1. Topological quantum computing ▬ Toric code
Duanlu Zhou
Institute of Physics, CAS
July 2007, Shanxi

2. Outline Motivation The system Ground states Excited states
Topological protection
Discussion

3. Motivation
Make quantum computation robust in the presence of local perturbations. (Avoid error).
Characterize many-body states beyond the picture of order parameter.
Today I will give a detailed analyses of the toric code model (simple, but containing rich physics).

4. The system Periodic boundary condition s p Torus
Edge: 8*6+8*6=96 (2 N_1 N_2)
Vertex: 8*6= (N_1 N_2)
Plaquette: 8*6=48 (N_1 N_2)
An edge (NOT a vertex) denotes a qubit (a ½ spin) , described by
A. Kitaev, quant-ph/ , Annals of physics 303, 2 (2003).

5. Vertex operator and plaquette operator
For each vertex s,
For each plaquette p,
The properties of the vertex operator and plaquette operator:
Eigenvalues: +1 or -1
These operators have common eigenvectors.

6. The Hamiltonian The ground states satisfy
Note that for any set S of vertexes and any set P of plaquettes,
The stabilizer group
The number of the generators of the stabilizer group is 94.
(2 N_1 N_2 -2)

7. The geometric picture
The subgroup
Reducible loop

8. Winding number =2 independent?
Irreducible loop
Winding number =2 independent?

9. Dual lattice
lattice dual lattice
vertex plaquette
plaquette vertex

10. Irreducible loop on dual lattice

11. Properties of irreducible loop operators
The generator of a new stabilizer group:
The 4-fold degenerate ground states:

12. Operations among ground states
Local spin flip
Any operator on the subspace of ground states can be expanded as
Except 1, the length of all the terms is not less than min{N_1,N_2}.

13. Excited states “Electric charge” type “magnetic flux” type
Always creates pairs of excitation for both types
Excited energy spectrum Degeneracy 2 4

14. How to create excitations? --- open string operator1 2
The same effect
A pair of “electric charges” are excited.

15. How to move excitation? 2 1 3
Similarly, we can excite and move “magnetic flux” type of excitation
by the open string operator on the dual lattice.

16. Statistical properties of excitation --- Braiding
2+1 dimension

17. Braiding of electric charge5 3 1 2 4
Note that
“Electric charge” type excitations are bosons.

18. Electric charge go around magnetic flux3 2 1
Note that

19. Fusion rules (Splitting rules)a b c

20. Braiding of composite excitation5 3 1 2 4
Composite excitations are fermions.

21. Topological protection
Energy gap between ground states and excited states ensures that perturbation has effects on the subspace of ground states.
Any local perturbation has little effect on the subspace of ground states.

22. Discussions Boundary condition of the toric code model
Abelian and non-abelian anyons
Non-abelian anyons are necessary to process universal quantum computation.
Possible physical realizations:
Quantum Hall system
Kitaev model 2 (beyond)

23. Edge state s p
The topological excitation types can be obtained by measuring the edge states.

24. Summary
We analyze the toric code model in detail. Especially, we emphasize some basic concepts in topological quantum computation.

25. Thank you!