  We show that universal holonomic quantum computation (HQC) can be achieved
fault-tolerantly by adiabatically deforming the gapped stabilizer Hamiltonian
of the surface code, where quantum information is encoded in the degenerate
ground space of the system Hamiltonian. We explicitly propose procedures to
perform each logical operation, including logical state initialization, logical
state measurement, logical CNOT, state injection and distillation,etc. In
particular, adiabatic braiding of different types of holes on the surface leads
to a topologically protected, non-Abelian geometric logical CNOT. Throughout
the computation, quantum information is protected from both small perturbations
and low weight thermal excitations by a constant energy gap, and is independent
of the system size. Also the Hamiltonian terms have weight at most four during
the whole process. The effect of thermal error propagation is considered during
the adiabatic code deformation. With the help of active error correction, this
scheme is fault-tolerant, in the sense that the computation time can be
arbitrarily long for large enough lattice size. It is shown that the frequency
of error correction and the physical resources needed can be greatly reduced by
the constant energy gap.
