Due to this, it is possible to stack Majoranas in the same quantum state, similar to bosons, but they will annihilate each other in pairs. Therefore, after the annihilation, we are left with either zero or one Majorana at a definite quantum state, similar to fermions. This means that majoranas can only coexist if they are completely uncoupled, as overlapping or interaction between two of these quasi-particles will result in annihilation.
2. Majoranas in solid state physics
S. R. Elliot and M. Franz [3] demonstrated that any system of electrons can be described using Majorana fermion states. This is done by expressing conventional fermionic states as symmetric and antisymmetric combinations of particle and hole states:
\[|\gamma_1\rangle = \frac{1}{\sqrt{2}}(|E\rangle +|-E\rangle)\]
\[|\gamma_2\rangle = \frac{1}{\sqrt{2}}(|E\rangle -|-E\rangle)\]
Here, |E〉 represents a fermionic state with energy E (a particle state), and |-E〉 represents its negative-energy counterpart (a hole state). The states |γ1〉 and |γ2〉 are Majorana states, which are formed by combining these particle and hole states in a way that makes them real.
This transformation to Majorana fermion states naturally appears in many-body electron systems due to the Pauli exclusion principle, which prevents two electrons from occupying the same quantum state. As a result, electrons fill available energy levels up to the Fermi energy, which is typically set as the reference point (zero energy). States with positive energy represent excitations where an additional electron is introduced, while negative energy states correspond to the removal of an electron from the filled Fermi sea, effectively creating a hole.
In general, these transformations provide a different perspective but are not always useful for analyzing a system’s behavior. Majorana states remain stationary only at zero energy, and in typical systems, their wavefunctions overlap, making them hard to distinguish as separate states. However, in specific systems like 1D Majorana nanowires, zero-energy Majorana states can exist while being spatially separated—one at each end of the wire. In such cases, the Majorana basis offers the most natural and accurate description of the system’s physics.
2.1 Majorana States in Superconductors
The previous two equations (which define Majorana states as superpositions of electron and hole states) resemble the structure of Bogoliubov quasiparticles, which arise in superconductors. In a superconductor, electron and hole states mix due to Cooper pairing, meaning that excitations are no longer purely electron-like or hole-like but a combination of both. This structure suggests that Majorana fermions can be thought of as equal superpositions of an electron and its corresponding hole state—something that superconductivity naturally enables.
Topological states are protected by fundamental symmetries, meaning they cannot be easily changed or destroyed [6, 7, 8]. Each symmetry is a global property of the system, described by a special mathematical operator. One important symmetry in superconductors is particle-hole symmetry (PHS). This ensures that for every state with energy E, there is a corresponding state at -E. Because of this, Majorana states always appear at exactly zero energy and cannot be easily shifted or removed. This makes Majorana states very stable against local noise and disturbances. Their ability to stay at zero energy is one reason they are promising for topological quantum computing, where information is stored in a way that naturally resists errors [6, 9, 10].
In strictly one-dimensional (1D) systems with particle-hole symmetry (PHS), only two possible phases exist: normal superconductive state (trivial) and topological superconductive state (Majorana-supporting) [8]. As a result, a purely 1D nanowire cannot host more than one Majorana mode at each end. If two Majoranas were to form at the same end, they would mutually annihilate, preventing their existence.
This constraint explains why Majorana modes can only exist in spinless or effectively spinless nanowires. If spin degeneracy were present, two Majorana states would form at the same end, overlap, and combine into a regular fermionic state, lifting their energy from zero.
2.2 Key Properties of Majorana Modes
Majorana modes exhibit a mix of general topological properties and unique characteristics. Below is a summary of their defining features:
- Majoranas are edge states: Topological states typically emerge at the physical boundaries of a system or near imperfections and impurities. However, they are different from Anderson-localized states, which arise due to disorder in the system. In Anderson localization, the electron wavefunctions become trapped in small regions due to random impurities or irregularities, preventing them from conducting. In contrast, topological states are more stable and robust. In 1D and quasi-1D nanowires, Majorana modes specifically appear at the ends of the nanowire.
- Majoranas are zero-energy states: In a system with particle-hole symmetry, every positive-energy state has a corresponding negative-energy state. To satisfy the condition that Majoranas are their own antiparticles, they must exist at zero energy, see Figure 4..
- Majorana modes are spinless: These modes are formed by an equal superposition of electron and hole states, which results in zero total spin. In 1D nanowires, they can emerge when spin degeneracy is broken by strong spin-orbit coupling, which locks the spin direction to the momentum direction. Without spin degeneracy, Majoranas can appear at opposite edges of the system. However, if spin degeneracy is present, two Majoranas would overlap at the same edge, no longer preserving their zero-energy nature, and instead behave like regular fermions with nonzero energy.
