My primary research focus at the moment is in quantum complexity theory. In general, I am presently interested in persuing research that explores the nature of the "landscape" between quantum and classical computation, by way of the algebraic structures manifested by quantum computation in various kinds of computational models.
Beyond quantum complexity theory, I have several research interests more broadly in the union of computer science, quantum foundations, and pure mathematics. There is a more-or-less complete list of my published works here.
Below are descriptions of some of the questions that I'm currently interested in. Please note however, this account of my research interests need not be taken as representative; many problems that are even tenuously related to these topics are likely to catch my interest nonetheless! Furthermore, I'd also recommend taking a look at the website of my friend and colleague Matthew Fox; we are interested in many of the same things, and I could safely endorse my interest in most (if not all!) of the problems he has indicated there as well.
In order for a quantum gate set to be meaningfully universal for computational tasks, it needn't be able to approximate any unitary to arbitrary precision. Rather it suffices for the gate set to simulate any unitary to within arbitrary statistical distance. What then, determines whether or not a given quantum gate set is universal for computational tasks? More precisely, how might the emergence of computational universality in quantum gatesets be mathematically characterized? See more...
There are several "weak" models of quantum computation that have been defined, which are certainly not universal for quantum computation, but for which it is nonetheless unlikely for there the be a classical algorithm which can efficiently simulate these models. How weak can these models get? How might these different restricted models relate to each other? What can these restricted models of quantum computation tell us about what the purported "landscape" between classical and quantum computation? See more...
Can we make progress towards determining the Ramsey numbers of double star graphs, and their generalizations? See more...