# In Search of Majorana

The long-awaited promise of topological quantum computing (TQC), a subject often discussed in the abstract as a possibility at some unspecified future date, suddenly seems imminent. Thanks in large part to an explosive revival of interest in a formerly obscure hypothetical entity called the Majorana fermion, a number of new and surprisingly practical proposed TQC designs are now in circulation, and benchtop experiments could begin very soon.

As nearly everyone knows by now, the anticipated benefits of an eventual quantum computer are both numerous and profound, from the ability to factor huge numbers for data security to ultra-fast searches of unstructured databases to finding the optimal ligand structure of a new drug molecule. But many of the key quantum processes under consideration as the basis for such a device are notoriously fragile, and error rates may far exceed the capabilities of correction protocols.

As a result, many quantum-information scientists – including an active group at JQI and the University of Maryland’s Condensed Matter Theory Center (CMTC) – have been exploring ways to realize a quantum computer that would be fault-tolerant at the hardware level because it manipulates information topologically.

Topology is the study of shapes with spatial properties that do not vary when an object is stretched, twisted or otherwise continuously deformed. (The classic illustrative example is the fact that a coffee cup and a donut are topologically identical because one can be transformed into the other without tearing, cutting or joining.) Information embodied in topological form would thus be robust in the face of local imperfections, perturbations or “noise” in the system. In TQC, data would be stored as shapes and patterns of shapes formed by quantum states, and performing an operation on the data would amount to changing those shapes and patterns in a controllable way.

Not surprisingly, it has been enormously difficult to imagine, let alone find, a physical system that has the requisite properties for TQC. Nor is it easy to identify an object that can be used to embody the kinds of durable patterns that TQC requires. However, over the past 20 years, scientists including JQI Fellow and CMTC Director Sankar Das Sarma have determined that certain kinds of exotic particle-like objects which arise in particular two-dimensional configurations at low temperature might fill the bill.

This effort has been accelerated recently by the realization that these candidate TQC states are, in fact, the same sort of entities predicted in 1937 by Italian theoretical physicist Ettore Majorana. In the course of working with Paul Dirac’s famous mathematical description of the electron – one of the class of particles called fermions – Majorana produced an equation that posited the existence of a special sort of particle-like state, equivalent in some respects to half of a fermion and serving as its own antiparticle. At that time, Majorana was interested in explaining the nature of an elusive particle named the “neutrino” by his friend and colleague Enrico Fermi. But now theorists have come to realize that the bizarre states needed for TQC involve “Majorana fermions.”

TQC cannot be realized in systems of either of the two canonical classes of particles that together make up all the ordinary stuff in the three-dimensional universe, bosons and fermions, because swapping them around or exchanging their order does not produce a topologically protected condition. Rather, TQC demands – and Majorana’s equation describes – an unusual sort of “quasiparticles,” that is, phenomena or excitation states that behave like, and can usefully be regarded as, particles.

Even then, only one special subset of 2D quasiparticles can serve to realize TQC. That type is called “non-Abelian,” a term that refers to mathematical operators that do not commute; that is, those for which A X B does not equal B X A. Thus in a non-Abelian scheme, the final state of the system depends critically on the order in which operations were performed. Obviously, this is a necessary property for TQC, which requires a situation in which swapping object A with object B in one fashion (for example, clockwise) produces one sort of durable topological pattern, but exchanging them in another way (such as counterclockwise) produces a recognizably different and equally robust shape. Topological computing would thus be a process of interchanging or “braiding” collections of such quasiparticle states.

Majorana fermions have precisely that sort of “handedness,” and their specific quantum properties have the perfect characteristics to serve as components of qubits – the fundamental units of information in a quantum computer, analogous to the bits in a classical electronic computer.

