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In this paper, the degenerate ground states of Z2 topological order on aplane with holes (the so-called surface codes) are used as the protected codesubspace to build a topological quantum computer by tuning their quantumtunneling effect. Using a designer Hamiltonian - the Kitaev toric-code model asan example, we study quantum tunneling effects of the surface codes and obtainits effective theory. Finally, we show how to do topological quantumcomputation including the initialization, the unitary transformation and themeasurement.

Realization of Topological Quantum Computation with surface codes