Statistical Mechanical Models and Topological Color Codes
aa r X i v : . [ qu a n t - ph ] N ov Statistical Mechanical Models and Topological Color Codes
H. Bombin and M.A. Martin-Delgado
Departamento de F´ısica Te´orica I, Universidad Complutense, 28040. Madrid, Spain.
We find that the overlapping of a topological quantum color code state, representing a quantummemory, with a factorized state of qubits can be written as the partition function of a 3-bodyclassical Ising model on triangular or Union Jack lattices. This mapping allows us to test thatdifferent computational capabilities of color codes correspond to qualitatively different universalityclasses of their associated classical spin models. By generalizing these statistical mechanical modelsfor arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-basedquantum computation with a color code state and we find that their classical simulatability remainsan open problem. We complement the meaurement-based computation with the construction of acluster state that yields the topological color code and this also gives the possibility to representstatistical models with external magnetic fields.
PACS numbers: 03.67.Lx, 03.67.-a, 75.10.Hk, 05.50.+q
I. INTRODUCTION
Recently, a very fruitful relationship has been estab-lished between partition functions of classical spin modelsand a certain class of quantum stabilizer states with topo-logical protection [1], [2]. The topological quantum codestates considered so far in these studies correspond to thetoric code states introduced by Kitaev [3], [4]. The clas-sical spin model that emerges when a planar toric code isprojected onto a product state of single-qubits with veryspecific coefficients is the standard classical Ising modelin two dimensions with homogeneous real couplings andzero magnetic field.Single-qubit measurements also appear naturally in ameasurement-based computation (MQC) scheme [5], [6].Thus, these connections between classical spin modelsand topological quantum states are also useful to testwhether those topological states are efficiently classicallysimulable with MQC. It has been shown that MQC witha planar Kitaev code state as input can be efficientlysimulated in a classical computer if at each step of thecomputation, the sets of measured qubits form simplyconnected subsets of the two-dimensional lattice [2]. Theconnection of classical spin models with measurement-based quantum computation has been shown to be usefulto prove the completeness of the classical 2D Ising modelwith suitably tuned complex nearest-neighbor couplingsin order to represent the partition function of the classicalIsing model on arbitrary lattices, with inhomogeneouspairwise interactions and local magnetic fields [7].Topological color codes (TCC) were introduced to im-plement the set of quantum unitary gates of the wholeClifford group by means of a topological a stabilizer codein a two dimensional lattice [8], and then generalizedto three dimensional lattices in order to achieve a uni-versal set of topological quantum gates [9]. These 2Dand 3D realizations of TCC are instances of general D-dimensional realizations. We call those lattices relatedto these codes as D-colexes (for color complexes), andthey are D-dimensional lattices with coordination num-ber D + 1 and certain colorability properties. Moreover, this codes can also appear as the ground state of suitableHamiltonians, and the corresponding quantum systemsare brane-net condensates. [10].Given these nice properties exhibited by the topolog-ical color codes, it is natural to ask what type of clas-sical spin models can be constructed out of them andsee whether they belong or not to the same universalityclass of the classical Ising model arising in the Kitaevmodel. In this work we address this issue and find thatthe the overlapping of a TCC state with a product stateof single qubits with appropriate coefficients is mappedonto the partition function of the 3-body classical Isingmodel on the dual lattice of the original lattice where thecolor code is defined. For concreteness, we consider thetriangular and the ‘Union Jack’ lattice for these classicalmany-body spin systems. This represents a sharp differ-ence with the result obtained with the topological statesin the Kitaev code. In fact, the universality classes of the3-body classical Ising model in several lattices are quitedifferent from the corresponding universality class of thestandard 2-body Ising model.Moreover, we also study the topological color codestates in a MQC scenario to test their classical simu-lability. We find that the current state of knowledge instatistical mechanical models with 3-body interactions,arbitrary inhomogeneous complex couplings and latticeshapes is much less developed than the 2-body Isingmodel which is relevant for the case of the toric codestates. Thus, we conclude that the classical simulabilityof TCC states with MQC remains an open problem.In a MQC, the usual initial many-particle entangledstate is a cluster state [5], [6] instead of a topological code.Then, we also show how to construct a color code statefrom a certain cluster state. Interestingly enough, thisconstruction turns out to be useful for the description ofstatistical mechanical systems with 3-body interactionsin the presence of an external field.This paper is organized as follows: in Sect.II we give anintroduction to the topological color code states neededto present in Sect.III the mapping onto the classical 3-body Ising model in the triangular and Union Jack lat- Figure 1: An example of 2-colex. Both edges and face are 3-colorable and they are colored in such a way that green edgesconnect green faces and so on and so forth for red and blueedges and faces. tices. In Sect.IV we study the measurement-based quan-tum computation with topologocial color code states bygeneralizing the results of the previous section. In Sect.Vwe show how to prepare a topological color code from acluster state as those introduced in MQC. This is alsouseful for studying partition functions of statistical me-chanical models with 3-body interactions and externalmagnetic fields. Sect.VI is devoted to conclusions.
