Topological Color Codes and Two-Body Quantum Lattice Hamiltonians
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Topological Color Codes and Two-Body Quantum LatticeHamiltonians
M. Kargarian , H. Bombin and M.A. Martin-Delgado Physics Department, Sharif University of Technology, Tehran 11155-9161, Iran Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA Departamento de F´ısica Te´orica I,Universidad Complutense, 28040 Madrid, Spain (Dated: October 11, 2018) bstract Topological color codes are among the stabilizer codes with remarkable properties from quantuminformation perspective. In this paper we construct a lattice, the so called ruby lattice, withcoordination number four governed by a 2-body Hamiltonian. In a particular regime of couplingconstants, in a strong coupling limit, degenerate perturbation theory implies that the low energyspectrum of the model can be described by a many-body effective Hamiltonian, which encodes thecolor code as its ground state subspace. Ground state subspace corresponds to vortex-free sector.The gauge symmetry Z × Z of color code could already be realized by identifying three distinctplaquette operators on the ruby lattice. All plaquette operators commute with each other and withthe Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-netstructures. Non-contractible closed strings winding the space commute with Hamiltonian but notalways with each other. This gives rise to exact topological degeneracy of the model. Connection to2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbativelevel. The particular structure of the 2-body Hamiltonian provides a fruitful interpretation interms of mapping to bosons coupled to effective spins. We show that high energy excitations ofthe model have fermionic statistics. They form three families of high energy excitations each ofone color. Furthermore, we show that they belong to a particular family of topological charges.The emergence of invisible charges related to the string-net structure of the model. The emergingfermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermionscan see the background fluxes in the ground state. Also, we use Jordan-Wigner transformation inorder to test the integrability of the model via introducing of Majorana fermions. The four-valentstructure of the lattice prevents the fermionized Hamiltonian to reduce to a quadratic form dueto interacting gauge fields. We also propose another construction for 2-body Hamiltonian basedon the connection between color codes and cluster states. The corresponding 2-body Hamiltonianencodes cluster state defined on a bipartite lattice as its low energy spectrum, and subsequentselective measurements give rise to the color code model. We discuss this latter approach alongthe construction based on the ruby lattice. PACS numbers: 03.65.Vf,75.10.Jm,71.10.Pm,05.30.Pr ontents I. Introduction II. Quantum Lattice Hamiltonian with Two-Body Interactions
III. String Operators and Integrals of Motion IV. A Gapped Phase: The Topological Color Code
V. Bosonic Mapping
VI. Fermionic Mapping VII. Conclusions A. 2-Body Hamiltonian for Color Codes using Cluster States References References I. INTRODUCTION
Topological color codes (TCC) are a whole class of models that provide an instance ofan interdisciplinary subject between Quantum Information and the physics of QuantumMany-Body Systems.Topological color codes were introduced [1] as a class of topological quantum codes thatallow a direct implementation of the Clifford group of quantum gates suitable for entangle-3ent distillation, teleportation and fault-tolerant quantum computation. They are definedon certain types of 2D spatial lattices. They were extended to 3D lattices [2] in order toachieve universal quantum computation with TCCs. This proposal of topological quantumcomputation relies solely on the topological properties of the ground state sector of certainlattice Hamiltonians, without resorting to braiding of quasiparticle excitations. In additionto these applications in Quantum Information, topological color codes have also a naturalapplication in strongly correlated systems of condensed matter with topological orders. In[3] was found that TCCs can be extended to arbitrary dimensions, giving rise to topologicalorders in any dimension, not just 2D. This is accomplished through the notion of D -colexes,which are a class of lattices with certain properties where quantum lattice Hamiltonians aredefined. This corresponds to a new class of exact models in D=3 and higher dimensionsthat exhibit new mechanisms for topological order: i/ brane-net condensation; ii/ existenceof branyons; iii/ higher ground-state degeneracy than other codes; iv/ different topologicalphases for D ≥ • there is an energy gap between the ground state and the excitations;5 topological degeneracy of the ground state subspace (GS); • this degeneracy cannot be lifted by local perturbations; • localized quasiparticles as excited states: anyons; • edge states;etc.These features reflect the topological nature of the system. In addition, a signature ofthe TO is the dependence of that degeneracy on topological invariants of the lattice wherethe system is defined, like Betti numbers [3].But where do we find topological orders? These quantum phases of matter are difficultto find. If we are lucky, we may find them on existing physical systems such as the quantumHall effect. But we can also engineer suitable quantum Hamiltonian models, e.g., using polarmolecules on optical lattices [14, 15, 20], or by some other means. There are methods fordemonstrating topological order without resorting to interferometric techniques [29].In this paper we present new results concerning the realization of 2-body Hamiltoniansusing cluster states techniques on one hand, and without measurement-based computationson the other. In this latter case, we present a detailed study of the set of integrals of motion(IOM) in a 2-body Hamiltoinan, fermionic mappings of the original spin Hamiltonian thatgive information about the physics of the system and which complements previous resultsusing bosonic mapping techniques [10].This paper is organized as follows: in Sect. II we present color codes as instances oftopological stabilizer codes with Hamiltonians based on many-body interacting terms andthen introduce the quantum Hamiltonian model based solely on 2-body interactions betweenspin- particles. The lattice is two-dimensional and has coordination number 4, instead ofthe usual 3 for the Kitaev model. It is shown in Fig. 3 and it is called ruby lattice. In Sect.III, we describe the structure of the set of exact integrals of motion (IOM) of the 2-Bodymodel. We give a set of diagrammatic local rules that are the building blocks to constructarbitrary IOMs. These include colored strings and string-nets constant of motion, which is adistinctive feature with respect to the Kitaev’s model. In Sect. IV, we establish a connectionbetween the original topological color code and the new 2-Body color model. This is donefirstly at a non-perturbative level using the colored string integrals of motion that are related6ith the corresponding strings in the TCC. Then, using degenerate perturbation theory inthe Green function formalism, it is possible to describe a gapped phase of the 2-Body colormodel that corresponds precisely to the topological color code. In Sect. V, we introducea mapping from the original spin- degrees of freedom onto bosonic degrees of freedom inthe form of hard-core bosons which also carry a pseudospin. This provides an alternativeway to perform perturbation theory and obtain the gapped phase corresponding to theTCC. It also provides a nice description of low energy properties of the 2-Body model andits quasiparticles. In Sect. VI, we introduce another mapping based on spinless fermionswhich is helpful to understand the structure of the 2-Body Hamiltonian and the presenceof interacting terms which are related to the existence of stringnets constants of motion.Sect. VII is devoted to conclusions and future prospects. A describes how to obtain 2-BodyHamiltonians for topological color codes based on cluster states and measurements usingancilla qubits. II. QUANTUM LATTICE HAMILTONIAN WITH TWO-BODY INTERACTIONSA. Color Codes as Topological Stabilizers
Some of the simplest quantum Hamiltonian models with topological order can be obtainedfrom a formalism based on the local stabilizer codes borrowed from quantum error correction[30] in quantum information [31, 32]. These are spin- local models of the form H = − X i S i , S i ∈ P n := h i, σ x , σ z , . . . , σ xn , σ zn i . (1)where the stabilizer operators S i constitute an abelian subgroup of the Pauli group P n of n qubits, generated by the Pauli matrices not containing −
1. The ground state is a stabilizercode since it satisfies the condition S i | GS i = | GS i , ∀ i, (2)and the excited states of H are gapped, and correspond to error syndromes from the quantuminformation perspective S i | Ψ i = −| Ψ i . (3)The seminal example of topological stabilizer codes is the toric code [33]. There arebasically two types of known topological stabilizer codes [13]. It is possible to study this type7f homological error correcting codes in many different situations and perform comparativestudies [34–42]. Topological color code (TCC) is the another relevant example of topologicalstabilizer codes, with enhanced computational capabilities [1–3]. In particular, they allowthe transversal implementation of Clifford quantum operations. The simplest lattice toconstruct them is a honeycomb lattice Λ shown in Fig.1, where we place a spin- system ateach vertex. There are two stabilizer operators per plaquette: B xf = τ x τ x τ x τ x τ x τ x ,B yf = τ y τ y τ y τ y τ y τ y , (4) H cc = − X f ( B xf + B yf ) , (5)where τ ν ’s ( ν = x, y ) are usual Pauli operators. There exist six kinds of basic excitations. FIG. 1: The hexagonal lattice is an example of 3-colorable lattice by faces, and also by edges. Atopological color code can be defined on it by associating two stabilizer operators for each plaquette(4).
To label them, we first label the plaquettes with three colors: Notice that the lattice is3-valent and has 3-colorable plaquettes. We call such lattices 2-colexes [3]. One can definecolor codes in any 2-colex embedded in an arbitrary surface. There exists a total of 15nontrivial topological charges as follows. The excitation at a plaquette arises because of theviolation of the stabilizer condition as in (3). Consider a rotation τ y applied to a certainqubit. Since τ y anticommutes with plaquette operators B xf of neighboring plaquettes, it willput an excitation at corresponding plaquette. Similarly, if we perform a τ x rotation on a8ubit, the plaquette operators B yf are violated. These are the basic excitations, two typesof excitations per each colored plaquette. Regarding the color and type of basic excitations,different emerging excitations can be combined. The whole spectrum of excitations is shownin Fig.2(a). Every single excitation is boson by itself as well as the combination of two basicexcitations with the same color. They form nine bosons. However, excitations of differentcolor and type have semionic mutual statistics as in Fig.2(b). The excitations of differentcolor and type can also be combined. They form two families of fermions. Each familyof fermions is closed under fusion, and fermions from different families have trivial mutualstatistics. This latter property is very promising and will be the source of invisible chargesas we will discuss in Sect.V. The anyonic charge sectors are in one to one correspondencewith the irreducible representations (irreps) of the underlying gauge group, and the fusioncorresponds to decomposition of the tensor product of irreps. FIG. 2: (a) Classification of excitations for the topological color code model (5), nine bosons andtwo families of fermions (b) The nontrivial phase arising from the braiding of different charges.
