Topological Insulators and Applications to Quantum Computing

Andrew Lowrance

Attempts to understand the phases of matter and their implications for application in the technological industry illustrate the current understanding of modern quantum mechanics and how it may pave the way for societal advancement. One such advancement is the topological insulator, a new phase of matter that stands on the boundary between insulation and conductance. This technology would allow for the eventual construction of the quantum computer, a revolutionary device that would use the dynamics of atomic-scale objects to store and manipulate information.  

The story of the topological insulator began with the discovery of the quantum spin Hall effect (QSH) in 1980. QSH was the first example of a quantum state with spontaneously broken symmetry, meaning it comprises of a symmetric probability distribution where any pair of outcomes has the same probability of occurring. In a paper published in Nature, this behavior was noted to depend on the topology, or physically-continuous geometric makeup, of a material and its geometry at quantum levels. An example of how the QSH effect occurs is when a strong magnetic field is applied to a two-dimensional gas of electrons in a semiconductor. At low temperature and high magnetic field, electrons travel in different regions along the edge of the semiconductor, whereas at normal temperatures they remain scattered below and about the surface. The separation of these electrons into separate regions provides the material with unique properties, such as the ability to simultaneously conduct and insulate.

The topological insulator is essentially a hybrid between an insulator and a conductor, where conduction is possible on the surface of the material while insulation occurs beneath the surface. Dr. Charles Kane and Dr. Eugene Mele from the University of Pennsylvania are leaders in discovering how the QSH state could be realized in certain theoretical models with spin-orbit coupling is revealed to be an effect in which electrons retain their angular momentum, or rotational motion, as a result of geometric symmetry about the material. In addition, this coupling causes electrons that are moving through a crystal to feel a force, without the presence of an external magnetic field, which by sight may be a contradiction of physical laws. These models predict a class of materials that fit the description of the topological insulator.

Various materials seem to exhibit the properties of a topological insulator. These materials include bismuth selenide as well as bismuth telluride, a gray, solid compound often used in refrigerators for its efficient thermoelectric properties. At high temperatures, these compounds exhibit properties that demonstrate the quantum Hall state and possess the simplest geometric configurations, as referenced in a paper published in the Physical Review Letters. Some labs have begun experimentation with these materials in an attempt to understand them fundamentally.

One such attempt is the ARPES (Angle-resolved photoemission spectroscopy) experiment, which uses high-energy photons to eject an electron from a crystal and then detects the expelled electron's momentum. This experiment has incited interest in studying more materials that may exhibit topological insulating characteristics due to its relative success in proving the theoretical models established for these materials, as with the recent advancements made by Drs. Kane and Mele.

In understanding these topological properties, it is equally important to study the mechanisms behind why topological insulators are ideal for applications of quantum computing. In particular, scientists have observed what is known as an "emergent" particle at the interface between two topological insulators, as referenced in an article published in Science Magazine. This particle is the Majorana fermion, a new class of matter that, unlike classical particles, possesses its own antiparticles (particles that annihilate their natural counterparts which possess the same mass but opposite electric or magnetic properties). These particles possess quantum numbers that are different from ordinary electrons. The Majorana fermion is one discovery that points to exciting developments in quantum research in the future.

A major goal for quantum mechanical physicists is the construction of a quantum computer: a computer that exceeds the theoretical computing limits of a modern digital computer. One might imagine a quantum computer to look like a desktop or a tablet. The quantum computer, however, could possess various forms ranging from ion traps, a combination of electric or magnetic fields used to capture charged particles, to cavity quantum electrodynamics and nuclear magnetic resonance techniques, methods used for the microscopic investigation of quantum-mechanical radiation and the study of the interaction between light confined in a reflective cavity and other particles.

A paper published in Science Direct demonstrates that the type of information that passes through a quantum computer is slightly different than the bit, the classic unit of information in a digital computer. Quantum computers use an informational unit called a "qubit." In a classical computer, bits may take one of many states, such as ones and zeroes, at any given time, which allows for the transmission of information in distinct forms. These forms are finite possibilities of finite states, bound by the laws of classical physics. The quantum computer, however, utilizes qubits that can take various states all at once, otherwise known as "superposition." In a computational sense, one can view these various states as graphical functions with probability amplitudes for each of the classical states within a classical computer.  

Quantum computing is a theoretical formulation that continues to pervade the modern discussions of advancing current technological capabilities for practical application, such as with current attempts from companies like IBM and their IBM Q initiative. Such practical application is even possible through natural compositions born from the environment. For instance, typical computers use random access memory to encode and interpret certain forms of information. In a similar sense, a single crystal could operate as a random access memory device, where electron spin resonance pulses decode and produce numbers that are significantly faster than a typical computer, as referenced in a paper published in Nature.

The mass production of such technological devices continues to be a thing of the near future, as evidenced by current initiatives by major businesses in the tech industry, and could even be a thing of today. As noted by Nella Ludlow from Cognitive World, “quantum computers exist, and access to them via the cloud is affordable, university- and industry-developed education is increasing, and government funding was approved to further research and focus on needed workforce development… the quantum computing tipping point is now.”


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