Collaborative Research: Topological Phases of Matter and Their Application to Quantum Computing

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In this collaborative project the Principal Investigators (PIs) willinvestigate mathematical models for topological phases of matter and theirapplication to the nascent field of quantum information. Examples oftopological phases of matter include Fractional Quantum Hall liquids andtopological insulators: materials subjected to extreme physical conditions(e.g. near zero Kelvin, under large magnetic forces) which appear toexhibit topological behavior. The main objectives of the project are tobetter understand the taxonomy of these exotic states of matter throughmathematical models, address physically and computationally motivatedmathematical problems, and apply topological and algebraic methods tostudy them. For example, the issue of quantum computational power(universality) may be analyzed by finding the images of the braid groupsunder representations associated to a particular model. Moreover, theextant hypothesized models for topological states of matter do not captureall of the subtleties (such as fermionic topological order) so the PIswill develop new models.The classical states of matter of solid, liquid and gas have more refinedclassifications: for example, solid crystals may be differentiated bytheir symmetries. Newly discovered topological materials have yet to befully 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 quantumcircuit model, requires challenging, if not impossible, accuracy on thehardware to be of practical value due to local interactions of the systemwith the surrounding environment. The topological model based on exoticstates of matter, while mathematically more complicated, has a built-intolerance for such interactions. The PIs will mathematically study theseexotic states of matter focusing on their application to new computationalparadigms with potentially significant benefit to society.