We describe all above excitations in terms of representation of the gauge group of theTCC. Before that, let us make a convention for colors which will be useful for subsequentdiscussions. We refer to colors by a bar operation ¯ c that transform colors cyclically as ¯r = g,9g = b and ¯b = r. The elements of the gauge group Z × Z are { e , r , b , g } . Each excitationcarries a topological charge. The corresponding topological charge can be labeled by the pair( q, χ ), where q ∈ Z × Z and χ an irrep of this group [10]. We label them as χ e ( c ) = χ c ( c ) = − χ c (¯ c ) = 1. Therefore, there are nine bosons labeled by ( c, χ e ), ( e, χ c ) and ( c, χ c ) and sixfermions ( c, χ ¯ c ) and ( c, χ ¯¯ c ). Taking into account the vacuum with trivial charge ( e, χ e ), colorcode has sixteen topological charges or superselection sectors. Regarding the fusion process,fusion of two charges ( q, χ c ) and ( q ′ , χ c ′ ) give rises to ( qq ′ , χ c χ c ′ ) charge. Additionally,the braiding of charge ( q, χ c ) around charge ( q ′ , χ c ′ ) produces the phase χ c ( q ′ ) χ c ′ ( q ). Anexcitation at a c -plaquette has ( c, χ e ) charge if − B x = B y = 1, ( e, χ c ) charge if B x = − B y =1 and ( c, χ c ) charge if B x = B y = − p − T has been mapped out using MonteCarlo, which in particular gives the value of the error threshold p c . This particular pointcan also be addressed using multicritical methods[49]. There is experimental realizationof topological error correction [50]. Without external active error correction, the effect ofthermal noise is the most challenging problem in toric codes [51–56]. Finite temperatureeffects of topological order in color codes has also been studied [57].In all, the type of entanglement exhibited by topological color codes is very remarkable[58–64]. A very illustrative way to see this is using the connection of the ground state oftopological codes with standard statistical models by means of projective measurements[7, 65–71]. For TCCs, this mapping yields the partition function of a 3-body classicalIsing model on triangular lattices [7]. This 3-body model is the same found in active er-ror correcting techniques [48], but without randomness since there is no noise produced byexternal errors. This type of statistical mapping allows us to test that different computa-tional capabilities of color codes correspond to qualitatively different universality classes oftheir associated classical spin models. Furthermore, generalizing these statistical mechan-10cal models for arbitrary inhomogeneous and complex couplings, it is possible to study ameasurement-based quantum computation with a color code state and we find that theirclassical simulatability remains an open problem. This is in sharp contrast with toric codeswhich are classically simulable within this type of scheme [66]. B. The Model
In nature, we find that interactions are usually 2-body interactions. This is becauseinteractions between particles are mediated by exchange bosons that carry the interactions(electromagnetic, phononic, etc.) between two particles.The problem that arises is that for topological models, like the toric codes and color codes,their Hamiltonians have many-body terms (5). This could only achieved by finding someexotic quantum phase of nature, like FQHE, or by artificially enginering them somehow.Here, we shall follow another route: try to find a 2-body Hamiltonian on a certain 2Dlattice such that it exhibits the type of topological order found in toric codes and colorcodes. In this way, their physical implementation looks more accessible.In fact, Kitaev [72] introduced a 2-body model in the honeycomb lattice that gives riseto an effective toric code model in one of its phases. It is a 2-body spin- model in ahoneycomb lattice with one spin per vertex, and simulations based on optical lattices havebeen proposed [17].The model features plaquette and strings constants of motion. Furthermore, it is exactlysolvable, a property that is related to the 3-valency of the lattice where it is defined [72–78].It shows emerging free fermions in the honeycomb lattice. If a magnetic field is added,it contains a non-abelian topological phase (although not enough for universal quantumcomputation). Interestingly enough, another regime of the model gives rise to a 4-bodymodel, which is precisely an effective toric code model. A natural question arises: Can weget something similar for color codes? We give a positive answer in what follows.Motivated by these physical considerations related to a typical scenario in quantum many-body physics, either condensed matter, AMO physics or the like, we will seek a quantumspin Hamiltonian with the following properties:i/ One of its phases must be the TCC.ii/ To have two sets of plaquette operators generating a Z × Z local, i.e. gauge, symmetry.11ii/ To have string-nets and colored strings IOM as in the TCC, but in all coupling regimes.Thus, the reasons behind demanding these properties are to guarantee that the soughtafter model will host the TCC. For instance, property i/ means that we must be able togenerate the 2D color code Hamiltonian consistently at some lowest order in perturbationtheory (PT). This we shall see in Sect.IV B. Likewise, properties ii/ and iii/ are demandedin order to have the fundamental signatures regarding gauge symmetry and constants ofmotions associated with TCCs. Notice that we have not demanded that the model beexactly solvable. This is a mathematical requisite, rather than physical. We leave the dooropen for considering larger classes of models beyond exactly solvable models, which maybe very interesting and contain new physics. For example, according to those properties, itwould be possible to have models with a number of IOMs that scales linearly with N , thenumber of spins or qubits. Thus, the Kitaev model has a number of IOMs of N .Our purpose is to present first the 2-body quantum Hamiltonian in 2D [10], and then toanalyze diverse possible mappings in later sections, like using bosonic and fermionic degreesof freedom. The analysis of the set of IOMs will play also a crucial role in the understandingof our model as we shall see in Sect.IIIIt is a 2-body spin-1/2 model in a ’ruby’ lattice as shown in Fig.3. We place one spin pervertex. Links come in 3 colors, each color representing a different interaction. H = − X h i,j i J w σ wi σ wj , w = x, red links y, green links z, blue links (6)For a suitable coupling regime, this model gives rise to an effective color code model.Furthermore, it exhibits new features, many of them not present in honeycomb-like models: • Exact topological degeneracy in all coupling regimes (4 g for genus g surfaces). • String-net integrals of motion. • Emergence of 3 families of strongly interacting fermions with semionic mutual statis-tics. • Z × Z gauge symmetry. Each family of fermions sees a different Z gauge subgroup.12 IG. 3: A lattice with coordination number 4 where the 2-body quantum lattice Hamiltonian forthe color codes is defined according to spin-spin interactions coded by the colors of the links, as in(6). A plaquette can be distinguished by an inner hexagon, an outer hexagon and six blue trianglesbetween them.
III. STRING OPERATORS AND INTEGRALS OF MOTION
We can construct integrals of motion (IOM), I ∈ P n , [ H cc , I ] = 0, following a pattern ofrules assigned to the vertices of the lattice, as shown in Fig.4. These rules are constructedto attach a Pauli operator of type σ xi , σ yi or σ zi to each of the vertices i . The lines around thevertices, either wavy lines or direct lines, are pictured in order to join them along paths ofvertices in the lattice that will ultimately translate into products of Pauli operators, whichwill become IOMs. Clearly, σ zi operators are distinguished from the rest. The contributionof each qubit in the string operator is determined in terms of how it appears in the string.Its contribution may be determined by the outgoing red and green links which have thequbit as their end point in the string. In this case the σ x or σ y Pauli operators contributein the string IOM . If a typical qubit crossed only by a wavy line as shown in Fig.4(a), itcontributes a σ z Pauli operator in the string. To have a clear picture of string operators, atypical example has also been shown in Fig.4(b). Part of string is shown on the left and itsexpression will be the product of Pauli operators which have been inserted in open circleson the right. With such definitions for string operators and their supports on the lattice,now we turn on to analyze the relevance of strings to the model. In particular, we will13onstruct elementary string operators with the local symmetry of the model. Therefore, inthis way we are representing the local structure of the IOMs of our 2-body Hamiltonian(5). We will illustrate them with several examples of increasing complexity. The ground
FIG. 4: (a) A diagrammatic representation of the local structure of the integrals of motion ofthe 2-body Hamiltonian (6). The colored links represent different spin-spin interactions. (b) Anexample of contribution of Pauli operators in a string. state of a lattice model described by the Hamiltonian (5) is a superposition of all closedcolored strings. Indeed, it is invariant under any deformations of colored strings as well assplitting of a colored string into other colors. In other words, the ground state is a string-netcondensed state and supports topological order. The gauge group related to this topologicalorder is Z × Z . Such symmetry of topological color code can be realized via defining aset of closed string operators on the ruby lattice. We shall verify the gauge symmetry byidentifying a set of string operators on the lattice of Fig.3.Let us start by constructing the elementary string IOM as shown in Fig.5. They aredenoted as I = A, B, C . They are closed since they have not endpoints left. The elementaryclosed strings are plaquettes. By a plaquette we mean an inner hexagon and an outer hexagonwith six triangles in between. For a given plaquette it is possible to attach three stringoperators. For each closed string, the contribution of Pauli operators are determined basedon outgoing red and green links or wavy lines as in Fig.4. Let V f stand for a set of qubitson a plaquette. Note that each plaquette contains 18 qubits corresponding to six trianglesaround it. For first plaquette operator in Fig.5 we can write its explicit expression in terms14f Pauli matrices as S Af = Q i ∈ V f σ νi , where f denotes the plaquette and ν = x, y , dependingon outgoing red or green links, respectively. Similarly the second plaquette operator has anexpression as S Bf = Q i ∈ V f σ νi . The third string is only a closed wavy string which coincides tothe inner hexagon of the plaquette. It’s expression is S Cf = Q i ∈ V h σ zi , where V h stands for sixqubits on the inner hexagon. The three closed strings described above are not independent.Using the Pauli algebra, it is immediate to check that they satisfy S Cf = − S Af S Bf . Thus,there exist 2 independent IOMs per plaquette: this is the Z × Z local symmetry of themodel Hamiltonian (6). FIG. 5: Schematic drawing of the plaquette IOMs according to the local rules in Fig.4. There are 3IOMs denoted as
A, B, C , but only 2 of them are independent. This corresponds to the symmetry Z × Z of the model. Plaquette operators commute with each other and with any other IOM. If a IOM corre-sponds to a nontrivial cycle c , it is possible to find another IOM that anticommutes withit, namely one that corresponds to a cycle that crosses once c ′ . Thus, IOMs obtained fromnontrivial cycles are not products of plaquette operators.Each string operator squares identity since we are working with qubits. Plaquette oper-ators corresponding to different plaquettes commute with each other and also with terms inHamiltonian in (6) since they share in zero or even number of qubits. Therefore, the closedstrings with the underlying symmetry obtained above define a set of integrals of motion.The number of integrals of motions is exponentially increasing. Let 3 N be the total numberof qubits, so the number of plaquettes will be N . Regarding to the gauge symmetry ofthe model, the number of independent plaquette operators is N . This implies that thereare 2 N integrals of motion and allow us to divide the Hilbert space into 2 N sectors being15igenspaces of plaquette operators. However, for closed manifold, for example a torus, allplaquette operators can not be independently set to +1 or − FIG. 6: An example of a stringnet IOM. Notice the presence of branching points located aroundblue triangles of the lattice. This is a remarkable difference with respect to honeycomb models likethe Kitaev model.
