Meeting 10.2: Topological quantum computing I

Continuing to follow Kitaev, we consider the toric code as the ground state space of a Hamiltonian constructed from the vertex and plaquette operators.  The other eigenspaces of this Hamiltonian are both topologically and physically meaningful, and allow us to intentionally introduce "particles" called anyons.  Braiding or "Dehn twisting" using  the anyons (i.e. applying string operators) gives a way to process quantum information.  These operations are "topologically protected", i.e. unlikely to be implemented wrong because errors can only occur when a homologically non-trivial loop operator accidentally occurs as noise. Unfortunately, when we use the toric code these operations are incapable of implementing a quantum universal gate set.  Next time we'll discuss a more potent example.