These are a few suggested projects and related papers. I will add more items to this list from time to time. Please feel free to pick a project not on this list, or drop by and see me if you want help in fleshing out an idea.
Adiabatic quantum algorithms - this provides an alternate paradigm for the design of quantum algorithms. A number of papers explore this subject: paper1.pdf paper2.pdf paper3.pdf
Kitaev's phase estimation algorithm has a number of applications. It gives an example of a quantum speedup without entanglement. And a recent paper claims that it leads to a significant speedup in solving classical differential equations: paper.pdf paper.pdf
Quantum techniques have been used to resolve purely classical problems. Here are a couple of examples: paper.pdf paper1.ps
There are special classes of quantum circuits that can be efficiently simulated classically. These include circuits involved in quantum error-correction: paper.pdf
The non-abelian hidden subgroup problem is the big challenge in quantum algorithms: paper1.ps paper2.pdf
Simulating quantum systems is a fundamental problem. Some ideas from quantum computation have lead to efficient classical algorithms for simulating special types of systems: paper1.pdf paper2.pdf
Quantum error-correction beyond the no-cloning bound: Streaming video MSRI