Topological Quantum Compiling

Quasiparticle excitations with nonabelian statistics are expected to arise in a variety of exotic two-dimensional quantum many- body systems. In some cases these systems can be used for topological quantum computation, in which quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are performed by moving quasiparticles around each other. The trajectories of quasiparticles in two space dimensions define world-lines in three dimensional space-time, and the resulting quantum gates depend only on the topology of the braids formed by these world-lines. Here, we find braiding patterns which yield a universal set of quantum gates for a specific kind of nonabelian statistics which is particularly promising for experimental realization. While it is known that quasiparticles with these specific statistics can be used for universal quantum computation, explicit braiding topologies for carrying out a universal set of quantum gates have not previously been constructed. In the language of computer science, translation of computer code from a high level computer language to a machine level set of instructions is known as compiling. In this sense, the translation of quantum computation operations into braids that can be physically performed in a topological quantum computer can be thought of as ``topological quantum compiling."