- Majoranas exist inside an energy gap: Like all topological states, Majorana modes are protected by an energy gap, ensuring they remain isolated from non-topological bulk states. This gap prevents external noise from introducing unwanted dynamics and helps maintain the system in its topological phase. This is also shown in Figure 4, where the available states are colored blue, with a gap in between the layers.
- Majoranas appear in pairs: In a finite system, Majorana modes always come in pairs, each located at opposite edges. A single, unpaired Majorana can only exist in a semi-infinite system where the second Majorana is infinitely far away.
- Majoranas obey non-Abelian statistics: When you swap two Majoranas, the system doesn’t just return to its original state; instead, it acquires a phase that’s dependent on the sequence of exchanges. This is different from typical particles, where swapping two particles just results in a sign change or no change at all. This non-Abelian property is essential for topological quantum computation, as it enables robust quantum information storage and manipulation.
2.3 Device design
Superconductivity, combined with the Rashba and Zeeman effects, can lead to the formation of a topological superconductor. The Rashba effect, a spin-orbit interaction, links the spin of an electron to its momentum. This interaction helps break spin degeneracy, making the system favorable for Majorana modes. The Zeeman effect, caused by an external magnetic field, further splits energy levels, breaking spin degeneracy and enabling the formation of topological states. These effects explain why devices like these require an external magnetic field and materials with strong spin-orbit coupling, such as InSb or InAs. For example, Microsoft’s Majorana 1 device uses InSb, a semiconductor with strong spin-orbit coupling, deposited on the surface of an s-wave superconductor like Al.
In this configuration, superconductivity is induced in the semiconductor through the proximity effect, where Cooper pairs tunnel from the superconductor into the semiconductor. If the semiconductor’s dimensions are smaller than the coherence length of the Cooper pairs, the semiconductor effectively becomes a superconductor. However, the presence of spin in a superconductor causes energy bands to be doubly degenerate, preventing the formation of uncoupled Majorana modes. Despite this, a hybrid semiconductor-superconductor nanowire retains the strong spin-orbit coupling characteristic of the semiconductor material. The external magnetic field breaks time-reversal symmetry, and in combination with spin-orbit coupling, induces spin precession in the system’s eigenstates. This process converts the states into spinless modes, creating ideal conditions for Majorana modes to emerge.
Rather than using a single nanowire, two parallel topological superconducting wires are used, connected by a trivial superconducting nanowire at their midpoints. The Majorana zero modes emerge at the ends of the two nanowires. This H-shaped configuration is advantageous for 2D connectivity.
The device operates inside a superconducting magnet (1.8 Tesla) to lift the degeneracy and is maintained at extremely low temperatures (50 mK) to reduce errors and enhance stability. In topological quantum computing, qubits are protected from local noise, which is influenced by the topological gap (Δ) and temperature (T). The ratio of the topological gap to temperature (Δ/k_B T) needs to be large to ensure the system stays in the ground state. This minimizes error correction overhead, thereby improving system efficiency.
3. References
[1] R. K. Pathria and P. D. Beale, Statistical Mechanics, third edition. Elsevier, 2011.
[2] J. J. Sakurai and S. F. Tuan, Modern Quantum Mechanics, Revised edition. Prentice Hall, 1993
[3] S. R. Elliott and M. Franz, “Colloquium : Majorana fermions in nuclear, particle, and solid-state physics,” Rev. Mod. Phys., vol. 87, pp. 137–163, 2015.
[4] R. M. Lutchyn, J. D. Sau, and S. D. Sarma, “Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures,” Phys. Rev. Lett. vol. 105, 077001, 2010.
[5] R. Aguado, and L. Kouwenhoven, “Majorana qubits for topological quantum computing,” Physics Today, vol. 73 (6) pp. 44-50, 2020.
[6] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys., vol. 80, pp. 1083–1159, 2008.
[7] A. Kitaev, “Periodic table for topological insulators and superconductors,” AIP Conference Proceedings, vol. 1134, no. 1, pp. 22–30, 2009.
[8] A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Phys. Rev. B, vol. 55, pp. 1142–1161, 1997.
[9] J. K. Pachos, Introduction to Topological Quantum Computation. Cambridge University Press, 2012.
[10] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. Fisher, “Non-abelian statistics and topological quantum information processing in 1d wire networks,” Nat Phys, vol. 7, pp. 131–136, 2011.
[11] Sau, J., Simon, S., Vishveshwara, S. et al. From anyons to Majoranas. Nat Rev Phys 2, 667–668 (2020).
[12] Microsoft Quantum, “Roadmap to fault tolerant quantum computation using topological qubit arrays,” Arxiv, February 19, 2025, URL: https://arxiv.org/pdf/2502.12252.
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