Demonstrating anyonic fractional statistics with a six-qubit quantum simulator
Chao-Yang Lu, Wei-Bo Gao, Otfried Gühne, Xiao-Qi Zhou, Zeng-Bing Chen, Jian-Wei Pan
aa r X i v : . [ qu a n t - ph ] O c t Demonstrating anyonic fractional statistics with a six-qubit quantum simulator
Chao-Yang Lu, Wei-Bo Gao, Otfried G¨uhne,
2, 3
Xiao-Qi Zhou, Zeng-Bing Chen, and Jian-Wei Pan
1, 4 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, 230026, China Institut f¨ur Quantenoptik und Quanteninformation,¨Osterreichische Akademie der Wissenschaften, Technikerstraße 21A, A-6020 Innsbruck, Austria Institut f¨ur theoretische Physik, Universit¨at Innsbruck, Technikerstraße 25, A-6020 Innsbruck Physikalisches Institut, Universit¨at Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany (Dated: October 24, 2018)Anyons are exotic quasiparticles living in two dimensions that do not fit into the usual categories offermions and bosons, but obey a new form of fractional statistics. Following a recent proposal [Phys.Rev. Lett. , 150404 (2007)], we present an experimental demonstration of the fractional statisticsof anyons in the Kitaev spin lattice model using a photonic quantum simulator. We dynamicallycreate the ground state and excited states (which are six-qubit graph states) of the Kitaev modelHamiltonian, and implement the anyonic braiding and fusion operations by single-qubit rotations. Aphase shift of π related to the anyon braiding is observed, confirming the prediction of the fractionalstatistics of Abelian 1 / Quantum statistics classifies fundamental particles inthree dimensions as bosons and fermions. Interestingly,in two dimensions the laws of physics permit the existenceof exotic quasiparticles—anyons—which obey a new sta-tistical behavior, called fractional or braiding statistics[1]. That is, upon exchange of two such particles, the sys-tem wave function will acquire a statistical phase whichcan take any value—hence this name. Anyons have beenpredicted to live as excitations in fractional quantum Hall(FQH) systems [2, 3, 4]. Alternatively, quantum stateswith anyonic excitations can be artificially designed inspin model systems that possess highly nontrivial groundstates with topological order. A prominent example is theKitaev spin lattice model [5, 6], which opened the avenueof fault-tolerant topological quantum computing [7, 8].It is an important goal to manipulate the anyons anddemonstrate their exotic statistics. Towards this goal,a number of theoretical schemes have been proposed,both in the FQH regime [9] and in the Kitaev models[10, 11, 12, 13, 14]. However, it has proved extremely dif-ficult to experimentally detect the fractional statistics as-sociated with anyon braiding. While recent experimentsin the FQH systems have indeed revealed some signa-tures of anyonic statistics [15, 16], resolving individualanyons remains elusive [7]. Here we take a different ap-proach to this challenge — exploiting the spin models tostudy the anyonic statistics. Following a recent proposal[14], we demonstrate the fractional statistics of anyonsby simulation of the Kitaev model on a six-photon graphstate. The method is to dynamically create the groundstate and excited state of the anyonic model Hamilto-nian, and implement the braiding and fusion operationsby single-qubit rotations.How can the statistical nature of elementary particlesbe experimental observable? First let us recall the con-cept of quantum statistics. The wave function of a two-particle system ψ ( r , r ) will acquire a statistical phase θ upon an adiabatic exchange of two particles, that is, ψ ( r , r ) = e iθ ψ ( r , r ), where θ = 0 for bosons, θ = π for fermions, and θ can be any value (0 θ π ) foranyons. It can be seen that a full circulation of a particlearound the other one is equivalent to two successive par-ticle exchanges [8]. After such a circulation, both bosonsand fermions show a trivial phase ( φ = 2 θ = 0 , π ), butanyons will