In this collaborative project the Principal Investigators (PIs) will investigate mathematical models for topological phases of matter and their application to the nascent field of quantum information. Examples of topological phases of matter include Fractional Quantum Hall liquids and topological insulators: materials subjected to extreme physical conditions (e.g. near zero Kelvin, under large magnetic forces) which appear to exhibit topological behavior. The main objectives of the project are to better understand the taxonomy of these exotic states of matter through mathematical models, address physically and computationally motivated mathematical problems, and apply topological and algebraic methods to study them. For example, the issue of quantum computational power (universality) may be analyzed by finding the images of the braid groups under representations associated to a particular model. Moreover, the extant hypothesized models for topological states of matter do not capture all of the subtleties (such as fermionic topological order) so the PIs will develop new models.
The classical states of matter of solid, liquid and gas have more refined classifications: for example, solid crystals may be differentiated by their symmetries. Newly discovered topological materials have yet to be fully understood, but potentially can be used to build (quantum) computational devices out-performing standard micro-chip based computers. The most commonly encountered model for quantum computation, the quantum circuit model, requires challenging, if not impossible, accuracy on the hardware to be of practical value due to local interactions of the system with the surrounding environment. The topological model based on exotic states of matter, while mathematically more complicated, has a built-in tolerance for such interactions. The PIs will mathematically study these exotic states of matter focusing on their application to new computational paradigms with potentially significant benefit to society.