II. TOPOLOGICAL COLOR CODESA. Construction
Let us start by recalling the notion of a TopologicalColor Code (TCC) in order to see what type of classi-cal spin models we obtain from them with appropriateprojections onto factorized quantum states and specificlattices.A TCC, denoted by C , is a quantum stabilizer errorcorrection code constructed with certain class of two-dimensional lattices called 2-colexes. The word colex isa contraction that stands for color complex, where com-plex is the mathematical terminology for a rather generallattice [11]. A 2-colex, denoted by C , is a 2D trivalentlattice which has 3-colorable faces and is embedded ina compact surface of arbitrary topology like a torus ofgenus g . A trivalent lattice is one for which three edgesmeet at every vertex. The property of being 3-colorablemeans that the faces (or plaquettes) of the lattice can becolored with these colors in such a way that neighbor-ing faces never have the same color. We select as colorsred (r), green (g) and blue (b). An example of a 2-colexconstruction is shown in Fig. 1.Edges can be colored in according to the coloring ofthe faces. In particular, we attach red color to the edges PSfrag replacements f γ
11 2 3456 a b c d e fFigure 2: An hexagonal lattice is an instance of 2-colex. Num-bered vertices belong to the face f . Vertices labeled with let-ters correspond to the red string γ displayed. γ is an openstring, because it has an endpoint in a red face. that connect red faces, and so on and so forth for the blueand gree edges/faces. When studying higher dimensionalcolexes, it turns out that the coloring of the edges is thekey property of D -colexes: all the information about a D -colex is encoded in its 1-skeleton, i.e., the set of edgeswith its coloring[10].Given a 2-colex C , a TCC C is constructed by placingone qubit at each vertice of the colored lattice. Let usdenote by V , E and F the sets of vertices v , edges e and faces f , respectively, of the given 2-colex. Then, thegenerators of the stabilizer group, denoted by S , are givenby face operators only. For each face f , they come intotwo types depending whether they are constructed withPauli operators of X - or Z -type: X f := O v ∈ f X v ,Z f := O v ∈ f Z v , (1)and there are no generators associated to lattice vertices.For example, an hexagonal lattice is an instance of a 2-colex, see Fig. 2. The operators for the face f displayedin the figure take the form X f = X X X X X X , Z f = Z Z Z Z Z Z . A given state | Ψ c i ∈ C is left triviallyinvariant under the action of the face operators, X f | Ψ c i = | Ψ c i , Z f | Ψ c i = | Ψ c i , ∀ f ∈ F . (2)An erroneous state | Ψ i e is one that violates conditions(2) for some set of face operators of either type. As thegenerator operators X f , Z f ∈ S satisfy that they square tothe identity operator, ( X f ) = 1I = ( Z f ) , ∀ f ∈ F , then anerroneous state is detected by having a negative eigen-value with respect to some set of stabilizer generators: X f | Ψ i e = −| Ψ i e and/or Z f | Ψ i e = −| Ψ i e .Interestingly enough, it is possible to construct a quan-tum lattice Hamiltonian H c such that its ground state isdegenerate and corresponds to the TCC C , while the er-roneous states are given by the spectrum of excitationsof the Hamiltonian [8]. Such Hamiltonian is constructedout of the generators of the topological stabilizer group S , H c = − X f ∈ F ( X f + Z f ) . (3)The ground state of this Hamiltonian exhibits what iscalled a topological order [12], as opposed to a more stan-dard order based on an spontaneous symmetry breakingmechanism. One of the signatures of that topological or-der is precisely the topological origin of the ground statedegeneracy: the number of degenerate ground states de-pends on topological invariants like Betti numbers [10].In two dimensional lattices, the relevant Betti numbercorresponds to the Euler characteristic χ of the surfacewhere the 2-colex is embedded. B. String-net operators
In order to better understand both the ground stateand excitations of this Hamiltonian and their topolog-ical properties, it is rather convenient to introduce theset of string operators that can be defined on a 2-colex C . String operators are generalizations of face operators(1) that can be either open or closed, i.e., with or with-out end-points. These strings are topological and likein Kitaev model, the homology is defined on Z sincewe work with two-level quantum systems located at thesites of the lattice. However, in a TCC we have an addi-tional ingredient to play around: color. Let us split thesets of edges and faces into colored subsets denoted by E := E r ∪ E g ∪ E b and F := F r ∪ F g ∪ F b , where E r is thesubset of red edges, and similarly for the rest of subsets.A colored string γ is a collection of edges of a givencolor. Thus, a blue string γ takes the form γ = { e i } with e i ∈ E b . The definition of colored string operatorsis completely analogous to that of face operators: X γ := O e ∈ γ X e , Z γ := O e ∈ γ Z e , (4)where, in turn, X e = X v ⊗ X v if v and v are the sitesat the ends of the edge e , and similarly for Z e = Z v ⊗ Z v . For instance, consider the red string operator inFig. 