“There’s really no way to measure a single Majorana fermion,” JQI theorist Jay Sau explains. “But if you take a pair of them, you can form a [conventional] Dirac fermion that is fundamentally delocalized between those two states. It doesn’t live either ‘here’ or ‘there.’ And so any potential that is trying to mess around with it here or there can never do anything to it. That’s the real sense in which these things are topological. It’s a protected qubit.”

“Establishing that this possibly fictitious particle exists is like proving that the Higgs exists,” says JQI theorist Roman Lutchyn, referring to the particle component of the hypothesized Higgs field thought to confer mass on all other particles. “But if it can be identified, it is an ideal candidate for a qubit. Majorana fermions, because they are special particles, are very weakly coupled to everything else, so they have a sort of built-in protection against decoherence.”

Can such an unconventional system actually be constructed and tested? Can Majorana fermions really be generated and manipulated? Until quite recently, the outlook was not very encouraging. There is only one experimentally observed condition that appears to prompt the formation of non-Abelian quasiparticles – a phenomenon called the fractional quantum Hall effect (FQHE), the experimental discovery of which won the 1998 Nobel Prize in Physics.

FQHE leads to the emergence of various quasiparticles that carry only a fraction of the electron charge. Theory indicates that one of those particles is non-Abelian. However, FQHE has only been observed in arrangements of extraordinary custom-fabricated materials, extremely high magnetic fields, and very low temperatures – on the order of millikelvin. These are formidably difficult conditions in which to search for the possible implementation of TQC.

Nonetheless, in 2005, Das Sarma and coauthors Michael Freedman and Chetan Nayak of Microsoft Research Station Q in Santa Barbara, CA identified a FQHE state to propose the first topological qubit based on *any* topological phase.1

Now in a new paper2 , the same scientists plus Parsa Bonderson at Station describe how “how the necessary operations can be physically implemented” to build and test a model system “using currently available materials and techniques.”

The work builds on recent suggestions that a TQC platform might be devised by making sandwich-like structures out of layers of ordinary materials whose shared surface constitutes the right kind of two-dimensional state for Majorana fermions to form. There are several proposed designs, including two recently described by Das Sarma and JQI/CMTC researchers Roman Lutchyn and Jay Sau.3

All of them have common features: A sheet or layer of a superconductor (such as niobium) with a hole in the middle is placed against another material. At the interface, the superconductor induces a thin region of superconductivity in the second material by a process called the “proximity effect.” If a magnetic field is applied perpendicular to the layers of the sandwich, magnetic flux lines passing through the hole in the superconducting layer cause vortices to arise in the proximity effect region. Each of those vortices, theory predicts, will trap a single Majorana fermion state.

A pair of such quasiparticles can be used as a qubit, which is the fundamental unit of information in a quantum computer just as a bit is the fundamental unit in a classical electronic computer.

In one of the JQI designs, the layer of superconductor is placed over a sheet of semiconductor, under which is a magnetic insulator. Another uses a one-dimensional wire of semiconductor strung across the junction gap of a superconductor device called a Josephson junction. (The JQI work is generously supported by Microsoft, the Defense Advanced Research Projects Agency and the National Security Agency.) A third design, proposed by Liang Fu and C. L. Kane at the University of Pennsylvania, features a layer of a material called a “topological insulator” under the superconductor. Topological insulators, which have yet to be fully realized, are insulators in bulk, but have robust metallic surface or edge states.

All of the designs, however, would permit the creation of Majorana fermion bound states which could then be measured, transported (e.g., “braided”) and manipulated with magnetic or electric fields that alter the state of the quasiparticles and conditions in the 2D boundary layer. If observed, this would be a landmark event.

“This may serve as a starting point for attempts to construct a fault-tolerant quantum computer,” and colleagues write, “which would have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and – most importantly – those applications which no on in our classical era has the prescience to foresee.”

*arXiv*,

**1003**, 2856 (2010)

*Phys. Rev. Lett.*,

**105**, 077001 (2010)

*Phys. Rev. Lett.*,

**104**, 040502 (2010)

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