The most general configuration that we may have is shown in Fig.6. We call them string-nets IOM since in the context of our model, they can be thought of as the string-netsintroduced to characterize topological orders [79]. The key feature of these IOMs is thepresence of branching points located at the blue triangles of the lattice. This is remarkableand it is absent in honeycomb 2-body models like the Kitaev model. When the string-nets IOM are defined on a simply connected piece of lattice they are products of plaquetteoperators. More generally, they can be topologically non-trivial and independent of plaquetteoperators.As a special case of IOMs we have string configurations, i.e., paths without branchingpoints. They correspond to the different homology classes of the manifold where the latticeis embedded , and are needed for characterization of the ground state manifold. Someexamples are shown in Fig.7. They may be open or closed, depending on whether theyhave endpoints or not, respectively. Strings IOM are easier to analyze. String-nets IOM are16roducts of strings IOM. For a given path, there exist 3 different strings IOM. These aredenoted as
A, B, C in Fig.7. We must introduce generators for the homology classes defining
FIG. 7: Examples of standard string configurations of IOMs, i.e., without branching points. Foreach path, we can in principle make 3 different assignments of IOMs, but again only 2 of them areindependent as with plaquette IOMs. This is another manifestation of the Z × Z symmetry ofthe model. closed manifold. Homology classes of the torus are determined by realizing two nontrivialloops winding around the torus. In the Kitaev’s model there are only two independent suchnontrivial closed loops. However, the specific construction of the lattice and contribution ofthe color make it possible to define for each homology class of the torus two independentnontrivial loops. These closed strings are no longer combination of plaquettes defined above.Let S Aµ stand for such string, where A and µ denote the type and homology class of thestring. For each homology class of the manifold we can realize three different types of stringoperators depending on how the vertices of the lattice are crossed by the underlying string.Each qubit crossed by the string contributes a Pauli operator according to the rules in Fig.4.Again, using Pauli algebra we can see that only two of them are independent, as with theplaquette IOMs. ( − t S Aµ S Bµ S Cµ = 1 , (7)17here t is the number of triangles on the string. To distinguish properly the three types wehave to color the lattice. We could already use the colors to label strings. Strings are thenred, green or blue. This is closely related to the topological color code [1, 10]. The latterrelation shows that each string can be constructed of two other homologous ones, which isexactly the expression of the Z × Z gauge symmetry. Each non-contractible closed stringoperator of any homology commutes with all plaquette operators and with terms appearingin the Hamiltonian, so they are constant of motions. But, they don’t always commute witheach other. In fact, if the strings cross once then (cid:2) S rµ , S rν (cid:3) = 0 , (8)but n S rµ , S r ′ ν o = 0 . (9)This latter anticommutation relation is a source of exact topological degeneracy [80] of themodel independent of phase we are analyzing it. IV. A GAPPED PHASE: THE TOPOLOGICAL COLOR CODEA. Non-Perturbative Picture
In this subsection we discuss the ruby lattice is connected to the 2-colex even at thenon-perturbative level. Then, in the subsequent sections we verify it using quantitativemethods. From the previous discussion on IOMs, we have already seen a connection withthe topological color codes. Now, we want to see how different strings introduced above arerelated to coloring of the lattice. To this end, consider the closed strings
A, B, C in Fig.5.The closed strings A and B can be visualized as a set of red and green links, respectively.With such visualization, we put forward the next step to color the inner hexagons of theruby lattice: a colored link, say red, connect the red inner hexagons. Accordingly, otherinner hexagons and links can be colored, and eventually we are left with a colored lattice.The emergence of the topological color code is beautifully pictured in Fig.8. Geometrically,it corresponds to shrinking the blue triangles of the original lattice into points, which willbe referred as sites of a new emerging lattice, see Fig.8 (left). Thus, we realize that theinner hexagons and vertices of the model are colorable, see Fig.8 (middle): if we regard blue18 IG. 8: (color online) The three stages showing the emergence of the topological color code: (left)the original lattice for the 2-body Hamiltonian (6). The colors in the links denote the type of spin-spin interactions; (middle) a different coloring of the lattice is introduced based on the propertythat the hexagons are 3-colorable, as well as the vertices; (right) the hexagonal lattice obtainedby shrinking to a point the blue triangles of the original lattice, which become sites in the finalhexagonal lattice. This corresponds to the strong coupling limit in (15). triangles as the sites of a new lattice, we get a honeycomb lattice, see Fig.8 (right). In fact,the model could be defined for any other 2-colex, not necessarily a hexagonal lattice.Connection to the 2-colexes can be further explored by seeing how strings on the rubylattice correspond to the colored strings on the effective honeycomb lattice. To this end,consider a typical string-net on the ruby lattice as shown in Fig.9(a). This corresponds to anon-perturbative picture of the IOMs of the model. The fat parts of the string-net connecttwo inner hexagons with the same color. In this way, the corresponding string-net on theeffective lattice can be colored as in Fig.9(b). The color of each part of the string-net onthe effective honeycomb lattice is determined by seeing which colored inner hexagons on theruby lattice it connects. Three colored strings cross each other at a branching point, and itsexpression in terms of Pauli matrices of sites are given by product of Pauli operators writtenadjacent to the sites. How they are determined, will be clear soon.It is possible to use colors to label the closed strings on the honeycomb lattice. Beforethat, let us use a notation for Pauli operators acting on effective spins of honeycomb lattice τ α , where α = x, y, z . We indicate the labels α as c | c := z, ¯ c | c := x, ¯¯ c | c := y , where we areusing a bar operator. To each c -plaquette, we attach three operators each of one color. Let19 c ′ f denotes such operators, where low and up indices stand for c -plaquette f and color ofthe closed string attached to the plaquette, respectively. With these notations, the plaquetteoperators read as follows: B c ′ f = Y v ∈ f τ c ′ | cv , (10)where the product runs over all vertices of the c -plaquette f in the honeycomb lattice inFig.8(c). Thus, we can write the explicit expression of operators as follows: B xf = B ¯ cf = Y v ∈ f τ ¯ c | cv B yf = B ¯¯ cf = Y v ∈ f τ ¯¯ c | cv B zf = − B cf = − Y v ∈ f τ c | cv . (11)All these plaquette operators are constant of motions. Again, We can realize the gaugesymmetry Z × Z through the relation B xf B yf B zf = 1. On a compact manifold, for exampleon the torus, all plaquettes are not independent. They are subject to the following constraint: Y f ∈ Λ B cf = ( − N/ , (12)where the product runs over all plaquettes f in the lattice Λ, and N is the total number ofplaquettes.We can also realize noncontractible strings on the effective lattice which are rooted inthe topological degeneracy of the model. They are just the IOMs in Fig.7 when reduced onthe effective honeycomb lattice. Once the inner hexagons of ruby lattice are colored, theycorrespond to colored strings as in Fig.9. Let S cµ stands for such string, where indices µ and c denote the homology and color of the string, respectively. This string operator is tensorproduct of Pauli operators of qubits lying on the string. Namely, the string operator is S cµ = Y v τ c ′ | cv . (13)The contribution of each qubit is determined by the color of the hexagon that the stringturns on it, see Fig.10. For example in the string S shown in Fig.10, the color of theplaquettes appearing in (13) marked by light circles. With this definition for string operators,the contribution of Pauli operators in the string-net on the effective lattice in Fig.9(b) are20 IG. 9: (color online) An illustraion of correspondence between (a) strings on the ruby lattice,corresponding to a non-perturbative picture, and (b)colored strings on the effective honeycomblattice. reasonable. Non-contractible colored strings are closely related to the topological degeneracyof the model, since they commute with color code Hamiltonian(5) being integrals of motion,but not always with each other. In fact, two strings differing in both homology and coloranticommute, otherwise they commute. For example let us consider two non-contractibleclosed strings S and S corresponding to different homologies of the torus. As shown inFig.10, they share two qubits. First, suppose both strings are of blue type. The contributionof Pauli operators of these two qubits in string S is τ y τ x , while for string S the contributionis τ x τ y implying [ S b , S b ] = 0. Then, let S be a green string. In this case the contributionof qubits will be τ y τ z , which explicitly shows that { S b , S g } = 0. The interplay in (7) can betranslated into an interplay between color and homology as follows.( − s S cµ S ¯ cµ S ¯¯ cµ = 1 , (14)where s is the number of sites on the string. This interplay makes the ground state subspaceof the color code model be a string-net condensed phase. B. Degenerate Perturbation Theory: Green Function Formalism
In this subsection we put the above correspondence between original 2-body lattice Hamil-tonian and color code model on a quantitative level. In fact, there is a regime of couplingconstants in which one of the phases of the 2-body Hamiltonian reproduces the TCC many-21
IG. 10: (color online) A piece of effective lattice. The strings S and S correspond to differenthomology classes of the manifold. Their expression in terms of Pauli operators are given by theirassociated color and the fact that how they turn on plaquettes on the lattice. body structure and physics. In particular, we show that this corresponds to the followingset of couplings in the original 2-body Hamiltonian J x , J y , J z > , J x , J y ≪ J z , (15)that is, a strong coupling limit in J z . The topological color code effectively emerges inthis coupling regime. This can be seen using degenerate perturbation theory in the Greenfunction formalism. Let H = H + V be a Hamiltonian describing a physical system withtwo-body interaction, and we regard the k V k , the norm of V , be very small in comparisonwith the spectral gap of unperturbed H . We also suppose that H has a degenerate ground-subspace which is separated from the excited states by a gap ∆. The effect of V will beto break the ground state degeneracy in some order of perturbation. Now the interestingquestion is whether it is possible to construct an effective Hamiltonian, H eff , which describesthe low energy properties of the perturbed Hamiltonian H . The effective Hamiltonianarises at orders of perturbation that break the ground state degeneracy. From the quantuminformation perspective the Hamiltonian H acts on the physical qubits while the effectiveHamiltonian acts on the logical qubits projected down from the physical qubits.We will clarify that how many-body Hamiltonian in (5) will present an effective descrip-tion of low lying states of the 2-body Hamiltonian (6). We use the perturbation about theHamiltonian in (6) considering the coupling regim (15). Here, the qubits on the triangles are22hysical qubits, and logical qubits are those living at the vertices of the 2-colex. We refer totriangles as sites, since they correspond to the vertices of the 2-colex. Thus a triangle willbe shown by index v and its vertices by Latin indices i, j . In fact the low lying spectrum of2-body Hamiltonian encodes the following projection from the physical qubits to the logicalones at each site: P v = | ⇑ih↑↑↑ | + | ⇓ih↓↓↓ | , (16)where | ⇑i and | ⇓i stands for the two states of the logical qubit at site v , and | ↑i ( | ↓i ) isusual up (down) states of a single spin in computational bases.To this end, we split the 2-body Hamiltonian into two parts. The unperturbed part is H = − J z P b − link σ zi σ zj . In the limit of strong Ising interaction the system is polarized. Theinteractions between neighboring qubits on different triangles are included in V . They are σ xi σ xj and σ yi σ yj corresponding to red and green links in Fig.3, respectively. So, the transversepart of the Hamiltonian is V = − J x X r − link σ xi σ xj − J y X g − link σ yi σ yj . (17)In the case when J z ≫ J x , J y the low lying excitations above the fully polarized state canbe treated perturbatively.The unperturbed part of the Hamiltonian, H , has a highly degenerate ground spacebecause, for each triangle, two fully polarized states | ↑↑↑i and | ↓↓↓i have same energy. Theground state subspace is spanned by all configurations of such polarized states. Let N bethe number of triangles of the lattice. The ground state energy is E (0)0 = − N J z and thedimension of the ground space of the H or ground state degeneracy reads g = 2 N . The firstexcited state is produced by exciting one of triangles and has energy E (0)1 = ( − N + 4) J z with degeneracy g = 6 N N − . The second excited