2, where we have X γ = X a X b X c X d X e X f · · · , Z γ = Z a Z b Z c Z d Z e Z f · · · .Colored strings are open if they have endpoints. Theseendpoints are localized at faces which share color withthe string. In particular, a face f is an endpoint of γ ifthe number of edges of γ meeting at f is odd, see Fig. 2.In terms of string operators, a face f is an endpoint of γ if { X γ , Z f } = 0 or, equivalently, if { Z γ , X f } = 0. Thus,open string operators do not commute with those faceoperators in their ends. In other words, a string op-erator that commutes with all the face operators must Figure 3: A 4-8 lattice is an instance of 2-colex. A closedstring-net is displayed, composed of 3 strings of different col-ors meeting at a branching point. The string-net is closedbecause at each face we find an even number of its vertices. correspond to a closed string, that is, a string withoutendpoints. In terms of the Hamiltonian (3), string op-erators produce quasi-particle excitations at their endswhen applied to the ground state. These quasi-particleexcitations are Abelian anyons.Closed strings are mainly of two types. They can behomologically trivial, meaning that they are the bound-ary of certain area of the surface, or homologically non-trivial. In terms of operators, a string is a boundary iffits string operators belong to the stabilizer group S ofthe color code C . Such boundary string operators arethus products of face operators. In fact, face operatorsthemselves are the basic boundary string operators.Although we have introduced colored strings and thecorresponding operators for illustrative purposes, in factin a TCC we have to deal with more general objects,namely string-nets. A string-net is a collection of stringsmeeting at certain branching or ramification points. Anexample of these types of configurations are shown inFig. 3.A string-net γ is a collection of vertices γ ⊂ V . Equiv-alently, γ is a formal sum of lattice vertices v ∈ V withcoefficients γ v ∈ Z , i.e., γ = X v ∈ V γ v v , (5)where γ v = 1 if v ∈ γ and γ v = 0 otherwise. Given astring-net γ , we define the string-net operators X γ := O v X γ v , Z γ := O v Z γ v (6)Just as in the case of colored strings, we can talk aboutopen and closed string-nets, and about trivial and non-trivial closed string-nets. In terms of operators, the sit-uation is exactly the same as with strings. That is, astring-net γ has an endpoint at a face f if { X γ , Z f } = 0,it is closed if its string-net operators commute with allthe face operators, and it is a boundary if it is a prod-uct of face operators. In order to translate these ideasinto purely geometric terms, we can define the boundaryoperator ∂ c γ := X f x f f , x f = ( , | γ ∩ f | is even,1 , | γ ∩ f | is odd, (7)where | γ ∩ f | is the number of vertices that γ and f share.Thus ∂ c γ is the formal sum of the endpoints of γ . It isalso natural to define an operator ∂ c for faces ∂ c f := X v x v v , x v = ( , v f , , v ∈ f , (8)so that ∂ c f is the string-net composed of the vertices of f . With this definitions, γ is closed if and only if ∂ c γ = 0 , (9)and it is a boundary if and only if there exist a collectionof faces S = P f S f f such that γ = ∂ c S. (10)It is possible to give explicit expressions for the statesof the TCC or, equivalently, for the ground states of theHamiltonian (3). The states are superpositions of all pos-sible closed string-nets, a typical feature of the groundstates of systems with topological order [12], [13]. Thefollowing is an un-normalized ground state for any given2-colex [8], [10] | Ψ c i := Y f (1 + X f ) | i ⊗| V | = X γ ∈ Γ X γ | i ⊗| V | =: X γ ∈ Γ | γ i , (11)where | V | is the number of vertices in the 2-colex C ,Γ denotes the set of boundary string-nets and | i is theeigenstate Z | i = | i .The degeneracy of the ground state or, equivalently,the number of logical states encoded in the color code, de-pends on the topology of the lattice. For a general 2-colex C with Euler characteristic χ ( C ) := | V | − | E | + | F | , thenumber k of encoded qubits is given by k = 4 − χ ( C ) :=2 h [8], where h is the first Betti number of the surfacewhere the 2-colex is embedded [10]. These additionalground states can be obtained from the one given by (11)by the action of the encoded logical operators ¯ X i , ¯ Z i with i = 1 , . . . , k . These, in turn, take the form of string-netoperators of non-trivial closed string-nets [8].For the purpose of this work, we shall be interestedonly in a representative ground state like (11). Thus, wewill have to consider suitable surface topologies such thatthe corresponding TCC is unique. We will return overthis issue later when we consider particular lattices. III. CONNECTION WITH CLASSICAL SPINSYSTEMSA. Overlap and Partition Function