get an observable non-trivial phase φ . To re-alize this idea, we need a system where anyons can becreated and braiding operations can be carried out ex-perimentally. The Kitaev model is well suited for this.The first Kitaev model was designed on a spin latticewith qubits living on the edges (see Fig. 1a). For eachvertex v and face f , we consider operators of the form A v = Y j ∈ star( v ) X j , B f = Y j ∈ boundary( f ) Z j , (1)where X ( Z ) denotes the standard Pauli matrix σ x ( σ z ).These operators A v , B f are put together to make up themodel Hamiltonian H = − X v A v − X f B f . (2)The ground state | ψ g i of this Hamiltonian (2) is givenby A v | ψ g i = | ψ g i and B f | ψ g i = | ψ g i for all verticesand faces. Violations of these conditions cost energy andgenerate excited states | ψ e i . A quasiparticle is createdon the vertex v i (face f i ) if A v i ( B f i ) acting on the ex-cited state | ψ e i , yields an eigenvalue − − / v f m e e m xx x x xxx zzza b FIG. 1: The first Kitaev model [5] and anyonic braiding op-erations [14]. ( a ). In this spin lattice, the qubits live on theedges and the stabilizer operators A v , B f (1) represent thefour-body interactions as illustrated. ( b ). Creation of quasi-particles and braid of an m-particle around an e-particle. Apair of e-particles (m-particles) are created on vertices (faces)by applying a Z ( X ) operation on the edge qubit. The quasi-particles can be moved horizontally and vertically by repeatedapplications of Z or X operations. The figure shows an ex-ample of how an m-particle forms a closed loop around ane-particle through a series of moves. Recently, Han, Raussendorf and Duan [14] exploitedthe fact that the statistical properties of anyons are man-ifested by the underlying ground and excited states [17].So, instead of direct engineering the interactions of theHamiltonian H (2) and ground-state cooling which areextremely demanding experimentally, an easier way isto dynamically create the ground state and the excita-tions of this model Hamiltonian, encoding the underlyinganyonic model in a multiparticle entangled state whichcan used to simulate the dynamics of the anyonic sys-tem. The quasiparticles are then defined by the nega-tive outcome of a stabilizer element A v or B f . Specif-icaly, as illustrated in Fig. 1b, with the ground state | ψ g i prepared, one can first create a pair of e-particlesby applying a single-qubit Z rotation. The system wavefunction will be in the excited state | ψ e i . To make frac-tional phase experimentally detectable in a later stage,we apply a √ Z rotation and get a superposition state(1 / √ | ψ g i + | ψ e i ). Then we create another pair of m-particles and move one of them around an e-particle alonga closed loop, and finally annihilate the m-particles. Af-ter doing so, it is predicted that a fractional phase π will be added to | ψ e i , thus the superposition state willbecome (1 / √ | ψ g i − | ψ e i ).As the anyons are perfectly localized quasiparticles inthis model Hamiltonian, a small spin lattice containingsix qubits shown in Fig. 2a allows for a proof-of-principleexperimental demonstration [5, 14]. The Hamiltonian ofthis model is H = − A − A − B − B − B − B ,where A = X X X , A = X X X X , B = Z Z Z , B = Z Z Z , B = Z Z , B = Z Z (the subscripts ofthe Pauli matrices label the qubits). The ground state | ψ i of this Hamiltonian H is locally equivalent to a BBOUV pulse BBO cdab
BBO ef a bc v v f D D D D D D ƒ⁄ d1 ƒ⁄ d2 ƒ⁄ d3 polarizer filterHWP PBS 1 1 2 3 =X X X =Z Z =Z Z Z =Z Z =Z Z =X X X X A = A B = B
22= 3 5
B = B
4= 5
HHH H ggg ggggg gg ggg g g f f f HWP@0HWP@45QWP@0
O OO
ZXZ ¡(cid:204)