state has energy E (0)1 = ( − N + 8) J z with degeneracy g = 18 N ( N − N − , and so on and so forth.We analyze the effect of V on the ground state manifold by using the degenerate pertur-bation theory[81] in couplings J x and J y . We are interested in how ground state degeneracyis lifted by including the interaction between triangles perturbatively. Let L stand for theground state manifold with energy E (0)0 and let P be the projection onto the ground statemanifold L . The projection is obtained from the degenerate ground states as follows: P = Y v ∈ Λ P v , P v = | ⇑ih↑↑↑ | + | ⇓ih↓↓↓ | . (18)23sing the projection and Green’s function we can calculate the effective Hamiltonian at anyorder of Perturbation theory. The eigenvalues of the effective Hamiltonian H eff appear asthe poles of the Green function G ( E ) = P [1 / ( E − H )] P . The effect of perturbation can berecast into the self-energy Σ( E ) by expressing the Green’s function as 1 / ( E − E (0)0 − Σ( E )).So, the effective Hamiltonian will be H eff = ∞ X l =0 H ( l )eff = E (0)0 + Σ( E ) . (19)The self-energy can be represented in terms of Feynman diagrams and can be computed forany order of perturbation: Σ( E ) = P V ∞ X n =0 U n P, (20)where U = [1 / ( E − H )](1 − P ) V . The energy E can also be expanded at different ordersof perturbation, E = E (0)0 + P ∞ l =1 E ( l )0 . Now, we are at the position to determine differentorders of perturbation. Each term of V acts on two neighboring physical qubits of differenttriangles. At a given order of perturbation theory, there are terms which are product of σ x and σ y acting on the ground state subspace. Each term when acts on the ground statemanifold brings the ground state into an excited state. However, there may be a specificproduct of the σ x and σ y which takes the ground state into itself, i.e. preserve the polarizedconfigurations of triangles.At zeroth-order the effective Hamiltonian will be trivial H (0)eff = E (0)0 . The first-ordercorrection is given by the operator H (1)eff = P V P. (21)The effect of V is to move the states out of the ground state manifold because each term either σ x σ x or σ y σ y flip two qubits giving rise to two triangles being excited, i.e V P = P V P , wherethe operator P is the projection to second excited state manifold. Therefore, P V P = 0,and there is no first-order correction to the ground state energy.The second-order correction to the ground state will be the eigenvalues of the followingoperator. H (2)eff = P V G ′ ( E (0)0 ) V P + P V P, (22)24here the operator G ′ ( E ) = 1 / ( E − H ) is the unperturbed Green’s function and the super-script prime stands for the fact that its value be zero when acts on the ground state. Thesecond-order correction only shifts the ground state energy, and therefore, the second-ordereffective Hamiltonian acts trivially on ground state manifold, H (2)eff = 3 N J x + J y − J z P. (23)In fact the first V flips the qubits and the second V flips them back. As we go to higherorder of perturbation theory the terms become more and more complicated. However, ifthe first-order is zero as in our case, the terms becomes simpler. Thus, the third-order ofperturbation will be zero and will leave corrections to energy and ground state intact: H (3)eff = P V (cid:16) G ′ ( E (0)0 ) V (cid:17) P = 0 . (24)The forth-order of perturbation theory contributes the following expression to the correctionof ground state manifold: H (4)eff = P V (cid:16) G ′ ( E (0)0 ) V (cid:17) P − E (2)0 P V (cid:16) G ′ ( E (0)0 ) (cid:17) V P, (25)where E (2)0 is the second order correction to the ground state energy obtained in (23). Thefirst term includes four V and must act in the ground state in which the last V returnsthe state to the ground state manifold. The second term is like the second-order. Thereare many terms which must be calculated. However, since the forth-order only gives ashift to the ground state energy, we don’t need them explicitly. So, we can skip the forth-order. Fifth-order correction yields terms each containing odd number of V , so it gives zerocontribution to the effective Hamiltonian.The sixth-order of perturbation leads to the following long expression. H (6)eff = P V (cid:16) G ′ ( E (0)0 ) V (cid:17) P − E (4)0 P V (cid:16) G ′ ( E (0)0 ) (cid:17) V P + (cid:16) E (2)0 (cid:17) P V (cid:16) G ′ ( E (0)0 ) (cid:17) V P − E (2)0 P V (cid:16) G ′ ( E (0)0 ) (cid:17) V (cid:16) G ′ ( E (0)0 ) V (cid:17) P − E (2)0 P V G ′ ( E (0)0 ) V (cid:16) G ′ ( E (0)0 ) (cid:17) V G ′ ( E (0)0 ) V P − E (2)0 P V (cid:16) G ′ ( E (0)0 ) V (cid:17) (cid:16) G ′ ( E (0)0 ) (cid:17) V P. (26)25part from the first term, other terms contain two or four V and as we discussed in thepreceding paragraphs they only contribute a shift in the ground state energy. However, thefirst term gives the first non-trivial term breaking in part the ground state degeneracy. Inthe sixth order correction, there are some terms which are the product of σ x σ x and σ y σ y associated to the red and green links of the ruby lattice. Some particular terms, as seenbelow, may map ground state subspace into itself. For instance, consider the followingproduct of links around an inner hexagon Y σ wi σ wj = ± Y i ∈ V h σ zi , (27)where the first product runs over three red and three green links making an inner hexagon, V h stands for the set of its vertices and the prefactor ± depends on the ordering of links inthe product. The action of a σ z on one vertex (qubit) of a triangle encodes an logical τ zv operator acting on the associated vertex of lattice Λ. This can explicitly be seen from thefollowing relation: τ zv = P v σ z P v = | ⇑ih⇑ | − | ⇓ih⇓ | , (28)where σ z acts on one of the vertices of a triangle and P v is the projection defined in (16).Thus, the expression of (27) can be related to the plaquette operator B zf = − Q τ zv , wherethe index f denotes a plaquette of effective lattice Λ as in Fig.1 and product runs over sixsites around it. Now we go on to pick up the sixth-order correction to the ground statemanifold. There are many terms which must be summed. Sixth-order correction up to anumerical constant contributes the following expression to the effective Hamiltonian: H (6)eff = constant − δ J x J y J z X f B zf , (29)where δ is a positive numerical constant arising from summing up 720 terms related tothe order of product of six links around an inner hexagon of ruby lattice. Although, itsexact numerical value is not important, but knowing its sign is essential for our subsequentdiscussions. As it is clear from the first term in (26), five Green’s functions and six V inthe perturbation contribute a minus sign to the expression. This minus sign together withthe sign appearing in (27) enforce the coefficient δ be a positive constant. Now it is simpleto realize how the vectors in the ground state manifold rearranged. Trivially, all plaquette26perators B zf commute with each other and their eigenvalues are ±
1. All polarized vectorsin L are the eigenvector of the effective Hamiltonian emerging at sixth-order. But, thoseare ground states of the effective Hamiltonian in (29) which are eigenvectors of all plaquetteoperators B zf with eigenvalue +1. Thus, highly degenerate ground state of the unperturbedHamiltonian is broken in part. The same plaquette operators B zf also appear at higherorder of perturbation. For example, at eighth-order. Instead of giving the rather lengthyexpression of the eighth-order correction, we only keep terms resulting in the plaquetteoperators as follows: H (8)eff = constant − β J x J y + J x J y J z X f B zf , (30)where β >
0. This term is added to the one in (29) to give the effective Hamiltonian upto eighth-order, but the ground state structure remains unchanged. Further splitting in theground state manifold is achieved by taking into account the ninth-order of perturbation.The expression of ninth-order is very lengthy. However, the first term of the expressioncontaining nine V gives some terms being able of mapping the ground state manifold intoitself in a nontrivial way. These terms map a polarized triangle, say up, to a down one.Indeed, when one or two qubits of the polarized triangle gets flipped, its state is excited.However, flipping three qubits of the triangle returns back the ground state onto itself. Thisprocess encodes τ x and τ y logical operators acting on logical qubits arising through theprojection. Let σ x , σ x and σ y act on three qubits of a triangle. The encoded τ y operatorwill be τ yv = P v σ x σ x σ y P v = − i | ⇑ih⇓ | + i | ⇓ih⇑ | . (31)If σ x , σ y and σ y act on three qubits of a triangle, the encoded τ x logical operator will be − τ xv = P v σ x σ y σ y P v = −| ⇑ih⇓ | − | ⇓ih⇑ | . (32)As we already pointed out a plaquette of the ruby lattice is made up of an inner hexagon,an outer hexagon and the six blue triangles. It is possible to act on the polarized space ofthe blue triangles by making two different combinations of 9 link interactions: i/ Applying6 link interactions on the outer hexagon (three of them of XX type and another three of YYtype), times 3 link interactions of red type on the inner hexagon. Notice that every vertex ofthe blue triangles in the plaquette gets acted upon these 9 link interactions. The resulting27ffective operator is of type τ y due to (31). ii/ Applying 6 link interactions on the outerhexagon, times 3 link interactions of green type. Then, the resulting effective operator is oftype τ x due to (32).The effective Hamiltonian at this order then reads H (9)eff = P V (cid:16) G ′ ( E (0)0 ) V (cid:17) P + ... (33)= constant − γ J x J y J z X f B yf − γ J x J y J z X f B xf . Again, the sign of coefficient γ is important. Nine V ’s, six τ x or τ y , and eight Green’sfunction imply that the γ must have positive sign.Putting together all above obtained corrections lead to an effective Hamiltonian encod-ing color code as its ground state[1, 58]. Therefore, up to constant terms, the effectiveHamiltonian reads as follows H eff = − k z X f B zf − k x X f B xf − k y X f B yf , (34)where the k z , k x and k y are positive coefficients arising at different orders. Since B xf B yf = B zf ,the above effective Hamiltonian is just the many-body Hamiltonian of the color code as in(1). The terms appearing in the Hamiltonian mutually commute, so the ground state willbe the common eigenvector of plaquette operators. Since each plaquette operator squaresidentity, the ground state subspace, C , spanned by vectors which are common eigenvectorsof all plaquette operators with eigenvalue +1, i.e C = {| ψ i : B xf | ψ i = | ψ i , B yf | ψ i = | ψ i ; ∀ f } . (35)The group of commuting boundary closed string operators can be used as an alternativeway to find the terms appearing in the effective Hamiltonian[82]. As we pointed out inthe preceding section, the non-zero contribution from various orders of perturbation theoryresults from the product of red and green links which preserve the configurations of thepolarized triangles, i.e maps ground state manifold onto itself. For instance consider theelementary plaquette operator A corresponding to a closed string in Fig.5. Each trianglecontributes σ y σ y σ x to the expression of operator which is projected to τ x as in (32). Thus,the effective representation of plaquette operator reads as follows: P S Af P → B xf . (36)28laquette operators S Bf and S Cf can also be recast into the effective forms as follows: P S Bf P → B yf , P S Cf P → − B zf . (37)These are lowest order contributions to the effective Hamiltonian as we obtained in (34).Higher order of perturbation will be just the product of effective plaquette operators. Thenontrivial strings winding around the torus will have also effective representations and appearat higher orders of perturbation. In general, every string-net IOM on the ruby lattice areprojected on an effective one as in Fig.9. V. BOSONIC MAPPING
As we stated in Sect.III, one of the defining properties of our model is the existence ofnon-trivial integrals of motion IOM, called string-nets. As a particular example, the Kitaev’smodel on the honeycomb lattice has strings IOMs, but not string-nets. We are interested inmodels in which the number of IOMs is proportional to the number of qubits in the lattice,i.e., I = ηN q , (38)where I is the number of IOMs, N q is the number of qubits (spins) in the lattice, and η is a fraction: η = for the Kitaev model [72], η = for our model in (6) [10]. The factthat η is a fraction η ≤ J z = 1 /
4, and consider the extreme case of J x = J y = 0. Inthis case the system consists of isolated triangles. The ground states of an isolated triangleare polarized states | ↑↑↑i and | ↓↓↓i with energy − /
4. The excited states that appear byflipping spins are degenerate with energy 1 /