Now, we come to the issue of what type of classical spinmodels may arise from the color code state (11) when weproject it onto a product state of a number of qubits givenby | V | . In this section we shall not consider the mostgeneral factorized state, but one specifically adapted forthe purpose of this connection in its most simple form,namely, | Φ P i := O v ∈ V | φ i v ; | φ i v := cosh( βJ ) | i v + sinh( βJ ) | i v , (12)with β := 1 /k B T the inverse temperature parameter.The classical spin model arises when computing theoverlapping between the ground state of the color codeHamiltonian (11) and this factorized state (12), O ( βJ ) := h Ψ c | Φ P i . (13)Using (11) and (12) we get the following expression forthis overlapping, O ( βJ ) = X γ ∈ Γ h γ | O v ∈ V | φ i v = (cosh( βJ )) | V | X γ ∈ Γ u | γ | , (14) u := tanh( βJ ) and | γ | is the number of vertices of γ .We want to relate (14) to the partition function of aclassical spin system. So let C be an arbitrary 2-colex.Consider the dual lattice Λ. The vertices of Λ correspondto the faces of C , and the faces of Λ are vertices in C . In particular, Λ is a lattice in which all faces aretriangular and vertices are 3-colorable. Moreover, for anysuch lattice Λ there exist a suitable dual 2-colex C .So let us associate a classical system to Λ by attach-ing classical spin variables σ i = ± i (equivalently, to each face f of C ). The classical Hamil-tonian is H := − J X h i,j,k i σ i σ j σ k , (15)where J is a coupling constant and the sum P h i,j,k i is over all triangles with spins σ i σ j σ k at their vertices.Thus, we have a classical Ising model with 3-body inter-actions. The case J >
J < Z ( βJ ) := X { σ } e βJ P h i,j,k i σ i σ j σ k , (16)where the sum P { σ } is over all possible configurations ofspins. The point then is that we have Z ( βJ ) = 2 N O ( βJ ) , (17) PSfrag replacements (a)(b)Figure 4: Two instances of dual lattices of a 2-colex, whichhave triangles as faces and have 3-colorable sites. The trian-gular lattice (a) is dual to the hexagonal one. The Union Jacklattice (b) is dual to the square-octogonal or 4-8 lattice. where N is the number of sites.Before we show why this identity holds, let us give apair of representative examples of dual lattices C andΛ. First, if the 2-colex is an hexagonal lattice then thedual lattice Λ is a triangular lattice , Fig. 4(a). Second,if the 2-colex is a square-octogonal lattice (also denotedby 4-8 lattice), then its dual is a Union Jack lattice , seeFig. 4(b). The relevance of these examples is two-fold.On the one hand, the hexagonal lattice is the simplestlattice for a 2-colex and the 4-8 lattice is the simplest onewhen we want to obtain TCC with certain transversalityproperties for quantum computation (see below). On theother hand, 3-body classical Ising-models on both latticeshave been studied in statistical mechanics to some extent.To prove (17), let us start expanding the partition func-tion Z ( βJ ) (16) using the following identity,e βJσ i σ j σ k = cosh( βJ ) + σ i σ j σ k sinh( βJ ) . (18)Inserting it into (16), we may expand the partition func-tion as Z ( βJ ) = (cosh( βJ )) N X { σ } Y h i,j,k i (1 + uσ i σ j σ k ) . (19)Let us rewrite (45) in the form Z ( βJ ) = (cosh( βJ )) N X δ u | δ | X { σ } Y h i,j,k i ( σ i σ j σ k ) δ ijk , (20) Figure 5: A typical chain of triangles δ ∈ ∆ in a triangularlattice. It is understood that only part of the lattice is dis-played. Black triangles represent the elements of δ . The factthat δ ∈ ∆ means that at each vertex always meet an evennumber of triangles. where δ = P h i,j,k i δ ijk △ ijk is a chain of triangles, that is,a formal sum over triangles with binary coefficients, and | δ | is the number of triangles in δ . Using the identities X σ = ± σ n o = 0 , X σ = ± σ n e = 2 , (21)where n o and n e are odd and even numbers, respectively,we get Z ( βJ ) = (2 cosh( βJ )) N X δ ∈ ∆ u | δ | , (22)where ∆ contains those chains of triangles such that atany given site i an even number of triangles meet, asshown in Fig. 5. In fact, this type of expansion is calleda high-temperature expansion of the partition functionof a statistical mechanical model [16].In order to compare (40) and (52), we simply observethat triangles △ ijk in Λ correspond to vertices of the 2-colex v ∈ V . This correspondence relates in an obviousway a string-net γ with a triangle chain δ , in such away that ∆ is identified with Γ . Therefore, we havethe desired relationship between the overlapping and thepartition function (17).Although the previous derivation was performed for amodel with uniform couplings, it is possible to obtain acompletely analogous result for triangle dependent cou-plings J ijk . For simplicity we have preferred to do theexposition with uniform couplings because, in fact, thecase of non-uniform couplings is contained in the moregeneral case of non-uniform couplings with non-uniformexternal field, to be considered in section V. B. Consequences
