Quantum computation and information is a vibrant field with many exciting questions still to be answered. My research is about the theory of quantum systems, architectures for quantum computers and quantum algorithms.
I want to discover more about where the power of quantum computers comes from and what differentiates quantum and classical mechanics. To this end, I have worked on topics such as entanglement theory, quantum state tomography and the geometry of quantum state space.
Is it possible to distribute a quantum computer across many sites and still be able to run powerful quantum algorithms with a minimal overhead? One might expect unwanted interference if say, two processor nodes tried to access the same memory location. Recent work  with coworkers from Bristol, Princeton and Seattle has demonstrated that the answer is in fact yes. We have provided algorithms for efficiently addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with low overhead by the more realistic model of a distributed quantum computer
Proof of principal experiments connecting two trapped ion processor nodes with an optical fibre have already been successful. Our work provides a theoretical basis for this architecture that could one day be used to build a full scale quantum computer.
My coworkers and I have discovered new algorithms to run on a quantum computer (once one is built!). The first is called the MultiGrover algorithm and allows you to perform parallel Grover searches of a single database stored in quantum memory (or a quantum accessible memory). We also found improved algorithms for the element distinctness and collision problems.
Mutually unbiased bases for quantum degrees of freedom are crucial to many applications in quantum information that exploit complementary properties or use generalizations of the familiar two qubit Bell states. S. Weigert and I have championed a variety of approaches to one of the main open problems regarding the existence of maximal sets in Hilbert spaces of non-prime-power dimension [2,3]. Investigating mutually unbiased bases has lead to an interest in commutative algebraic geometry and I hope to find other applications of techniques such as Gröbner bases in problems involving quantum systems of finite dimension.
The existence of a SIC POVM in every dimension is a big open question of interest to Bayesian quantum mechanics. The solution to this problem will almost certainly involve some deep mathematics and yield useful results along the way. A good example of this is that on working on this problem my co-authors and I found a strikingly simple representation of the Clifford group  - a group that is used extensively by the quantum information community. It helped us simplified the SIC problem in square dimensions and allowed us to find exact solutions in dimension sixteen.
Described by Schrödinger as not one but rather the characteristic trait of quantum mechanics, entanglement distinguishes quantum states from states with purely classical correlations and is a key resource in many quantum information applications. Detecting entangled states by way of entanglement criteria tends to fall into one of two categories; either the criterion is powerful, being necessary and sufficient; or it is practical, being only necessary but easy to use. I am particularly interested in the many non-linear functionals that have been proposed as practical entanglement criteria. I have successfully addressed a problem of the entanglement of four-qubit pure states. Using variational techniques, A. Higuchi and I were able to prove that a particularly interesting four-qubit pure state locally maximizes a suitable measure of entanglement .
The field of quantum information proposes many interesting optimization problems. For example, one might ask what the optimal measurements to determine an unknown density matrix are, or want to find the minimal resources needed to perform a particular task. These problems inevitably lead to the use of techniques such as semidefinite programming, numerical minimization, Lagrange multipliers and the calculus of variations; all of which I have experience with. In addition, suitable measures of distance have enabled me to find the states and measurements which maximize the disturbance of an eavesdropper in a particular quantum key distribution protocol.