4. In this limit, the spectrum of whole systemis made of equidistance levels being well-suited for perturbative analysis of the spectrum:Green’s function formalism as discussed in the preceding section, which may capture onlythe lowest orders of perturbation or another alternative approach based on the PCUT. Thechange from the ground state to an exited state can be interpreted as a creation of particleswith energy +1. This suggest an exact mapping from the original spin degrees of freedomto quasiparticles attached to effective spins. The mapping is exact, i.e. we don’t miss anydegrees of freedom. Such a particle is a hard-core boson. At each site, we attach such aboson and also an effective spin- . Let choose the following bases for the new degrees offreedom | a, d i = | a i ⊗ | d i , a = ⇑ , ⇓ , d = 0 , r , g , b , (39)where a and d stand for states of the effective spin and quasiparticle attached to it. TheHilbert space H C representing the hard-core bosons is four dimensional spanned by bases {| i , | r i , | g i , | b i} . Now the following construction relates the original spin degrees of freedomand new ones in (39) | ⇑ , i ≡ | ↑↑↑i , | ⇓ , i ≡ | ↓↓↓i| ⇑ , r i ≡ | ↑↓↓i , | ⇓ , r i ≡ | ↓↑↑i| ⇑ , g i ≡ | ↓↑↓i , | ⇓ , g i ≡ | ↑↓↑i| ⇑ , b i ≡ | ↓↓↑i , | ⇓ , b i ≡ | ↑↑↓i . (40)Within such mapping, the effective spins and hard-core bosons live at the sites of the effec-tive hexagonal lattice Λ in Fig. 8(c). Recall that this lattice is produced by shrinking thetriangles. At each site we can introduce the color annihilation operator as b c := | ih c | . Thenumber operator n and color number operator n c are n := X c n c , n c := b † c b c . (41)30nnihilation and creation operators anticommute on a single site, and commute at differentsites, that is why they are hard-core bosons. We can also label the Pauli operators of originalspins regarding to the their color in Fig. 8(b) as σ wc with c = r , g , b. The mapping in (40)can be expressed in operator form as follows σ zc ≡ τ z ⊗ p c , σ νc ≡ τ ν ⊗ ( b † c + b c + s ν r c ) , (42)where ν = x, y , s x := − s y := 1, the symbols τ denote the Pauli operators on the effectivespin and we are using the color parity operators p c and the color switching operators r c defined as p c := 1 − n ¯ c + n ¯¯ c ) , r c := b † ¯ c b ¯¯ c + b † ¯¯ c b ¯ c . (43)Now we can forget the original ruby lattice and work on the effective lattice in which thebosons are living at its sites. With the above identification for Pauli operators, the 2-bodyHamiltonian in (6) can be written in this language. Before that, let fix a simplified notation.All spin and bosonic operators act on the sites of the effective lattice. We refer to a siteby considering its position relative to a reference site: the notation O ,c means O applied atthe site that is connected to a site of reference by a c -link. The 2-body Hamiltonian thenbecomes H = − N/ Q − X Λ X c = c ′ J c ′ | c T c ′ c , (44)with N the number of sites, Q := P Λ n the total number of hardcore bosons, the firstsum running over the N sites of the reduced lattice, the second sum running over the 6combinations of different colors c, c ′ and T c ′ c = u c ′ c + t c ′ c + v c ′ c r c ′ c . c ., (45)a sum of several terms for an implicit reference site, according to the notation conventionwe are using. The meaning of the different terms in (45) is the following. The operator t c ′ c is a c -boson hopping, r c ′ c switches the color of two ¯ c - or ¯¯ c -bosons, u c ′ c fuses a c -boson with a¯ c -boson (or a ¯¯ c -boson) to give a ¯¯ c -boson (¯ c -boson) and v c ′ c destroys a pair of c -bosons. Theexplicit expressions are t c ′ c := τ c ′ c b c b † c,c ′ , r c ′ c := τ c ′ c r c r c,c ′ ,u c ′ c := s c ′ | c τ c ′ c b c r c,c ′ , v c ′ c := τ c ′ c b c b c,c ′ , (46)31here we are using the notation τ c ′ c := τ c ′ | c τ c ′ | c,c ′ . (47)We can also describe the plaquette IOM operators in Fig. 5 in terms of spin-boson degreesof freedom by means of the mapping in (42). For each plaquette f and color c , the plaquetteoperator is expressed as S cf := Y v ∈ f τ c | c ′ v p c ′ ⋆c , (48)where c ′ is the color of the plaquette f , the product runs through its sites and ⋆ is just aconvenient symmetric color operator defined by c ⋆ c := c, c ⋆ ¯ c := ¯ c ⋆ c := ¯¯ c. (49)The relation in (48) is just a generalization of plaquette operators in (11) to other sectorsof the system. In fact taking the zero particle sector, the expressions in (11) are recovered.In the same way the nontrivial string operators in Fig.7 can be described with the abovemapping as S cµ := Y v ∈ µ τ c | c ′ v p c⋆c ′ , (50)where µ denotes the homology class of the string. On closed surfaces, not all plaquetteoperators are independent. They are subject to the following constraints Y f ∈ Λ S cf = ( − N/ , Y c = r , g , b S cf = ( − s/ , (51)where s is the number of sites of a given plaquette f . The first equation can be furtherdivided into products over subsets of plaquettes giving rise to other constants of motion, theso called color charges, as Y f ∈ Λ ¯ c S ¯ cf Y f ∈ Λ ¯¯ c S ¯¯ cf = Y f ∈ Λ ¯¯ c S cf Y f ∈ Λ c S ¯ cf = Y Λ p c . (52)In these products the spin degrees of freedom are washed out, since they appear twice andconsequently square identity. By use of equation (43), the product over parity operators canbe written as χ Λ ( c ) = Y Λ p c = ( − Q ¯ c + Q ¯¯ c , (53)where Q c = P Λ n c is the total number of c -bosons. It is simple to check that the aboveequation commutes with Hamiltonian in (44). For each family of bosons we can attach a32harge. We suppose that each c -boson carries a charge as χ c , that is, an irrep of the gaugegroup. In particular, the Hamiltonian preserves the following total charge χ Λ := χ Q r r χ Q g g χ Q b b . (54) A. Emerging particles: anyonic fermions
Equation (52) could already suggest that the parity of vortices are correlated to the parityof the number of bosons. In particular, creation of a c -boson changes the vorticity contentof the model.The statistics of vortices depend on their color and type as in Fig.2. But what about thestatistics of c -bosons? As studied in [86], the statistics of quasiparticles can be examinedusing the hopping terms. These hopping terms are combined so that two quasiparticlesare exchanged. In addition to usual hopping terms, we also need for composite hopping ,that is, a c -boson hops from a c -plaquette to another, which is carried out by terms like t cc = u c ¯¯ c u c † ¯ c,c = u c ¯ c u c † ¯¯ c,c . Let us consider a state with two c -boson excitations located at twodifferent sites separated from a reference site by ¯ c and ¯¯ c links. An illustrative example for thecase of, say, blue bosons is depicted in Fig.11(a). Consider a process with the net effect ofresulting into the exchange of two bosons. Each step of process can be described by hoppingterms. Upon the combination of hopping terms, we are left with the following phase t ¯¯ cc t cc,c t ¯ cc t ¯¯ cc, ¯¯ c t cc t ¯ cc, ¯ c = ( τ y τ y, ¯¯ c τ z τ z,c τ x τ x, ¯ c ) = − , (55)which explicitly show that the quasiparticles made of hardcore-bosons and effective spinshave fermion statistics [10]. Thus we have three families of fermions each of one color.These are high energy fermions interact strongly with each other due to the fusion term inthe Hamiltonian. Fermions from different families have mutual semionic statistics, that is,encircling one c -fermion around a ¯ c -fermion picks up a minus sign. This can also be checkedby examining the hopping terms as in Fig.11(a). Thus we are not only dealing with fermionsbut also with anyons.The elementary operators in (46) have a remarkable property, and that, they all commutewill plaquettes and strings IOM. This naturally implies any fermionic process leaves thevorticity content of the model unaffected. A fermionic process may correspond to hopping,splitting, fusion and annihilation driven by the terms in the Hamiltonian (45) and (46). A33 IG. 11: (color online) (a)A step by step illustration of hopping of two blue fermions with the netresult of exchange of two fermions (b)A representation of a fermionic process in which a c -fermion(here red) at the origin site is annihilated and undergoes hopping, splitting and fusion processes,and then is created at another site. typical fermionic process is shown in Fig.11(b), with the net result of displacement of ar-fermion from a site at the origin to other one of the effective lattice. A very feature ofthis process is the existence of vertexes, which is essential to high-energy fermions, that is,three fermions with different color charge can fuse into the vacuum sector. At the vertex,three different colored strings meet. Notice that the colored strings shown here have nothingto do with the ones we introduced in Sect.III. Indeed, these are just product of some redand green links of the ruby lattice. When translated into the spin-boson language, they areresponsible for transportation of c -fermions through the lattice.Now we can think of constraints in (52) and (53) as a correlation between the low energyand high energy sectors of our model. They explicitly imply that the creation of a c -fermioncreates the vortices with net topological charge of (¯ c, χ ¯¯ c ). Alternatively, as suggested bymapping in (40), flipping of spin on a triangular can create or destroy the excitation, thatis, a high energy fermion can be locally transformed into low energy ones. This amountsto attach a topological charge from low energy sector to high energy excitations. Thus a c -fermion carries a topological charge. On the other hand, an open c -string commutes withall plaquettes except some of them, so they create or destroy a particular charge among the34harges in Fig.2. It is simple to check that which charge they carry at their open ends. In factthey carry (¯¯ c, χ ¯ c ) charges. These latter charges have trivial mutual statistics relative to thecharges carried by high energy fermions, since they belong to different families of fermions.Thus the charge carried by a c -string must be invisible to the high energy fermions. B. Perturbative continuous unitary transformation
The physics behind Hamiltonian in (44) can be further explored by resorting to approxi-mate methods. The specific form of the energy levels of our model in the isolated limit, theexistence of equidistant levels, makes it suitable for perturbative continuous unitary trans-formations. In this method the Hamiltonian is replaced by an effective one within unitarytransformations in which the resulting effective Hamiltonian preserves the total charges, i.e.[ H eff , Q ] = 0. Thus the analysis of the model relies on finding the effective Hamiltonianin a sector characterized by the number of charges at every order of perturbation. For ourmodel each sector are determined by the number of c -fermions. Each term of the effectiveHamiltonian in any sector is just a suitable combination of expressions in (46) in such a waythat respects the total color charge in that sector. For now we briefly analyze the lowestcharge sectors.In the zero-charge sector, only the effective spin degrees of freedom do matter. Theeffective Hamiltonian is just a many-body Hamiltonian with terms that are product ofplaquette operators as follows H eff0 = E − X { c } X { f } O c ,...,c n f ,...,f n S c f ...S c n f n , (56)where the first and second sum run over an arbitrary collection of colors and plaquettesof effective honeycomb lattice. The coefficients O ’s are determined at a given order ofperturbation. The product of plaquette operators is nothing but the string-net operators.Let us focus at lowest order of perturbation, where the model represents non-interactingvortices. First, let us redefine plaquette operators as B xf = j s/ x Y v ∈ f τ xv , B yf = j s/ y Y v ∈ f τ yv , (57)where j w := J w / | J w | . At ninth order of perturbation the effective Hamiltonian is H eff0 = − X f ∈ Λ (cid:0) k x B xf + k y B yf + k z B xf B yf (cid:1) , (58)35ith [10] k z = 38 | J x J y | + O ( J ) , k x | J y | = k y | J x | = 5548913824 | J x J y | . (59)This is exactly the many-body Hamiltonian of topological color code obtained in (34) usingdegenerate perturbation theory with the additional advantage of knowing the coefficientsexactly. Its ground state is vortex free and can be written explicitly by choosing a referencestate as 2 N/ − Y f (cid:18) B xf (cid:19) | ⇑i ⊗ | i b . (60)Other degenerate ground states can be constructed by considering the nontrivial stringoperators winding around the torus. Excitations above ground state don’t interact. Goingto higher order of perturbation, as equation in (56) suggests, the ground states remainunchanged, however the excitation spectrum changes and vortices interact with each other.The one-quasiparticle sector can also be treated by examining the expressions in (46).The effective Hamiltonian can be written as H eff1 = H eff0 − X { R } O R ˆ Rb † c,R b c . (61)What the second term describes is nothing but the annihilation of a c -fermion at a referencesite and then its creation at a site connected to the reference by a string-net R , as shownin Fig.11(b). Again notice that this string-net is just the product of green and red linksof original 2-body Hamiltonian, in its effective form is given in terms of spin-boson degreesof freedom. The coefficients O ’s are determined at any order of perturbation. Notice thatthese coefficients are different from those in (56). In the first order, only the hopping termdoes matter. Let us consider the sector containing a c -fermion. Up to this order, the fermioncan only hop around a c -plaquette. This implies that at first order the fermion perform anorbital motion around a plaquette of its color. Notice that the fermion can not hop froma c -plaquette to other c -plaquette at the first order, since it needs for a composite processwhich appears at second order. This composite process is a combination of splitting andfusion processes. This is a virtual process in the sense that the splitting of a c -fermion intotwo ¯ c - and ¯¯ c -fermion takes the model from 1-quasiparticle sector into the 2-quasiparticlesector, but the subsequent process fuses two particles into a single one turning back to 1-quasiparticle sector. Thus, at second order the c -fermion can jump from one orbit to otherone. 36t first order, for J = J x = J y , we get a − J contribution to the energy gap comingfrom orbital motion. Going to second order we get a non-flat dispersion relation. The gap,at this order, is given by 1 − J − J / J ≃ .