We hereby draw a series of very important conse-quences from these results, that will continue in the nextsection. i/ Interactions:
We see that there is a clear qualita-tive difference between topological color code states andKitaev’s toric code states since they yield quite differenttype of spin interactions: a TCC state yields a 3-bodyinteraction like in (16), while a Kitaev’s code producesthe standard 2-body classical Ising model, namely, Z Ising ( βJ ) := X { σ } e βJ P h i,j i σ i σ j . (23) ii/ Symmetry: A distinctive feature of our result (16) isthat the classical spin model associated to the TCC statedoes not posses the up-down Z spin-reversal symmetry.However, the partition function (16) exhibits a Z × Z symmetry. Recall that the lattice Λ is 3-colorable at sites,so that we can redundantly label our classical spin vari-ables as σ ci with c = r, g, b the color at site i . Then thechange of variables σ ci → s ( c ) σ ci , s ( r ) s ( g ) s ( b ) = 1 , s ( c ) = ± , (24)gives a global symmetry, which has symmetry group Z × Z because s ( b ) = s ( r ) s ( g ). The ground stateshave to display this symmetry, in fact. Consider statesin which the values of the spin variables only dependon the color, that is, for which σ ci = f c with f c = ± f r , f g , f b ).Then it is easy to check that for the ferromagnetic case J > , + , +) , (+ , − , − ) , ( − , + , − ) , ( − , − , +)whereas in the antiferromagnetic case J < − , − , − ) , ( − , + , +) , (+ , − , +) , (+ , + , − ). Thus, eachparity sector, or classical ground state of (15) is fourth-fold degenerate. Notice that the 3-body Ising model insuch 3-colorable lattices of triangles shows no frustration,as opposed to the standard Ising model (23) in such lat-tices which is indeed frustrated.Remarkably, the gauge group underlying the topologi-cal order related to Hamiltonian (3) is also Z × Z . Thesituation is the same also with toric codes, where theglobal symmetry of the classical system is Z and thegauge group for the toric code topological order is Z .This is certainly not a matter of chance, since one canrelate the types of domain walls in the classical systemto the types of condensed strings in the quantum system. iii/ Selfduality: The models in the triangular andUnion Jack lattices turn out to be self-dual like the usual2-body Ising model, with a critical temperature β c givenby the same condition,sinh 2 K c = 1 , K c := β c J c = 0 . . (25)Duality is a property between high-temperature andlow-temperature expansions of a statistical mechanical model like (16) or (23). A high-temperature expansionis a polynomial in the variable u = tanh( βJ ) that issmall when T → ∞ , while a low temperature expan-sion is another polynomial in the variable u ∗ := e − βJ ∗ that is small in the limit T →
0. Then, a self-duality isa relationship between the high-temperature expansionof one classical spin model in a given lattice Λ and thelow-temperature expansion of the same lattice. This isprecisely the case of the 3-body Ising model (16) on boththe triangular and Union Jack lattices [21], [22] and thestandard Ising model (23) [20]. iv/ Universality Classes:
Interestingly enough, the3-body Ising model on the triangular lattice [17], [18]and the Union-Jack lattice [19] are exactly solvable mod-els under certain circumstances and this fact allows usto check their criticality properties when compared withthose of the standard Ising model solution.The critical exponent for the specific heat in the 3-bodyIsing model on the triangular lattice is α = . This rep-resents a power law behaviour which is in sharp contrastwith the well-known logarithmic divergence ( α = 0) ofthe specific heat in the standard Ising model (23). Otherrepresentative exponents are also different: the correla-tion length exponent is ν = (vs. ν = 1), the magneti-zation exponent is β = (vs. β = ), while they sharethe same two-point correlation function exponent at thecritical point η = .For the 3-body Ising model on the Union Jack lattice,the specific heat critical exponent is also remarkably dif-ferent α = . In fact, if the coupling constant J is allowedto be anisotropic, then even the critical exponent α maytake on a set of continuous values in (0 , ) depending ona parameter related to the coupling constants [19].The computational capabilities of a topological colorcode depends on the 2-colex lattices where it is defined.For a TCC on a square-octogonal lattice it is possibleto implement the whole Clifford group of unitary gatesgenerated by the set of gates { H, K , Λ } , where H isthe Hadamard gate, K the π/ the CNOTgate [8]. However, for a 2-colex like the hexagonal lat-tice the set of available gates is more reduced since the π/ C. Borders
If we want to consider classical systems of spins witha finite number of sites, then we have to introduce eitherborders or a nontrivial topology. Since in TCCs the non-trivial topology gives rise to degeneracy, it is preferableto have borders. Also, borders play a role for the ideasto be explained in the next section.