FIG. 2: ( a ). The small Kitaev spin lattice system with sixqubits used for demonstration of anyonic braiding operations.( b ). The six-qubit graph state which, after Hadamard (H)transformations on qubit 2, 3, 4, and 5, is equivalent to theground state of the system in Fig. 2a. The graph state isassociated with a graph, where each vertex denotes a qubitprepared in the state (1 / √ | i + | i ) and each edge repre-sents a controlled phase gate having been applied between thetwo connected qubits [18, 19]. The graph state is a commoneigenstate of the stabilizer operators g i , that is, g i | ψ i = | ψ i ,which describe the correlation in the state, and the graphstate is the unique state fulfilling this. Here we use the samelabel as Fig. 2a-b in ref. [14]. ( c ). Experimental set-up for thegeneration of graph state and demonstration of braiding oper-ations. A pulsed ultraviolet laser successively passes throughthree β -barium borate (BBO) crystals to generate three pairsof entangled photons [20]. The photons a , b , c , d , and f arecombined on the three polarizing beam splitters (PBSs) stepby step [21]. To achieve good spatial and temporal overlap, allphotons are spectrally filtered (∆ λ FWHW = 3 . , · · · , D ).The detector labels correspond to the qubit labels in the textand in Fig. 2a-b. The coincidence events are registered bya programmable multichannel coincidence unit. For single-qubit rotations and polarization analysis, quarter wave plates(QWPs), half wave plates (HWPs), together with polarizersor PBSs are used. six-qubit graph state [18, 19], which can be representedby the graph as depicted in Fig. 2b. This equivalencefollows from the fact that the operators A , · · · , B in theKitaev model can be uniquely derived from the stabilizeroperators g i (see Fig. 2b) of the graph state.Now we proceed with the experiment in three steps:(1) create and analyze of the ground state | ψ i , (2) ver-ify the anyonic excitations, (3) implement the braidingoperations and detect the anyonic phase. We use sin-gle photons as a real physical system to simulate thecreation and control the anyons. The quantum statesare encoded in the polarization of the photons whichare robust to decoherence. The experimental set-upis illustrated in Fig. 3c. We start from three pairsof entangled photons produced by spontaneous para-metric down-conversion (SPDC) [20]. The photons inspatial modes a - b and e - f are prepared in the states | φ + i ij = (1 / √ | H i i | H i j + | V i i | V i j ), while those inmode c - d are disentangled using polarizers and then pre-pared in the states | + i i = (1 / √ | H i i + | V i i ), where H ( V ) denotes horizontal (vertical) polarization, and i and j label the spatial modes. We then pass the photonsthrough a linear optics network (see Fig. 3c). A coinci-dence detection of all six outputs corresponds exactly tothe ground state | ψ i = 12 ( | H i | H i | H i | H i | H i | H i + | V i | V i | V i | H i | H i | H i + | H i | H i | V i | V i | V i | V i + | V i | V i | H i | V i | V i | V i ) . (3)To verify that the ground state | ψ i has been obtained,first we experimentally measure the expectation valuesof its stabilizer operators A , · · · , B . These stabilizeroperators describe the intrinsic correlations in the state | ψ i and uniquely define it, thus their expectation val-ues could serve as a good experimental signature. For anideal state | ψ i , all expectation values should give +1.In our experiment however, the ground state was createdimperfectly. Figure 3a shows the measurement results,with all expectation values being positive in a rang be-tween 0 . ± .
04 and 0 . ± .
03, in qualitative agreementwith the theoretical prediction. For a more completeand quantitative analysis, we aim to estimate the fidelityof the produced state, that is, its overlap with the de-sired one. This quantity is given by F ψ = h ψ | ρ exp | ψ i ,which is equal to one for an ideal state, and 1 /
64 for acompletely mixed six-qubit state. To do so, we considera special observable, which allows for lower bounds onthe fidelity, while being easily measurable with few cor-relation measurements [22, 23]. By making these mea-surements, we estimate the fidelity of the created groundstate to be F ψ > . ± .