45. This is just anapproximate estimation, since we are omitting all fermion interactions and, perhaps moreimportantly, we are taking J ≃ J z . However, it is to be expected that as the couplings J x ∼ J y grow in magnitude the gap for high-energy fermions will reduce, producing a phasetransition when the gap closes. Such a phase transition resembles the anyon condensationsdiscussed in [87–89]. There are three topological charges invisible to the condensed anyons.This means that in the new phase there exists a residual topological order related to thesecharges. They have semionic mutual statistics underlying the topological degeneracy in thenew phase. C. Fermions and gauge fields
The emerging high-energy c -fermions always appear with some nontrivial gauge fields[10,86], and carry different representation of the gauge symmetry Z × Z of the model. Beforeclarifying this, we can see that the plaquette degrees of freedom correspond to Z × Z gaugefields. This correspondence is established via introducing the following plaquette operators B ¯¯ cf := j s/ x S ¯ cf ,B ¯ cf := j s/ y S ¯¯ cf ,B cf := ( − j x j y ) s/ S cf . (62)The gauge element q f ∈ Z × Z that can be attached to the plaquette f is determined byfollowing eigenvalue conditions χ c ( q f ) = B cf , (63)which always has a solution due to( B cf ) = B r f B g f B b f = 1 . (64)The ground state of color code Hamiltonian (58) is vortex free and corresponds to χ c = 1.The fact that for a 2-colex with hexagonal plaquettes, the gauge fields can be related torepresentation of the group is immediate. One way to see this is to check the phase picked37 IG. 12: (color online) A piece of 2-body lattice corresponding to a 2-colex composed of octagonand square plaquettes. The lattice is 3-colorable as in Fig.8. up by a c -fermion when it moves around a plaquette. Turning on a plaquette is done bycombination of hopping operators yielding the phases as B ¯¯ cf , B ¯ cf and B cf that are consistentwith (64). However, this is not generic for all 2-colexes. For 2-colexe plaquettes that thenumber of their edges is a multiple of four, we see that the ground state carries fluxes.Perhaps the most important of such lattices is the so called 4-8-8 lattice shown in Fig.12. Itcontains inner octagons and squares. Once degenerate perturbation theory is applied aboutthe strong limit of the system, the effective color code Hamiltonian in terms of plaquetteoperators in (62) at 12th order of perturbation is produced, as follows H eff = − X f ( k x B xf + k y B yf + k z B zf ) + multiplaquette terms , (65)where sum runs over all squares and octagons. Notice that at 12th order of perturbationmultiplaquette terms that are product of square plaquette operators are also appeared. It issimple to check that the coefficients k ’s have positive sign. As we can relate the plaquettesto the representation of gauge group, the ground state corresponds to vortex free sector.In fact, the ground state of all 2-colexes with plaquettes of any shape is vortex free andcorrespond to χ c = 1 of gauge group. What is able to differentiate between ground statesof 2-colex plaquttes with 4 n ( n an integer) edges from others is related to the gauge fields38 IG. 13: (color online) A c -fermion process. A red fermion goes around a region λ . The phasethat it picks up depends on vortex configurations shown by spiral lines, blue and green fermionsand number of plaquettes that the number of their edges is of multiple four. attached to a fermion. In particular, there is a background of π -fluxes in the ground states ofsuch lattices, and the emerging c -fermions can detect them. To make sense of the existenceof such fluxes, let us consider a simple fermionic process as explained above. When a c -fermion turns on a plaquette, the combination of hopping terms yield − B ¯¯ cf , − B ¯ cf and − B cf ,which clearly imply that ( − B ¯¯ cf )( − B ¯ cf )( − B cf ) = − . (66)This result exhibit that the ground states of color code models defined on lattices with 4 n -plaquettes carry fluxes. Thus, in its representation in terms of Z × Z gauge group, thefluxes must be subtracted away.Our derivation for ground states with 0- and π -flux can be compared with Liebtheorem[90], which states for a square lattice the energy is minimized by putting π fluxin each square face of the lattice. The connection to our models makes sense when we con-sider how 2-colexes with hexagonal and 4-8-8 plaquettes can be constructed from a square39attice by removing some edges. Then the total π fluxes in a set of square faces correspond-ing to a 2-colex plaquette amounts to the flux that it carries. It is simple to see that eachhexagonal plaquette composed of two (imaginary) square faces, and flux π in each squareface then implies flux 0 in the hexagon. The same strategy holds for fluxes carried by the4-8-8 plaquette(and in general for all 4 n -plaquettes). Once again we see that each plaquettesof latter 2-colxes is composed of odd number of square faces, thus they carry flux π in theirplaquettes.Now we can give a general expression for the gauge fields seen by emerging high-energyfermions. To do so, let us consider a process in which a c -fermion is carried around a region λ , as in Fig.13. The hopping process yields a phase φ cλ = χ c ( q λ ) ( − n λ ¯ c + n λ ¯¯ c + n λ (67)where q λ = Q λ q f , n λc denotes the number of c -fermions inside λ and n λ the number of 2-colex plaquettes inside λ with a number of edges that is a multiple of four. Thus, we can seethat each family of fermions carries a different representation of gauge group given by valuesof q f inside the region. Moreover, it emphasizes that fermions with different color chargeshave mutual semionic statistics. Clearly for hexagonal lattices n λ = 0, and the ground statecarries no fluxes. VI. FERMIONIC MAPPING
In this section we will come back to the original Hamiltonian of (6) on the lattice in orderto use another approximate method based on fermionic mappings. This Hamiltonian canbe fermionized by Jordan-Wigner transformation [73]. To do so, firstly it is convenient topresent a lattice which is topologically equivalent to the lattice of Fig.3. This is a new typeof ”brick-wall” lattice as shown in Fig.14. The black and white sites are chosen such that,at the effective level, the lattice be a bipartite lattice, since the effective spins are located atthe vertices of hexagonal lattice representing a bipartite lattice. Note that neither originallattice in Fig.3 nor the brick-wall one in Fig.14 are bipartite in their own. Also as we willsee, the fermionization of the model needs some ordering of sites in the ”brick-wall” lattice.The unit cell of the brick-wall lattice is comprised of two triangles as shown in Fig.14 in ayellow ellipse. The translation vectors ~n and ~n connect different unit cells of the lattice.40he deformation of the original lattice into a ”brick-wall” lattice allows one to performthe one dimensional Jordan-Wigner transformation. The one dimensional structure of thelattice is considered as an array of sites on a contour as shown in Fig.15. The sites on thecontour can be labeled by a single index and the ordering of the sites is identified by thedirection of the arrows in Fig.15. The expression of the Pauli operators in terms of spinlessfermions will be: σ + j = a † j exp iπ X l 1. This isresemblance of vertex interaction in high energy fermions that we have seen in Sect.V withthe bosonic mapping. However, this latter relation doesn’t coincide with the symmetry ofthe model as they do not commute with each other. This will be considered next.Thus far, we have considered fields that are present in the Hamiltonian. In what follows,we introduce another set of fields which have the Z × Z symmetry commuting with eachother and with the Hamiltonian. To this end, consider a plaquette f . As before, by aplaquette we mean an outer and an inner hexagon with six triangles between them. Let V f V h stand for sets of vertices of plaquette and inner hexagon, respectively. It is naturalthat V h ⊂ V f . To each plaquette we attach the three following fields: φ f = Y j ∈ V f c j , φ f = Y j ∈ f \ h c j Y v ∈ V h d j , φ f = Y j ∈ V h c j d j , (78)where by f \ h we simply mean V f − V h . Each φ f squares identity. They commute with eachother and with Hamiltonian and ˆ F I : h φ kf , φ k ′ f ′ i = h φ kf , ˆ F I i = (cid:2) H, φ kf (cid:3) = 0 (79)Also, the fields φ f are responsible of the Z × Z gauge symmetry since φ f ⊗ φ f ⊗ φ f = − S Af = φ f ˆ F I , S Bf = φ f ˆ F I , S Cf = φ f . (80)Although the above gauge fields make it possible to divide the Hilbert space into sectorsin which be eigenspaces of gauge fields (or eigenspaces of plaquette operators), they do notallow us to reduce the Hamiltonian in (76) into a quadratic form. The ˆ F I ’s can be fixedas they commute with the Hamiltonian. But, we are not able to reduce the four-bodyinteraction terms in the Hamiltonian into quadratic form. In fact, the anticommutation ofˆ U jk ’s on a blue triangle prevents them to be fixed consistently with gauge fixing. VII. CONCLUSIONS We have introduced a two-body spin-1/2 model in a ruby lattice, see Fig.3. The modelexhibits an exact topological degeneracy in all coupling regimes. The connection to thetopological color codes can be discussed on the non-perturbative level as well as confirmedby perturbative methods. In the former case, on the ruby lattice we realized plaquetteoperators with local Z × Z symmetry of the color codes. All plaquette operators commutewith the Hamiltonian and they correspond to integrals of motion. The plaquettes can beextended to more complex objects that can be considered as string-nets: non-trivial stringswith branching points. The nontrivial strings corresponding to the various homology classesof the manifold determine the exact degeneracy of the model. For the case of periodic44oundary conditions, i.e on a torus, and for each non-contractible cycle of the torus, wecan identify three nontrivial closed strings. Once each of them is colored, the plaquettes ofthe lattice can be correspondingly colored as in Fig.8. For each homology class, they arerelated to each other by the gauge symmetry of the model. The crucial property of thesestrings is that they commute with Hamiltonian but not always with each other. This isindependent of the regimes of coupling constants of the model. Being anticommuting closednontrivial strings, the model has exact topological degeneracy. To clarify this observation,we use perturbation theory to investigate a regime of coupling corresponding to a strongcoupling limit (triangular limit). In this limit the topological color code will be the effectivedescription of the model. The effective representation of the closed loop operators determinethe terms appearing in the effective Hamiltonian at different orders.Unlike the Kitaev’s model or any its variants, our model is not integrable in terms ofmapping to Majorana fermions, to the best of our knowledge. This model is a four-valentlattice