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Anyon and Topological Quantum Computation Su-Peng Kou Beijing Normal university

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Outline 1.Part I: Anyons and braiding group 2.Part II: Quantum computation of topological qubits in Z2 topological orders 3.Part III : Topological quantum computation by Ising anyons 4.IV: Topological quantum computation by Fibonacci anyons topological string operator, nonAbelian anyon Key words: topological string operator, nonAbelian anyon

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1997, Kitaev proposed the idea of topological quantum bit and fault torrent quantum computation in an Abelian state. 2001, Kitaev proposed the topological quantum compuation by braiding non- Abelian anyons. 2001, Preskill, Freedman and others proposed a universal topological quantum computation. Milestone for topological quantum computation

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(I) Anyons and braid groups

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Abelian statistics via non-Abelian statistics

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Exchange statistics and braid group Particle Exchange : world lines braiding

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Braid group

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General anyon theory 1. A finite set of quasi-particles or anyonic “charges.” Fusion rules 2. Fusion rules (specifying how charges can combine or split). Braiding rules 3. Braiding rules (specifying behavior under particle exchange).

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3. Associativity relations for fusion: F matrix 2. Braiding rules: R matrix

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Pentagon equation:

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Hexagon equation:

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Non-Abelian statistics Exchanging particles 1 and 2: Matrices M 12 and M 23 don’t commute; Matrices M form a higher-dimensional representation of the braid-group. Exchanging particles 2 and 3:

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(II) Quantum computation of topological qubits in Z2 topological orders

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1. There are four sectors : I (vacuum), ε(fermion), e (Z2 charge), m (Z2 vortex) ; 2. Z2 gauge theory 3. U(1)×U(1) mutual Chern-Simons theory 4. Topological degeneracy : 4 on torus SP Kou, M Levin, and XG Wen, PRB 78, 155134 (2008). 1. Z2 topological order

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flux Mutual semion statistics between Z2 vortex and Z2 charge Z2 vortex Z2 charge Fermion as the bound state of a Z2 vortex and a Z2 charge Mutual Flux binding

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Fusion rule A. Yu. Kitaev, Ann. Phys. 303, 2 2003.

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Toric-code model A ． Y ． Kitaev ， Annals Phys. 303, 2 (2003)

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Wen-plaquette model X. G. Wen, PRL. 90, 016803 (2003)

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The energy eigenstates are labeled by the eigenstates of Because of, the eigenvalues are Solving the Wen-plaquette model

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For g>0, the ground state is The ground state energy is E 0 =Ng The elementary excitation is The energy gap for it becomes The energy gap

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The statistics for the elementary excitations There are two kinds of Bosonic excitations: Z2 vortex Z2 charge Each kind of excitations moves on each sub- plaquette: Why?

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There are two constraints (the even-by-even lattice): One for the even plaquettes, the other for the odd plaquettes The hopping from even plaquette to odd violates the constraints : You cannot change a Z2 vortex into a Z2 charge

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On an even-by-even lattice, there are totally states Under the constaints, the number of states are only For the ground state, it must be four-fold degeneracy. Topological degeneracy on a torus (even-by-even lattice) :

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Z2 vortex (charge) can only move in the same sub- plaquette: The hopping operators for Z2 vortex (charge) are The dynamics of the Z2 Vortex and Z2 charge

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X. G. Wen, PRD68, 024501 (2003). The mutual semion statistics between the Z2 Vortex and Z2 charge When an excitation (Z2 vortex) in even-plaquette move around an excitation (Z2 charge) in odd- plaquette, the operator is it is -1 with an excitation on it This is the character for mutual mutual semion statistics Fermion as the bound state of a Z2 vortex and a Z2 charge.

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The hopping operators of Z2 vortex and charge are Controlling the hopping of quasi-particles by external fields The hopping operator of fermion is So one can control the dynamics of different quasi- particles by applying different external.

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Closed strings Open strings

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String net condensation for the ground states The string operators ： For the ground state, the closed-strings are condensed

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The toric-code model There are two kinds of Bosonic excitations: Z2 vortex Z2 charge Fermion as the bound state of a Z2 vortex and a Z2 charge.

