  Topological quantum computing promises error-resistant quantum computation
without active error correction. However, there is a worry that during the
process of executing quantum gates by braiding anyons around each other, extra
anyonic excitations will be created that will disorder the encoded quantum
information. Here we explore this question in detail by studying adiabatic code
deformations on Hamiltonians based on topological codes, notably Kitaev's
surface codes and the more recently discovered color codes. We develop
protocols that enable universal quantum computing by adiabatic evolution in a
way that keeps the energy gap of the system constant with respect to the
computation size and introduces only simple local Hamiltonian interactions.
This allows one to perform holonomic quantum computing with these topological
quantum computing systems. The tools we develop allow one to go beyond
numerical simulations and understand these processes analytically.
