NSF Award
1108725
In this collaborative project the Principal Investigators (PIs) will<br/>investigate mathematical models for topological phases of matter and their<br/>application to the nascent field of quantum information. Examples of<br/>topological phases of matter include Fractional Quantum Hall liquids and<br/>topological insulators: materials subjected to extreme physical conditions<br/>(e.g. near zero Kelvin, under large magnetic forces) which appear to<br/>exhibit topological behavior. The main objectives of the project are to<br/>better understand the taxonomy of these exotic states of matter through<br/>mathematical models, address physically and computationally motivated<br/>mathematical problems, and apply topological and algebraic methods to<br/>study them. For example, the issue of quantum computational power<br/>(universality) may be analyzed by finding the images of the braid groups<br/>under representations associated to a particular model. Moreover, the<br/>extant hypothesized models for topological states of matter do not capture<br/>all of the subtleties (such as fermionic topological order) so the PIs<br/>will develop new models.<br/><br/>The classical states of matter of solid, liquid and gas have more refined<br/>classifications: for example, solid crystals may be differentiated by<br/>their symmetries. Newly discovered topological materials have yet to be<br/>fully understood, but potentially can be used to build (quantum)<br/>computational devices out-performing standard micro-chip based computers.<br/>The most commonly encountered model for quantum computation, the quantum<br/>circuit model, requires challenging, if not impossible, accuracy on the<br/>hardware to be of practical value due to local interactions of the system<br/>with the surrounding environment. The topological model based on exotic<br/>states of matter, while mathematically more complicated, has a built-in<br/>tolerance for such interactions. The PIs will mathematically study these<br/>exotic states of matter focusing on their application to new computational<br/>paradigms with potentially significant benefit to society.