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The hopping operator of Z2 vortex is Controlling the hopping of quasi-particles by external fields The hopping operator of fermion is So one can control the dynamics of different quasi-particles by applying different external fields. The hopping operator of Z2 charge is

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A. Yu. Kitaev, Annals Phys. 303, 2 (2003) |0> and |1> are the degenerate ground-states of a (Z2) topological order due to the (non-trivial) topology. Advantage No local perturbation can introduce decoherence. 2. Topological qubit Ioffe, &, Nature 415, 503 (2002)

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Topology of Z2 topological order E CylinderTorus E Disc E 124 Hole on a Disc

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Topological closed string operators on torus – topological qubits

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Degenerate ground states as eigenstates of topological closed operators Define pseudo-spin operators: Algebra relationship:

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Topological closed string operators On torus ， pseudo-spin representation of topological closed string operators: S.P. Kou, PHYS. REV. LETT. 102, 120402 (2009). J. Yu and S. P. Kou, PHYS. REV. B 80, 075107 (2009). S. P. Kou, PHYS. REV. A 80, 052317 (2009).

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Degenerate ground states as eigenstates of topological closed operators

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topological qubits on torus Toric codes : topological qubits on torus There are four degenerate ground states for the Z2 topological order on a torus: m, n = 0, 1 label the flux into the holes of the torus.

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How to control the topological qubits? A. Y. Kitaev : Unfortunately, I do not know any way this quantum information can get in or out. Too few things can be done by moving abelian anyons. All other imaginable ways of accessing the ground state are uncontrollable “Unfortunately, I do not know any way this quantum information can get in or out. Too few things can be done by moving abelian anyons. All other imaginable ways of accessing the ground state are uncontrollable.” A ． Y ． Kitaev ， Annals Phys. 303, 2 (2003)

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3. Quantum tunneling effect of topological qubits : topological closed string representation Tunneling processes are virtual quasi-particle moves around the periodic direction.

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Topological closed string operator as a virtual particle hopping

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Topological closed string operators may connect different degenerate ground states S.P. Kou, PHYS. REV. LETT. 102, 120402 (2009). J. Yu and S. P. Kou, PHYS. REV. B 80, 075107 (2009). S. P. Kou, PHYS. REV. A 80, 052317 (2009).

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Higher order perturbation approach Energy splitting : lowest order contribution of topological closed string operators L 0 is the length of topological closed string operator

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The energy splitting from higher order (degenerate) perturbation approach L : Hopping steps of quasi-particles t eff : Hopping integral : Excited energy of quasi-particles J. Yu and S. P. Kou, PHYS. REV. B 80, 075107 (2009).

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Topological closed string operators of four degenerate ground states for the Wen-plaquette model under x- and z-component external fields

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Effective model of four degenerate ground states for the Wen-plaquette model under x- and z-component external fields

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External field along z direction In anisotropy limit, the four degenerate ground states split two groups, 2×6 lattice on the Wen- plaquette model under z direction field

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External field along z direction Isotropy limit ， the four degenerate ground states split three groups 4×4 lattice on Wen- plaquette model under z- direction

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External field along x direction Under x-direction field, the four degenerate ground states split three groups: 4×4 lattice on Wen- plaquette model under x- direction

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Ground states energy splitting of Wen-plaqutte model on torus under a magnetic field along x-direction

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Ground states energy splitting of Wen-plaqutte model on torus under a magnetic field along z-direction

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topological qubits on surface with holes Planar codes : topological qubits on surface with holes L. B. Ioffe, et al., Nature 415, 503 (2002). Fermionic based

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Effective model of the degenerate ground states of multi-hole The four parameters Jz, Jx, hx, hz are determined by the quantum effects of different quasi-particles. S.P. Kou, PHYS. REV. LETT. 102, 120402 (2009). S. P. Kou, PHYS. REV. A 80, 052317 (2009).

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Unitary operations A general operator becomes : For example, Hadamard gate is

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CNOT gate and quantum entangled state of topological qubits S. P. Kou, PHYS. REV. A 80, 052317 (2009).

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III. Topological quantum computation by braiding Ising anyons

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initialize create particles operation braid output measure ComputationPhysics Eric Rowell Topological Quantum Computation

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(I) Ising anyons Fusion rules:

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flux Ising anyons SU(2) 2 non-Abelian statistics between π-flux with a trapped majorana fermion. Another anyon Majorana fermion σ:π-Flux binding a Majorana Fermion

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px+ipy-wave superconductor : an example of symmetry protected topological order µ>0, non-Abelian Topologial state µ<0, Abelian Topologial state Read, Green, 2000. S. P. Kou and X.G. Wen, 2009.

