Presentation on theme: "Anyon and Topological Quantum Computation Beijing Normal university"— Presentation transcript:
1Anyon and Topological Quantum Computation Beijing Normal university Su-Peng KouBeijing Normal university
2Outline Part I: Anyons and braiding group Part II: Quantum computation of topological qubits in Z2 topological ordersPart III : Topological quantum computation by Ising anyonsIV: Topological quantum computation by Fibonacci anyonsKey words: topological string operator, nonAbelian anyon
3Milestone for topological quantum computation 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.
9General anyon theory1. A finite set of quasi-particles or anyonic “charges.”2. Fusion rules (specifying how charges can combine or split).3. Braiding rules (specifying behavior under particle exchange).
14Non-Abelian statistics Exchanging particles 1 and 2:Exchanging particles 2 and 3:Matrices M12 and M23 don’t commute;Matrices M form a higher-dimensional representation of the braid-group.
15(II) Quantum computation of topological qubits in Z2 topological orders
16SP Kou, M Levin, and XG Wen, PRB 78, 155134 (2008). 1. Z2 topological orderThere are four sectors : I (vacuum), ε(fermion), e (Z2 charge), m (Z2 vortex) ;Z2 gauge theoryU(1)×U(1) mutual Chern-Simons theoryTopological degeneracy : 4 on torusSP Kou, M Levin, and XG Wen, PRB 78, (2008).
17Mutual semion statistics between Z2 vortex and Z2 charge fluxZ2 vortexMutual Flux bindingFermion as the bound state of a Z2 vortex and a Z2 charge
21Solving the Wen-plaquette model The energy eigenstates are labeled by the eigenstates ofBecause of , the eigenvalues are
22The energy gap For g>0, the ground state is The ground state energy is E0=NgThe elementary excitation isThe energy gap for it becomes
23The statistics for the elementary excitations There are two kinds of Bosonic excitations:Z2 vortexZ2 chargeEach kind of excitations moves on each sub-plaquette:Why?
24There are two constraints (the even-by-even lattice): One for the even plaquettes, the other for the odd plaquettesThe hopping from even plaquette to odd violates the constraints :You cannot changea Z2 vortex into a Z2 charge
25Topological degeneracy on a torus (even-by-even lattice) : On an even-by-even lattice, there are totallystatesUnder the constaints,the number of states are onlyFor the ground state , it must be four-fold degeneracy.
26The dynamics of the Z2 Vortex and Z2 charge Z2 vortex (charge) can only move in the same sub-plaquette:The hopping operators for Z2 vortex (charge) are
27The 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 isit is -1 with an excitation on itThis is the character for mutual mutual semion statisticsFermion as the bound state of a Z2 vortex and a Z2 charge.X. G. Wen, PRD68, (2003).
28Controlling the hopping of quasi-particles by external fields The hopping operators of Z2 vortex and charge areThe hopping operator of fermion isSo one can control the dynamics of different quasi-particles by applying different external.
30String net condensation for the ground states The string operators：For the ground state, the closed-strings are condensed
31The toric-code model There are two kinds of Bosonic excitations: Z2 vortexZ2 chargeFermion as the bound state of a Z2 vortex and a Z2 charge.
32Controlling the hopping of quasi-particles by external fields The hopping operator of Z2 vortex isThe hopping operator of Z2 charge isThe hopping operator of fermion isSo one can control the dynamics of different quasi-particles by applying different external fields.
332. Topological qubit 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.AdvantageNo local perturbation can introduce decoherence.Ioffe, &, Nature 415, 503 (2002)
34Topology of Z2 topological order CylinderTorusDisc124Hole on a Disc
35Topological closed string operators on torus – topological qubits
36Degenerate ground states as eigenstates of topological closed operators Algebra relationship:Define pseudo-spin operators:
37Topological closed string operators On torus，pseudo-spin representation of topological closed string operators:S.P. Kou, PHYS. REV. LETT. 102, (2009).J. Yu and S. P. Kou, PHYS. REV. B 80, (2009).S. P. Kou, PHYS. REV. A 80, (2009).