and gauge fields not always commute with each other. However, we have emphasizedin Sect.III that the existence of exact integrals of motion (IOMs) at a non-perturbative levelis far more enriching than demanding exact-solvability of a model. In fact, if the numberof IOMs is large enough, the model can turn out to be solvable. Thus, fixing plaquetteoperators can not give rise to fix all gauge fields.The description of our model in terms of hard core bosons yields very fruitful and interest-ing physics of the model. Using a bosonic mapping, it is possible to discuss the emergence ofstrongly interacting anyonic fermions. They form three classes each of one color. Fermionsfrom different classes have mutual semionic statistics. A very intriguing feature of thesefermions is related to the topological color charges they carry. They carry charges from aparticular family of low energy fermions. Thus the charges created by open strings are in-visible to high energy fermions. Moreover, there are some experimental proposals to realizehard-core bosons with optical lattices[91] and it would be a nice challenge to implement aHamiltonian like (44) and (45).We have shown that this new model exhibits enough novel interesting and relevantproperties so as to justify further research. Some of these possible lines of study are asfollows: We have only studied a particular phase of the system, although we are ableto study non-perturbative effects as well. The fact that all phases show a topologicaldegeneracy anticipates a rich phase diagram. In this regard, one may explicitly break the45olor symmetry that the model exhibits and still keep the features that we have discussed.It would be particularly interesting to check whether any of the phases displays non-abeliananyons. The model has many integrals of motion, although not enough to make it exactlysolvable. This becomes another appealing feature of the model since other methods ofstudy, like numerical simulations and experimental realizations will help to give a completeunderstanding of all its phases. Acknowledgements We acknowledge financial support from a PFI grant of EJ-GV, DGSgrants under contracts, FIS2006-04885, and the ESF INSTANS 2005-10. Appendix A: 2-Body Hamiltonian for Color Codes using Cluster States A topological color code can be constructed from a graph state defined on a bipartitelattice by means of a set of measurements on certain subsystems. This bipartite lattice isshown in Fig.16(a), where the black vertices correspond to plaquettes and the white verticescorrespond to the vertices of a 2-colex. To this graph we can attach a set of stabilizers asfollows K α = X α Y ≺ α,β ≻ Z β , (A1)where α and β stand for vertices of the graph and the product runs over all vertices thatare connected to α by black links. Let us set V = U ∪ U , where U and U stand forthe set of white and black vertices of the bipartite graph in Fig.16(a). Note that whiteand black vertices corresponds to the vertices and plaquettes of the 2-colex. This bipartitegraph is exactly what we need to construct color codes. To this end, we first impose aunitary transformation on the sublattices that allows us to have a more symmetric form ofthe stabilizer operators, i.e ∀ v ∈ U K v = X N ( v ) ∀ f ∈ U K f = Z N ( f ) , (A2)where N ( v ) denotes the site v and its neighbors, and the same goes for N ( f ). The corre-sponding cluster state denoted by | G i will be the common eigenvector of the above stabilizer46 IG. 16: (color online) (a) The graph needed for obtaining color codes from graph states. Thegraph is bipartite. Black and white vertices correspond to the plaquettes and vertices of a 2-colex.Black solid links are edges of the graph (b) The corresponding graph state can be approximatedas a low energy description of a lattice with 2-body Hamiltonian. The lattice is obtained from thegraph by replacing its vertices with some hexagons and triangles. The interactions σ z σ z and σ x σ z are associated to the solid and dashed links, respectively. operators. Thus we have: ∀ v ∈ U X N ( v ) | G i = | G i∀ f ∈ U Z N ( f ) | G i = | G i . (A3)Finally, a graph state can be related to a color code within a set of measurements in the Z basis on all qubits corresponding to the set U .We suppose there is a two dimensional lattice of physical qubits that is governed by a2-body Hamiltonian. Physical qubits of the lattice are projected to logical qubits. The pointis that this projection is achieved by going to some order in perturbation theory. We thinkof vertices of the graph in Fig.16(a) as logical qubits. The lattice with 2-body interactionis shown in Fig.16(b), where the number of physical qubits corresponding to the vertices ofthe graph equals the number of links crossing the vertex. The new resulting lattice consistsof triangles and hexagons and physical qubits live on their vertices. Triangles and hexagonsare in one to one correspondence with the white and black vertices ( U and U ) of the graph47n Fig.16(b), respectively. Note that each triangle is linked with three neighboring hexagonsand each hexagon is linked with six neighboring triangles.The low lying spectrum of a well constructed 2-body Hamiltonian defined on the latticecomposed of hexagons and triangles may describe a cluster state. To this end, we need forthe following projection from the physical qubits to the logical ones: P v = | ⇑ v ih↑↑↑ | + | ⇓ v ih↓↓↓ | , P f = | ⇑ f ih↑↑↑↑↑↑ | + | ⇓ f ih↓↓↓↓↓↓ | (A4)where | ⇑ L i and | ⇓ L i with L = v, f stand for the two states of the logical qubits obtainedwithin the above projections, or alternatively they are states of logical qubits of the graphin Fig.16(a). We set the following Hamiltonian: H = H + λV (A5)where H is the unperturbed Hamiltonian which can be treated exactly and λ is a smallquantity which allow us to treat the term λV perturbatively. We refer to each triangle(hexagon) and its vertices by a site index v ( f ) and indices i, j , respectively. The unperturbedpart of the Hamiltonian included in H is as follows. H = − X L X σ zL,i σ zL,j (A6)where the first sum runs over all triangles and hexagons (sites) and stand for thenearest-neighbor qubits around the corresponding triangle or hexagon connected by the solidlines as in the Fig.16(b). The interaction between qubits of triangles and qubits of hexagonsare included in V : V = − X 6) + N v ( − 3) = − N in terms of the energy scale of the problem. The firstexcited state is produced by exciting one of the triangles or hexagons and has energy: E (0)1 = N f ( − 6) + N v ( − 3) + 4 = − N − 1) with degeneracy g = 14 N N . The secondexcited state has energy E (0)1 = N f ( − 6) + N v ( − 3) + 8 = − N − 2) with degeneracy: g = 4( N + 5 N )2 N − , and so on and so forth.Using degenerate perturbation theory as in Sec.IV B, the effect of perturbation V onthe ground sate subspace can be investigated, and see if it breaks the degeneracy. It issimple to see that first order perturbation does not have anything to do with the groundstate subspace. The second order gives rise to a trivial effect as a shift in energy, since eachoperator related to dashed links appears twice. The third order perturbation theory howevergives rise to a nontrivial effect. It causes a partial lift of the ground state degeneracy, but notcomplete. The initial degeneracy 2 N gets reduced down to 2 N f . This nontrivial effect arisesfrom the product of three dashed links crossing a typical triangle, namely the ground statevectors are grouped into the 2 N v states, each containing 2 N f vectors. The product of three(six) σ x operators around a triangle (hexagon) is equivalent to an X operator acting on thelogical qubit which is projected down from the three(six) qubits of the triangle(hexagon) ,since X L = | ⇑ L ih⇓ L | + | ⇓ L ih⇑ L | . (A10)Also the action of a σ z on one qubit of a triangle or hexagon is equivalent to an Z operatoracting on the related logical qubit, since Z L = P L σ z P † L = | ⇑ L ih⇑ L | − | ⇓ L ih⇓ L | . (A11)Now we can go on in order to calculate the third order perturbation: H (3)eff = − E (0)0 − E (0)1 ) X v K v = − δ X v K v , (A12)where K v = X v Y f Z f , (A13)49he product runs over three black vertices linked to the v and δ = . The operator K v is astabilizer for the logical qubits which are projected down from the triangles. Since K v = 1,the ground states correspond to the values of k v = +1. We skip the forth and fifth order ofperturbation because they have trivial effects.Like in 3 rd order perturbation, we are faced with a nontrivial term in the 6 th orderperturbation theory. We will see that by considering this order, the ground state degeneracyis lifted completely. This nontrivial effect arises from the product of terms in the perturbation λV corresponding to the links around a hexagon. Finally, for the 6 th order perturbation wehave: H (6)eff = − γ X f K f , (A14)where K f = X f Y v Z v , (A15)and the product runs over six white vertices linked to the hexagon f . The coefficient γ haspositive sign and its precise value is unimportant. We would like to emphasize that at sixorder in perturbation theory some other terms appear which are product of two distinct K v .However, we skip them as they all commute. Equations (A13) and (A15) provides all weneed to adopt the cluster state in (A3) as ground state of the low energy effective theory ofHamiltonian in (A5), which up to six order of perturbation can be written as follows H eff = constant − δ X v K v − γ X f K f . (A16)We see that the above effective Hamiltonian is completely different from that of in (34). Thelatter equation gives rise directly to the topological color code as its ground state, but theground state (cluster state) of former one needs further local measurements to encode thedesired color code. References [1] H. Bombin, M. A. Martin-Delgado; “Topological Quantum Distillation”; Phys. Rev. Lett. ,180501 (2006); quant-ph/0605138. 2] H. Bombin, M.A. Martin-Delgado; “Topological Computation without Braiding”; Phys. Rev.Lett. , 160502 (2007); quant-ph/0610024.[3] H. Bombin, M.A. Martin-Delgado “Exact Topological Quantum Order in D=3 and Beyond:Branyons and Brane-Net Condensates”; Phys. Rev. B , 075103 (2007); cond-mat/0607736.[4] H. J. Briegel, R. Raussendorf, Phys. Rev. Lett. , 910 (2001).[5] R. Raussendorf, H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001); Quantum Inf. Comput. ,443 (2002).[6] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A , 022312 (2003).[7] H. Bombin, M.A. Martin-Delgado, “Statistical mechanical models and topological colorcodes”; Physical Review A 77, 042322 (2008). arXiv:0711.0468.[8] S. D. Bartlett and T. Rudolph, Phys. Rev. A , 040302(R) (2006).[9] M. Van den Nest, K. Luttmer, W. D¨ur and H. J. Briegel, Phys. Rev. A , 012301 (2008).[10] H. Bombin, M. Kargarian, M.A. Martin-Delgado, “Interacting Anyonic Fermions in a Two-Body ‘Color Code’ Model”; Phys. Rev. B , 075111 (2009). arXiv:0811.0911.[11] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (CambridgeUniversity Press, Cambridge, England, 2000).