Figure 6: Here both an hexagonal 2-colex C and its dualtriangular lattice Λ (dashed) are displayed to illustrate howborders are introduced in the 2-colex if Λ has borders. Allvertices of the 2-colex which are not triangles in Λ have beenremoved, and also all the faces which keep no vertices. Thefaces that only keep part of their vertices remain, but onlytheir Z face operators are kept in the stabilizer. In TCCs, borders can be of several types. For exam-ple, in [8] it was shown how to build borders of a givencolor. Here our guide to construct the border must bethe dual lattice Λ, which now has a border, along withthe properties of the classical system. Then, as shownin Fig. 6, in order to construct the stabilizer for a TCCwith border in such a way that (17) remains true is tostart with an infinite 2-colex C and then keep only partof it following certain criteria. (i) Keep those vertices v of C which correspond to triangles in Λ. (ii) For thosefaces f of C which keep all their vertices, we keep theface operators X f and Z f . (iii) For those faces f whichonly keep a subset f ′ of their vertices, we introduce aface operator Z f ′ acting on those qubits. Condition (i)ensures the correspondence between triangle chains in Λand string-nets in C . Conditions (ii) and (iii) ensure thecorrespondence between Γ and ∆ . IV. MEASUREMENT-BASED QUANTUMCOMPUTATION WITH COLOR CODES
In a measurement-based quantum computer (MCQ)[5], information processing is carried out via a sequenceof one-qubit measurements on an initialized entangledquantum register. This is an alternative to the stan-dard gate-based quantum computation that can simulatequantum networks efficiently.An interesting problem is to study the performance ofthe MQC when the initial entangled multiparticle stateis a topological toric code like in Kitaev’s model [2]. Inparticular, under which general circumstances the MQC based on the planar Kitaev code can be efficiently simu-lated by a classical computer. The answer to this ques-tion is that the planar code state can be efficiently sim-ulated on a classical computer if at each step of MQC,the sets of measured and unmeasured qubits correspondto simply connected subsets of the lattice [2].Likewise, another very interesting problem is whethera topological color code state is classically sumulable inan scenario of measurement-based quantum computation(MQC). By extending the results of sect. III, it is possibleto address this problem here. Our aim is to see whatconclusions can we learn from the statistical mechanicalmodels in order to test the classical simulability by MQCof the topological code states.The results of sect. III can be interpreted as a com-plete projective measurement of the planar topologicalcolor code state (11) onto a very specific product of sin-gle qubit states (12). This type of global measurementsare not enough for doing a MQC based on the color codestate. Instead, we need to allow for more general one-qubit measurements and to perform them in an adaptivefashion as the computation proceeds from the startingpoint till the end.To be more specific, let us consider a generic qubitstate with complex coefficients | ϕ i v := c v | i v + c v | i v , c v , c v ∈ C . (26)Then, a MQC starts with a planar color code (11) andwe apply a series of projective measurements M v := | x i v h x | , x = 0 ,
1, from the first qubit v = 1 in the code un-til the last one v = | V | . The order in which this sequenceof measurements is carried out through the 2-colex latticeis arbitrary. After each measurement, the correspondingqubit at the vertex v gets projected onto one the statesin (26) and the result is a value for the outcome denotedby m v = 0 , m , . . . , m v , . . . , m | V | with a certain probability distribu-tion P ( m , . . . , m | V | ). We adopt the definition [2] that aMQC is classically simulable in an efficient way if thereexists a classical randomized algorithm that allows oneto sample the outputs m , . . . , m | V | from the probabilitydistribution P ( m , . . . , m | V | ) in a time poly( | V | ).Then, after a complete run of a MQC with (11) we getthe following generalized overlapping O MQC := h Ψ c | O v ∈ V | ϕ i v . (27)Using the same type of computations that led to (14), wearrive at the following expression O MQC = X γ ∈ Γ h γ | O v ∈ V | ϕ i v = Y v ∈ V c v X γ ∈ Γ Y v ∈ V : x v =1 (cid:18) c v c v (cid:19) . (28)This result can also be turned into the partition functionof a statistical model with 3-body Ising interactions (16)but with complex and inhomogeneous Boltzmann weights w ijk = e βJ ijk ∈ C , (29)depending on each triangular plaquette < i, j, k > . Thegeneralized overlapping (28) is proportional to a gener-alized partition function of a 3-body Ising model withinhomogeneous and complex coupling constants Z ( β, { J ijk } ) := X { σ } e β P h i,j,k i∈ Λ J ijk σ i σ j σ k , (30)where the lattice Λ can be the complete triangular or thecomplete Union Jack lattice.At an intermediate stage of a run of a MQC with thecolor code state, the 2-colex will split into two subsets C := M ∪ M , with M the set of measured qubits and M the set of unmeasured qubits [2]. Thus, during therunning of the MQC starting with the color code state,we shall find