041 [24]. The imperfectionof this state is mainly caused by the high-order photonemissions during the SPDC and the partial distinguisha-bility of independent photons [25].We now move to the step (2). With the ground state | ψ i created, we apply a Z (X) rotation on qubit 3 (4),creating an excited state | ψ em i on which a pair of e-particles live on the vertices v and v , and another pairof m-particles on faces f and f (see Fig. 2a). The Z and X rotations are experimentally realized using HWPsoriented at 0 ◦ and 45 ◦ , respectively. As discussed be- X1X2X3X3X4X5X6Z1Z3Z4Z2Z3Z5Z4Z6Z5Z6 A A B B B B E x pe c t a t i on v a l ue X1X2X3X3X4X5X6Z1Z3Z4Z2Z3Z5Z4Z6Z5Z6 -0.8-0.6-0.4-0.20.00.20.40.60.8 A A B B B B a Ground state b f f Excited state mm e v e X z A A B B B B A A B B B B FIG. 3: The measured expectation values of the operators A , · · · , B of the ground state | ψ i ( a ) and the excited state | ψ em i ( b ). The excited state | ψ em i has a pair of e-particleson v , v and a pair of m-particles on f , f , thus the values for A , A , B , B become negative. Each expectation value isderived from a complete set of 64 six-fold coincidence eventsin 15h in measurement basis Z ⊗ or X ⊗ . The error barsrepresent one standard deviation, deduced from propagatedPoissonian statistics of the raw detection events. fore, the anyonic excitations are signaled by violationsof stabilizer conditions, that is, A v i | ψ em i = −| ψ em i , B f i | ψ em i = −| ψ em i [5]. Thus in our case, theoretically,the expectation values of A and A should become − B and B due to the m-particles. To verify this, we measure theexpectation values of the operators A , · · · , B . The re-sults are shown in Fig. 3b, where the values of A , A , B and B flip compared to those shown in Fig. 3a whichsupports the presence of anyonic excitations [5, 14].Now we proceed to the step (3). On the ground state | ψ i , first we apply a √ Z operation using a QWP ori-ented at 0 ◦ on the qubit 3 of the ground state | ψ i , yield-ing a superposition state | ψ s i = (1 / √ | ψ i + | ψ e i ),where | ψ e i is the excited state with a pair of e-particleson v and v . With an X rotation on the qubit 4 we fur-ther create a pair of m-particles on f and f . Then weperform four X operations on the qubits 6-5-3-4 to imple-ment the braiding operation, that is, the m-particle on f is moved around the e-particle on v along an anticlock-wise closed loop. We note that the crossing at the qubit3, which is unavoidable in two dimensions, is relevant forthe unusual statistics. Finally, the pair of m-particles isannihilated with an X operation on qubit 4 (fusion).After these, if there is a fractional phase φ acquired, thestate | ψ s i will become | ψ f i = (1 / √ | ψ i + e iφ | ψ e i ).To determine this φ , we look at the correlation mea-surement outcomes of the six photons where the pho-tons 1 and 2 are fixed at | + i polarization, 4, 5 and6 at | H i and the photon 3 is measured in the basis( | + i + e iα |−i ) with α varying in π/ C ( φ, α ) ∝ φ − α ) for the state | ψ f i , thus X1X2X3X3X4X5X6Z1Z3Z4Z2Z3Z5Z4Z6Z5Z6 A A B B B B E x pe c t a t i on v a l ue S i x f o l d c o i n c i den c e ( hou r s ) Angle of (degree)
Before braiding After braiding a b A A B B B B FIG. 4: ( a ). Measured fringes for the state | ψ s i and | ψ f i .The measurements in basis ( | + i + e iα |−i ) are done using acombination of HWPs, QWP and PBS. ( b ). The expectationvalues of the operators A , · · · , B of the state | ψ f i after the √ Z transformation. an unknown phase φ , if occurs, can be revealed. Fig-ure 4a shows the measurement results for both the state | ψ s i and | ψ f i , before and after the process of m-particlecreation, braiding and fusion. These two curves clearlyexhibit a phase difference of π , confirming the predictionof the fractional statistics.For a more complete proof, we implement a √ Z trans-formation on the qubit 3 of the remaining