In this paper, the degenerate ground states of
Z2 topological order on a plane with holes (the so-called planar codes) are used as the protected code subspace to build a topological quantum computer by tuning their quantum tunneling effect. Using a designer Hamiltonian---the Kitaev toric-code model as an example, we study quantum tunneling effects of the planar codes and obtain its effective theory. Finally, we show how to do topological quantum computation including the initialization, the unitary transformation and the measurement.
This is a comprehensive review on fault-tolerant topological quantum computation with the surface codes. The basic concepts and useful tools underlying fault-tolerant quantum computation, such as universal quantum computation, stabilizer formalism, and measurement-based quantum computation, are also provided in a pedagogical way. Topological quantum computation by brading the defects on the surface code is explained in both circuit-based and measurement-based models in such a way that their relation is clear. The interdisciplinary connections between quantum error correction codes and subjects in other fields such as topological order in condensed matter physics and spin glass models in statistical physics are also discussed. This manuscript will be appeared in SpringerBriefs.