generalized partition functions of the type(30) but for a lattice that is the dual of the subset ofmeasured qubits: Λ = M ∗ .Therefore, we arrive at the situation that in order toclassically simulate a topological color code state in aMQC we need to simulate the conditional probabilities P ( m v +1 | m , . . . , m | v | ) (at step v + 1 knowing the prob-abilities of previous steps) and for these we need to beable to compute efficiently in a time poly(L) on the size L of the intermediate lattice at step v + 1.At this point there is a sharp difference between theclassical simulation with MQC of Kitaev states and colorcode states. Kitaev states can be classically simulatedunder very general conditions: the subsets M and M must be simply connected. The basic ingredient toachieve this result in the 2-body Ising model is that eventhough a generalized standard Ising model (with arbi-trary complex and inhomogeneous couplings) may not betranslationally invariant, nevertheless there always exista technique allowing it to be mapped onto a dimer cover-ing problem (DCP) which in turn can be solved efficientlythrough the Paffian method in polynomial time [2], [23],[24].However, the dimer problem technique is applicable to2-body interactions but not for the 3-body interactionsthat arise in the generalized statistical mechanical modelsfrom color codes. In the case of the 3-body Ising modelwith uniform and real couplings J ijk = J ∈ R in a tri-angular lattice, it can be exactly solved by mapping itonto the generating function of a suitable site-colouringproblem (SCP) on a hexagonal lattice [17], [18], whichcan be solved by the Bethe ansatz. In order for this site-coloring mapping to work, certain restrictive conditionson the triangular lattice must be fulfilled. In particular,the partition function (16) has to be defined on a periodictriangular lattice with L rows in the horizontal directionand N columns in the vertical direction. Let us denoteit as Z (3) LN . Let us also denote by Z SCP MN the generating function of a site-coloring problem on a hexagonal latticewith M = L and N columns. Then, the aforementionedmapping works in the limit N → ∞ as Z (3) LN = Z SCP MN , N ≫ . (31)Furthermore, the SCP is solved by Bethe ansatz tech-nique. This technique also poses another fundamentalproblem in this situation since it is used to compute theeigenvalues of the associated transfer matrix of the SCPand then the issue about the completeness of that spec-trum in terms of Bethe ansatz eigenfunctions arises. Thisissue is always a difficult question and, strictly speaking,it is a conjeture. Quite on the contrary, these difficul-ties are absent in the standard Ising model case since theDCP is more versatile.The situation becomes even more difficult if we con-sider the generalized partition function (30) in the frame-work of an intermediate step in the MQC. Then, it is notknown how to solve it efficently with a mapping to a SCPin an hexagonal lattice without restrictions.This site-coloring mapping plays a similar role thanthe dimer covering mapping in the standard 2-body Isingmodel. However, the known solutions to this site-coloringproblem demand more restrictive conditions on the typeof lattices and they are less powerful than the dimer map-ping technique.As for the topological color code on a 2-colex like theUnion Jack lattice, similar conclusions apply: in the thecase of real couplings J ijk = J ∈ R , it is exactly solvablesince it can be mapped onto a 8-vertex model [19], whichin turn has to be solved by the Bethe ansatz.The fact that the 3-body classical Ising model is ex-actly solvable in the triangular and Union Jack lat-tices for real and isotropic couplings, is not enough soas to conclude that the corresponding topological colorcode states are classically simulable in a MQC scenario.Therefore, the classical simulability of topological colorcodes with MQC remains an open problem. V. CLUSTER STATES AND MODELS WITHAN EXTERNAL FIELD
In a MQC scheme of quantum computation, the in-put state is called a cluster state [5] which is a rathergeneral entangled state associated to a great variety ofgraph states, i.e., states constructed from qubit stateslocated at the vertices of a lattice specified by the inci-dence matrix of a graph. A very important property ofthese cluster states is that they can be created efficientlyin any system with a quantum Ising-type interaction be-tween two-state particles in the specified lattice configu-ration. In Sect.IV we have assumed that the input statefor the MQC was a TCC state, without caring about howit could be prepared. Here we show how such a topolog-ical color code state can be obtained from a appropriatecluster state. As a by-product, this construction will turnout to be useful for obtaining classical Ising models in
Figure 7: The graph needed to obtain a color code state froma cluster state. The graph is bipartite, and the vertices aredivided in black and white. Black vertices correspond to thefaces of the 2-colex. White vertices correspond to the verticesof the 2-colex. Black thick lines represent the edges of thegraph, and grey lines correspond to the edges of the 2-colex. an external magnetic field being associated to color codestates.
