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

Outline Motivation The system Ground states Excited states Topological protection Discussion

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).

The system Periodic boundary condition s p Torus Edge: 8*6+8*6=96 (2 N_1 N_2) Vertex: 8*6=48 (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/9707021, Annals of physics 303, 2 (2003).

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.

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)

The geometric picture The subgroup Reducible loop

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

Dual lattice lattice dual lattice vertex plaquette plaquette vertex

Irreducible loop on dual lattice

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

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}.

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

How to create excitations? --- open string operator 1 2 The same effect A pair of “electric charges” are excited.

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.

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

Braiding of electric charge 5 3 1 2 4 Note that “Electric charge” type excitations are bosons.

Electric charge go around magnetic flux 3 2 1 Note that

Fusion rules (Splitting rules) a b c

Braiding of composite excitation 5 3 1 2 4 Composite excitations are fermions.

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.

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)

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

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

Thank you!