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Winding number in momentum space

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BdG equation of px+ipy superconductor Bogoliubov deGennes Hamiltonian: Eigenstates in +/- E pairs Spectrum with a gap Excitations: Fermionic quasiparticles above the gap

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BdG equation of vortex in px+ipy superconductor E = 0

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Whyπ vortex in px+ipy wave superconductors traps majorona fermion? The existence of zero mode in πflux for chiral superconducting state : cancelation between the π flux of vortex and edge chiral angle (winding numer in momentum space) Majorana fermion in chiral p-wave – mixed annihilation operator and generation operation

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Chiral edge state y x p+ip superconductor Edge state Edge Majorana fermion Chiral fermion propagates along edge Edge state encircling a droplet Antiperiodic boundary condition Spinor rotates by 2π encircling sample

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Vortex (πflux) in p x +ip y superconductor Single vortex Fermion picks up π phase around vortex: Changes to periodic boundary condition E=0 Majorana fermion encircling sample : an encircling vortex - a “vortex zero mode”

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E = nω

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“5/2” FQHE states Pan et al. PRL 83,1999 Gap at 5/2 is 0.11 K Xia et al. PRL 93, 2004 Gap at 5/2 is 0.5K, at 12/5: 0.07K

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Moore-Read wavefunction for 5/2 FQHE state Moore, Read (1991) Greiter, Wen, Wilczek (1992) “Paired” Hall state Pfaffian:

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Moore/Read = Laughlin × BCS

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Ising anyons in the generalized Kitaev model Gapped B phase are SU(2) 2 non- Abelian topological order for K>0. Boundaries: Vortex-free: J=1/2 Full-vortex: J=1/√2 Sparse: 1/2 ≤ J ≤ 1/√2 (J z = 1 and J = J x = J y )

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px+ipy SC for generalized Kitaev model by Jordan-Wigner transformation Y. Yue and Z. Q. Wang, Europhys. Lett. 84, 57002 (2008)

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Topological qubits of Ising anyons Pairs of Ising anyons : each anyon binds to a Majorana fermion, the fermion state of two anyons is described by a regular fermion which is a qubit. A qubit

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Braiding operator for two-anyons The braiding matrices are (Ivanov, 2001) :

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Braiding matrices for the degenerate states of four Ising anyons Two- qubit

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N matrices R matrices F matrices

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i = f i f time Topological Quantum Computation

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Topological quantum computation by Ising anyons Two pairs of Ising anyons R matrices of two pairs of anyons : braiding operators

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X gate and Z gate L.S.Georgiev, PRB74,235112(2006)

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Hadmard gate L.S.Georgiev, PRB74,235112(2006)

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CNOT gate L.S.Georgiev, PRB74,235112(2006)

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No π/8 gate Toffoli gate ? L. S. Georgiev, PRB74,235112(2006)

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IV. Topological quantum computation by braiding Fibonacci anyons

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(2) Fibonacci anyon There are two sectors : I and τ. Two anyons ( τ) can “fuse” two ways. Fusion rules

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Fibonacci anyon Fib(n) = Fib(n–1) + Fib(n–2)

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Fibonacci anyons =1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

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N matrices Fibonacci anyon R matrices F matrices

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Other examples of Fibonacci anyon

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Possible example of Fibonacci anyon in “12/5” FQHE state

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Read-Rezayi wave-function Para-fermion state : bound state of three fermions N. Read and E. H. Rezayi, Phys. Rev. B 59, 8084 (1999).

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Topological Qubit of Fibonacci anyons 1 × 1 = 0 + 1 Two Fibonacci span a 2-dimensional Hilbert space 01 01 0111 To do non-trivial operation we need three Fibonacci anyons

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P. Bonderson et. al

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Single qubit rotation Universalcomputation Ising anyons Fibonacci anyons

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P. Bonderson et. al

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(Bonesteel, et. al.) Topological Quantum Computation (Kitaev, Preskill, Freedman, Larsen, Wang) 01

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P. Bonderson et. al

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Thank you!

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