In collaboration with Earl Campbell at FU Berlin, Hussain and Dan have posted a new pre-print on magic state distillation to arxiv.org.
You can see Dan give a recent seminar on this work at the Perimeter Institute Research Seminar Archive:
A quantum computer would exploit the non-classical aspects of quantum
mechanics to achieve modes of calculation impossible on conventional
"classical" computers. However, constructing a device large enough to
perform complex calculations represents a huge challenge for modern
physics. One reason for this is that the delicate quantum states
encoding the information are easily corrupted by the noise induced by
the environment or imperfections in the computer itself. Fortunately,
methods of fault tolerant quantum computation have been developed
which enable reliable quantum computation in the presence of noise. A
key component in many fault tolerance schemes is the preparation of
magic states, the essential ingredient for non-classical computation.
The magic states are prepared by distillation, a process that
exchanges quantity for computational potency.
Quantum computing is usually conceived in terms of qubits (two-level
systems) but this need not be the case. In fact, systems with any
number of levels known as qudits can be the basis for a quantum
computer. Here we prove, for the first time, that magic state
distillation can be achieved in qudits with any prime number of
levels. In particular, we show that for qutrits (three-level systems)
magic state distillation works equally well as its qubit counterpart,
and in some aspects it is even better. To devise the distillation
process we employ quantum variants of ideas from classical computer
science, called Reed-Muller codes. The techniques we develop are
important in their own right, with the quantum Reed-Muller codes
having useful applications beyond magic state distillation.
This opens the door for the development of qudit-based fault tolerance
schemes based on magic state distillation, which may offer significant
advantages over their qubit counterparts, and hence bring the
development of a scalable quantum computer closer to our grasp.