[12] A. Galindo and M. A. Martin-Delgado, Information and Computation: Classical and QuantumAspects ; Rev. Mod. Phys. , 347 (2002). Electronic preprint arXiv:quant-ph/0112105.[13] J.E. Levy, A. Ganti, C.A. Phillips, B.R. Hamlet, A.J. Landahl, T.M. Gurrieri, R.D. Carr,M.S. Carroll, “The impact of classical electronics constraints on a solid-state logical qubitmemory”; arXiv:0904.0003.[14] A. Micheli, G.K. Brennen, P. Zoller, “A toolbox for lattice spin models with polar molecules”;Nat. Phys. , 341 (2006). quant-ph/0512222.[15] L. Jiang, G.K. Brennen, A.V. Gorshkov, K. Hammerer, M. Hafezi, E. Demler, M.D. Lukin, P.Zoller, “Anyonic interferometry and protected memories in atomic spin lattices”; Nat. Phys. , 482 (2008). arXiv:0711.1365.[16] M. M¨uller, L. Liang, I. Lesanovsky, P. Zoller, “Trapped Rydberg ions: from spin chains tofast quantum gates”; New J. of Phys. , 093009 (2008). arXiv:0709.2849[17] L.-M. Duan, E. Demler, M.D. Lukin, “Controlling Spin Exchange Interactions of UltracoldAtoms in Optical Lattices”; Phys. Rev. Lett. , 090402 (2003). arXiv:cond-mat/0210564.[18] J.J. Garcia-Ripoll, M.A. Martin-Delgado, J.I. Cirac, “Implementation of Spin Hamiltonians n Optical Lattices”; Phys. Rev. Lett. , 250405 (2004).[19] M.J. Hartmann, F.G.S.L. Brandao, M.B. Plenio, “Strongly interacting polaritons in coupledarrays of cavities”; Nat. Phys. , 849 (2006). quant-ph/0606097.[20] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, U. Sen, “Ultracold atomic gasesin optical lattices: mimicking condensed matter physics and beyond”; Adv. Phys. , 243(2007).[21] J. Du, J. Zhu, M. Shi, X. Peng, D. Suter, “Experimental observation of a topological phasein the maximally entangled state of a pair of qubits”; Phys. Rev. A 76, 042121 (2007).arXiv:0705.3566.[22] A.F. Albuquerque, H.G. Katzgraber, M. Troyer, G. Blatter, “Engineering exotic phases fortopologically protected quantum computation by emulating quantum dimer models”; Phys.Rev. B 78, 014503 (2008).[23] G.K. Brennen, M. Aguado, J.I. Cirac, “Simulations of quantum double models”;arXiv:0901.1345.[24] F. Wilczek, “Quantum Mechanics of Fractional-Spin Particles”. Phys. Rev. Lett. , 957(1982).[25] J.M. Leinaas, J. Myrheim, “On the theory of identical particles”. Il Nuovo Cimento B37 , 1(1977).[26] G. Moore, N. Read, “Nonabelions in the fractional quantum Hall effect”, Nucl. Phys. B Quantum Field Theory of Many-body Systems: From the Origin of Sound to anOrigin of Light and Electrons (Oxford Univ. Press, New York, 2004).[28] X.G. Wen, “Topological orders and edge excitations in fractional quantum Hall states”; Adv.Phys., , 405 (1995).[29] H. Bombin, M.A. Martin-Delgado, “Interferometry-free protocol for demonstrating topologicalorder”; Physical Review B 78, 165128 (2008). arXiv:0705.0007.[30] D. Gottesman, “Class of quantum error-correcting codes saturating the quantum Hammingbound”. Phys. Rev. A , 1862 (1996).[31] Z.W.E. Evans, A.M. Stephens, “Optimal decoding in fault-tolerant concatenated quantumerror correction”; arXiv:0902.4506.[32] Z.W.E. Evans, A.M. Stephens, “Accuracy threshold for concatenated error detection in one imension”; arXiv:0902.2658.[33] A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics , 2(2003). quant-ph/9707021.[34] H. Bombin, M.A. Martin-Delgado; “Homological Error Correction: Classical and QuantumCodes”; J. Math. Phys. , 052105 (2007); quant-ph/0605094.[35] H. Bombin, M.A. Martin-Delgado; “Topological quantum error correction with optimal en-coding rate”; Physical Review A 73, 062303 (2006). quant-ph/0602063.[36] C.D. Albuquerque, R. Palazzo, E.B. Silva, “Topological quantum codes on compact surfaceswith genus g ≥ , 023513 (2009).[37] S. Bullock, G.K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”;J. of Phys. A: Math. and Theor. , 3481 (2007).[38] H. Bombin, M.A. Martin-Delgado; “Optimal Resources for Topological 2D Stabilizer Codes:Comparative Study”, Phys. Rev. A , 012305 (2007). quant-ph/0703272.[39] M. Planat, M.R. Kibler, “Unitary reflection groups for quantum fault tolerance”;arXiv:0807.3650.[40] L.S. Georgiev, “Computational equivalence of the two inequivalent spinor representations ofthe braid group in the topological quantum computer based on Ising anyons”; arXiv:0812.2337.[41] D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, C.H. Oh, “Graphical nonbinary quantum error-correcting codes”; Phys. Rev. A 78, 012306 (2008). arXiv:0801.0831.[42] L.S. Georgiev, “Towards a universal set of topologically protected gates for quantum compu-tation with Pfaffian qubits”; Nuc. Phys. B , 095302 (2009). arXiv:0704.2540.[44] R. Raussendorf, J. Harrington, K. Goyal, “Topological fault-tolerance in cluster state quantumcomputation”; New J. of Phys. , 199 (2007). arXiv:quant-ph/0703143.[45] R. Raussendorf, J. Harrington, “Fault-Tolerant Quantum Computation with High Thresholdin Two Dimensions”; Phys. Rev. Lett. , 190504 (2007). quant-ph/0610082.[46] D. P. DiVincenzo, “Fault tolerant architectures for superconducting qubits”; arXiv:0905.4839.[47] E. Dennis, A. Kitaev, A. Landahl, J. Preskill, “Topological quantum memory”. J. Math. Phys. , 4452-4505 (2002).[48] H.G. Katzgraber, H. Bombin, M.A. Martin-Delgado, “Error Threshold for Color Codes and andom 3-Body Ising Models”; Phys. Rev. Lett. , 090501 (2009). arXiv:0902.4845.[49] M. Ohzeki, “Threshold of topological color code”; arXiv:0903.2102.[50] W.-B. Gao, A.G. Fowler, R. Raussendorf, X.-C. Yao, H. Lu, P. Xu, C.-Y. Lu, C.-Z. Peng, Y.Deng, Z.-B. Chen, J.-W. Pan; “Experimental demonstration of topological error correction”.arXiv:0905.1542v2.[51] R. Alicki, M. Fannes, M. Horodecki, “A statistical mechanics view on Kitaev’s proposal forquantum memories”, J. Phys. A: Math. Theor. 40 (2007) 6451-6467.[52] R. Alicki, M. Fannes, M. Horodecki, “On thermalization in Kitaev’s 2D model”;arXiv:0810.4584.[53] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, “On thermal stability of topologicalqubit in Kitaev’s 4D model”; arXiv:0811.0033.[54] S. Iblisdir, D. Perez-Garcia, M. Aguado, J. Pachos, “Thermal States of Anyonic Systems”;arXiv:0812.4975.[55] S. Iblisdir, D. Perez-Garcia, M. Aguado, J. Pachos, “Scaling law for topologically orderedsystems at finite temperature”; arXiv:0806.1853.[56] S. Bravyi, B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memorybased on stabilizer codes”; arXiv:0810.1983.[57] M. Kargarian, “Finite temperature topological order in 2D topological color codes”; PhysicalReview A 80, 012321 (2009). arXiv:0904.4492.[58] M. Kargarian, “Entanglement properties of topological color codes”; Physical Review A ,097 (2008). arXiv:0805.1891.[61] E. Rico, H.J. Briegel, “2D multipartite valence bond states in quantum anti-ferromagnets”;Ann. of Phys. , 2115 (2008). arXiv:0710.2349.[62] J. Eisert, M. Cramer, M.B. Plenio, “Area laws for the entanglement entropy - a review”;arXiv:0808.3773.[63] W. D¨ur, H.J. Briegel, “Entanglement purification and quantum error correction”; Rep. onProg. in Phys. , 1381 (2007). arXiv:0705.4165. 64] V. Karimipour, “A complete characterization of the spectrum of the Kitaev model on spinladders”; arXiv:0904.3554.[65] M. Van den Nest, W. D¨ur, and H. J. Briegel, “Classical Spin Models and the Quantum-Stabilizer Formalism”; Phys. Rev. Lett. , 117207 (2007); arXiv:quant-ph/0610157.[66] S. Bravyi and R. Raussendorf, “Measurement-based quantum computation with the toric codestates”; Phys. Rev. A , 022304 (2007); arXiv:quant-ph/0610162.[67] M. Van den Nest, W. D¨ur, and H. J. Briegel, “Completeness of the classical 2D Ising modeland universal quantum computation”; arXiv:0708.2275.[68] M. Van den Nest, W. D¨ur, R. Raussendorf, H.J. Briegel, “Quantum algorithms for spin modelsand simulable gate sets for quantum computation”; arXiv:0805.1214.[69] R. H¨ubener, M. Van den Nest, W. D¨ur, H.J. Briegel, “Classical spin systems and the quantumstabilizer formalism: general mappings and applications”; arXiv:0812.2127.[70] G. De las Cuevas, W. D¨ur, M. Van den Nest, H.J. Briegel, “Completeness of classical spinmodels and universal quantum computation”; arXiv:0812.2368.[71] G. De las Cuevas, W. D¨ur, H. J. Briegel, M. A. Martin-Delgado, “Unifying all classical spinmodels in a Lattice Gauge Theory”; Phys. Rev. Lett. , 230502 (2009). arXiv:0812.3583.[72] A. Yu. Kitaev; “Anyons in an exactly solved model and beyond”; Ann. of Phys. , 2 (2006).arXiv:cond-mat/0506438v3.[73] X.-Y. Feng, G.-M. Zhang, T. Xiang, “Topological Characterization of Quantum Phase Tran-sitions in a Spin-1/2 Model”; Phys. Rev. Lett. , 087204 (2007). cond-mat/0610626.[74] H. Yao, S.A. Kivelson; “Exact Chiral Spin Liquid with Non-Abelian Anyons”;Phys. Rev. Lett. , 247203 (2007). arXiv:0708.0040.[75] G. Baskaran, S. Mandal, R. Shankar, “Exact Results for Spin Dynamics and Fractionalizationin the Kitaev Model”; Phys. Rev. Lett. , 247201 (2007).[76] S. Yang, D.L. Zhou, C.P. Sun; “Mosaic spin models with topological order”; Phys. Rev. B76 ,180404 (R) (2007). arXiv:0708.0676.[77] T. Si, Y. Yu, “Anyonic loops in three-dimensional spin liquid and chiral spin liquid”; Nuc.Phys. B B 79, 024426 (2009).[79] M. Levin, X.-G. Wen, “String-net condensation: A physical mechanism for topological phases ; Physical Review B 71, 045110 (2005).[80] M. Oshikawa and T. Senthil,“Fractionalization, topological order, and quasiparticle statistics”;Phys. Rev. Lett. , 060601 (2006).[81] D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents, “Degenerate perturbation theory ofquantum fluctuations in a pyrochlore antiferromagnet”;Physical Review B 75, 094403 (2007).[82] G. Kells, A. T. Bolukbasi, V. Lahtinen, J. K. Slingerland, J. K. Pachos, and J.Vala,“Topological Degeneracy and Vortex Manipulation in Kitaev `Os Honeycomb Model”;Phys. Rev. Lett. , 240404 (2008).[83] C. Knetter, G. Uhrig; “Perturbation theory by flow equations: dimerized and frustrated S =1/2 chain”; Eur. Phys. J. B13, 209 (2000).[84] F. Wegner, “Flow-equations for Hamiltonians”; Ann. der Phys. 506, 77 (1994).[85] J. Vidal, K.P. Schmidt, S. Dusuel; “Perturbative approach to an exactly solved problem:Kitaev honeycomb model”; Phys. Rev. B 78, 245121 (2008). arXiv:0809.1553.[86] M. Levin, X.-G. Wen, “Fermions, strings, and gauge fields in lattice spin models”; PhysicalReview B 67, 245316 (2003).[87] H. Bombin, M.A. Martin-Delgado, “Family of non-Abelian Kitaev models on a lattice: Topo-logical condensation and confinement”; Physical Review B 78, 115421 (2008).[88] H. Bombin, M.A. Martin-Delgado, “Nested Topological Order”; arXiv:0803.4299.[89] F.A Bais, J.K. Slingerland, “Condensate-induced transitions between topologically orderedphases”; Physical Review B 79, 045316 (2009). arXiv:0808.0627.[90] Elliott H. Lieb, “The flux phase of the half-filled band”; Phys. Rev. Lett. , 2158 (2009).[91] B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, G. Pupillo; “Quantum Phasesof Cold Polar Molecules in 2D Optical Lattices”; arXiv:0906.2009., 2158 (2009).[91] B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, G. Pupillo; “Quantum Phasesof Cold Polar Molecules in 2D Optical Lattices”; arXiv:0906.2009.