state | ψ f i .The state | ψ f i will be converted to | ψ i if there is a frac-tional phase π , otherwise it will go to | ψ e i . To test this,again we measure the expectation values of the opera-tors A , · · · , B of the state after the √ Z transformation.The experimental results are shown in Fig. 4b, which arein agreement with that the final state is | ψ i and thusprove the fractional phase change of φ = π . Here we notethat the facts that the present setup use free-flying non-interacting photons and that the timescale of the braidingoperations is extremely small ( ∼ picoseconds for photonspassing through the HWPs and QWPs) implies that thisacquired phase cannot be a dynamical phase. Moreover,the creating, braiding and annihilating of the m-particleswhich corresponds to the operation X X X X do notgive rise to a phase either, as the ground state | ψ i is aneigenstate of this observable X X X X . Similar argu-ments also apply to the e-particles’ case. Consequently,this excludes possible geometrical phases [26] and provesthe observed phase is purely statistical.In summary, we have demonstrated the creation andmanipulation of anyons in the Kitaev spin lattice model,and observed the fractional statistics of the abelian 1 / [1] J.M. Leinaas, J. Myrheim, Nuovo Cimento , 1(1977); F. Wilczek, Phys. Rev. Lett. , 1144 (1982).[2] D.C. Tsui, H.L. St¨ormer, A.C. Gossard, Phys. Rev. Lett. , 1559 (1982).[3] R.B. Laughlin, Phys. Rev. Lett. , 1395 (1983).[4] X.-G. Wen, Quantum Field Theory of Many-body Sys-tems (Oxford Univ. Press, Oxford, 2004).[5] A.Y. Kitaev, Ann. Phys. , 2 (2003).[6] A.Y. Kitaev, Ann. Phys. , 2 (2006).[7] F. Wilczek, Phys. World , 22 (2006).[8] G.K. Brennen, J.K. Pachos, arXiv:0704.2241.[9] S. Das Sarma, M. Freedman, C. Nayak, Phys. Rev. Lett. , 166802 (2005); A. Stern, B. I. Halperin, Phys. Rev.Lett. , 016802 (2006); P. Bonderson, A. Kitaev, K.Shtengel, Phys. Rev. Lett. , 016803 (2006).[10] L.-M. Duan, E. Demler, M.D. Lukin, Phys. Rev. Lett. , 090402 (2003).[11] A. Micheli, G.K. Brennen, P. Zoller, Nature Phys. , 341(2006).[12] C.-W. Zhang, V.W. Scarola, S. Tewari, S.D. Sarma, Proc.Natl. Acad. Sci. USA , 18415 (2007).[13] L. Jiang et al. Nature Phys. , 482 (2008).[14] Y.-J. Han, R. Raussendorf, L.-M. Duan, Phys. Rev. Lett. , 150404 (2007)[15] F.E. Camino, W. Zhou, V.J. Goldman, Phys. Rev. B ,075342 (2005).[16] E.-A. Kim et al. Phys. Rev. Lett. , 176402 (2005).[17] see also M. Levin and X.-G. Wen, Phys. Rev. Lett. 96,110405 (2006).[18] H.J. Briegel, R. Raussendorf, Phys. Rev. Lett. , 910(2001).[19] M. Hein, J. Eisert, H.J. Briegel, Phys. Rev. A , 062301(2004).[20] P. G. Kwiat et al. , Phys. Rev. Lett. , 4337 (1995).[21] J.-W. Pan et al. , Phys. Rev. Lett. , 4435 (2001). [22] O. G¨uhne, et al. Phys. Rev. A , 030305 (2007).[23] see EPAPS on line for supplementary information.[24] Though some connection between the ground-state fi-delity of the Kitaev code and its topological order hasbeen explored in literatures (see e.g. A. Hamma, R. Ion-iciou, P. Zanardi, Phys. Lett. A , 22 (2005); M.Aguado, G. Vidal, Phys. Rev. Lett. , 070404 (2008);L. Amico et al. , Rev. Mod. Phys. , 517 (2008)), a strictquantitative criteria remains unclear concerning the fi-delity requirements to prove the fractional statistics. Nev-ertheless, our results shown in Fig. 3,4 display reasonablyhigh visibility for experimental confirmation. [25] T.J. Weinhold et al. arXiv:0808.0794.[26] M. Levin, X.-G. Wen, Phys. Rev. B , 245316 (2003).[27] S. Lloyd, Science , 1073 (1996); M. H. Freedman, A.Kitaev, Z. Wang arXiv:quant-ph/0001071.[28] W.-B. Gao, et al. arXiv: 0809.4277.[29] R. Raussendorf, J. Harrington, K. Goyal, New J. Phys. , 199 (2007); A. G. Fowler, K. Goyal, arXiv: 0805.3202(2008); S.J. Devitt, et al. arXiv:0808.1782.[30] Note added : After the submission of our results toarXiv (0710.0278), we became aware of two relatedwork: J.K. Pachos et al. arXiv:0710.0895; J. Du et al.et al.