38Degenerate ground states as eigenstates of topological closed operators
39Toric 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.
40How 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.”A．Y．Kitaev，Annals Phys. 303, 2 (2003)
413. Quantum tunneling effect of topological qubits : topological closed string representationTunneling processes are virtual quasi-particle moves around the periodic direction.
42Topological closed string operator as a virtual particle hopping
43Topological closed string operators may connect different degenerate ground states S.P. Kou, PHYS. REV. LETT. 102, (2009).J. Yu and S. P. Kou, PHYS. REV. B 80, (2009).S. P. Kou, PHYS. REV. A 80, (2009).
44Higher order perturbation approach Energy splitting : lowest order contribution of topological closed string operatorsL0 is the length of topological closed string operator
45The energy splitting from higher order (degenerate) perturbation approach L : Hopping steps of quasi-particlesteff : Hopping integral: Excited energy of quasi-particlesJ. Yu and S. P. Kou, PHYS. REV. B 80, (2009).
46Topological closed string operators of four degenerate ground states for the Wen-plaquette model under x- and z-component external fields
47Effective model of four degenerate ground states for the Wen-plaquette model under x- and z-component external fields
48External 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
49External field along z direction Isotropy limit， the four degenerate ground states split three groups4×4 lattice on Wen-plaquette model under z-direction
50External 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
51Ground states energy splitting of Wen-plaqutte model on torus under a magnetic field along x-direction
52Ground states energy splitting of Wen-plaqutte model on torus under a magnetic field along z-direction
53Planar codes : topological qubits on surface with holes Fermionic basedL. B. Ioffe, et al., Nature 415, 503 (2002).
54Effective model of the degenerate ground states of multi-hole S.P. Kou, PHYS. REV. LETT. 102, (2009).S. P. Kou, PHYS. REV. A 80, (2009).The four parameters Jz, Jx, hx, hz are determined by the quantum effects of different quasi-particles.
55Unitary operations A general operator becomes : For example , Hadamard gate is
56CNOT gate and quantum entangled state of topological qubits S. P. Kou, PHYS. REV. A 80, (2009).
57III. Topological quantum computation by braiding Ising anyons
60σ:π-Flux binding a Majorana Fermion Ising anyonsMajorana fermionAnother anyonσ:π-Flux binding a Majorana FermionfluxSU(2)2 non-Abelian statistics between π-flux with a trapped majorana fermion.
61µ>0, non-Abelian Topologial state µ<0, Abelian Topologial state px+ipy-wave superconductor : an example of symmetry protected topological orderµ>0, non-Abelian Topologial stateµ<0, Abelian Topologial stateRead, Green, 2000.S. P. Kou and X.G. Wen, 2009.
65Whyπ 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
66Chiral edge state Edge state y x p+ip superconductor Edge Majorana fermionp+ip superconductorChiral fermion propagates along edgeEdge state encircling a dropletSpinor rotates by 2πencircling sampleAntiperiodic boundary condition
67Vortex (πflux) in px+ipy superconductor Single vortexE=0 Majorana fermion encircling sample : an encircling vortex - a “vortex zero mode”Fermion picks up π phase around vortex:Changes to periodic boundary condition
72Ising anyons in the generalized Kitaev model Gapped B phase are SU(2)2 non- Abelian topological order for K>0.Boundaries:Vortex-free: J=1/2Full-vortex: J=1/√2Sparse: 1/2 ≤ J ≤ 1/√2(Jz = 1 and J = Jx = Jy )
73px+ipy SC for generalized Kitaev model by Jordan-Wigner transformation Y. Yue and Z. Q. Wang, Europhys. Lett. 84, (2008)
74Topological 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
75Braiding operator for two-anyons The braiding matrices are (Ivanov, 2001) :
76Braiding matrices for the degenerate states of four Ising anyons Two- qubit