A. Cluster state formulation of TCC
Instead of giving a general definition of cluster states,we will consider only bipartite cluster states. So let G bea finite bipartite graph, that is, a graph such that its setof vertices U is the disjoint union of two sets U = U ∪ U in such a way that neighboring vertices never belong tothe same U i . Consider the quantum system obtained byattaching a qubit to each of the vertices u . For each suchvertex, we denote by N ( u ) the set containing both u andits neighbors. The cluster state | κ i for the graph G isthen completely characterized by the conditions: ∀ u ∈ U , X N ( u ) | κ i = | κ i , ∀ u ∈ U , Z N ( u ) | κ i = | κ i . (32)In order to relate the TCC C of a 2-colex C to a clusterstate, we construct a bipartite graph G by setting U = V and U = F . Then the edges are defined so that u = v ∈ U is a neighbor of u = f ∈ U if v is a vertex of f in C ,see Fig. 7. Observe that the corresponding cluster state | κ i satisfies ∀ f ∈ F , X f | κ i = | κ i , (33)because X f = N v ∈ f X N ( v ) . In fact, if γ is a string-netwith ∂ c γ = 0, then X γ | κ i = | κ i . However, to keep thingsas simple as possible, let us just consider 2-colexes inwhich all closed string-nets are boundaries. If we measurein the Z basis all the qubits corresponding to vertices u ∈ U we will obtain a series of binary values x u = x f .The remaining qubits are then in a state characterizedby the conditions: ∀ f ∈ F , X f | κ i = | Ψ c ( x ) i , ∀ f ∈ F , Z f | κ i = x f | Ψ c ( x ) i . (34)Thus, if x f = 0 for all the faces f , the result is the TCCstate | Ψ c i (11). In the other cases the result is essentiallya TCC, in particular the state is | Ψ c ( x ) i := X γ ∈ Γ x | γ i (35)where x denotes the binary vector of the measurementresults and Γ x is the set of string-nets γ with ∂ c γ = P f x f f .In fact, the original cluster state can be written asfollows: | κ i = X x | x i ⊗ X γ ∈ Γ x | γ i (36)where | x i = N u ∈ U | x u i u is an element of the computa-tional basis of the subsystem of qubits in U . To checkthat this is indeed the correct expression for the clusterstate, it is enough to note that it satisfies (32). B. Models with an external field
Thus far, we have only considered statistical mechan-ical models with zero external magnetic field. Here wego beyond that situation, considering the formulation ofmodels with 3-body Ising interactions and arbitrary mag-netic fields from the projection of topological color codesonto appropriate product states.Let us define the product state | Φ P ( J , h ) i := O v ∈ U | φ ( J v ) i v O f ∈ U | φ ( h f ) i f , (37)where J = ( J v ) ∈ R | V | , h = ( h f ) ∈ R | F | and | φ ( s ) i := cosh s | i + sinh s | i , s ∈ R . (38)Consider the overlapping O ( β, J , h ) := h Ψ c | Φ p ( β J , β h ) i . (39)Its value is X x h x | O f ∈ U | φ ( h f ) i f X γ ∈ Γ x h γ | O v ∈ U | φ ( J v ) i v = Y f cosh( βh f ) Y v cosh( βJ v ) X x X γ ∈ Γ x Y f u f x f Y v u v γ v , (40)0where u f := tanh( βh f ) , u v := tanh( βJ v ) . (41)We want to relate (39) to the partition function ofa classical spin system. As in section III, we considerthe lattice Λ dual to the 2-colex C and we associate aclassical system to Λ by attaching classical spin variables σ i = ± i . This time however weinclude triangle dependent couplings J ijk in triangles anda site dependent external field h i . Thus, we want toderive the high temperature expansion for the partitionfunction Z ( β, J , h ) := X { σ } e − β H ( J , h ) , (42)where the classical Hamiltonian is H ( J , h ) := − X i h i σ i − X h i,j,k i J ijk σ i σ j σ k . (43)We start using the identitiese βh i σ i = cosh( βh i ) + σ i sinh( βh i ) , e βJ ijk σ i σ j σ k = cosh( βJ ijk ) + σ i σ j σ k sinh( βJ ijk ) , (44)so that, Z ( β, J , h ) = C X { σ } Y i (1 + u i σ i ) Y h i,j,k i (1 + u ijk σ i σ j σ k ) , (45)where C := Y i cosh( βh i ) Y h i,j,k i cosh( βJ ijk ) , (46) u i := tanh( βh i ) , u ijk := tanh( βJ ijk ) . (47)Let us rewrite (45) in the form Z ( β, J , h ) = C X x X δ X { σ } Y i ( u i σ i ) x i Y h i,j,k i ( u ijk σ i σ j σ k ) δ ijk , (48)where x = ( x i ) is a binary vector and δ = P h i,j,k i δ ijk △ ijk is a chain of triangles, that is, a formalsum over triangles with binary coefficients. Reorderingthe expression we get Z ( β, J , h ) = C X x X δ ǫ ( x , δ ) Y i u ix i Y h i,j,k i u ijkδ ijk , (49)where ǫ ( x , δ ) := X { σ } Y i σ ix i Y h i,j,k i ( σ i σ j σ k ) δ ijk , (50)From (21) it follows that ǫ ( x , δ ) = (cid:26) N , if ∀ i Q h j,k i i = x i , , in other case. (51) where h j, k i i denotes the pairs ( j, k ) which form a trianglewith i , and N is the number of sites. Finally we can givethe desired high temperature expansion of the partitionfunction: Z ( β, J , h ) = 2 N C X x X δ ∈ ∆ x Y i u ix i Y h i,j,k i u ijkδ ijk , (52)where ∆ x contains those chains of triangles such that atany given site i an even (odd) number of triangles meetif x i = 0 ( x i = 1).In order to compare (40) and (52), we first observethat sites i correspond to faces of the 2-colex f ∈ F = U and triangles △ ijk correspond to vertices of the 2-colex v ∈ V = U . This correspondence relates in an obvi-ous way x i with x v , h i with h v and so on and so forth.Also, there is an exact correspondence between chains oftriangle δ and string-nets γ . In particular, this correspon-dence identifies ∆ x with Γ x , so that we get the desiredrelationship between the overlapping and the partitionfunction Z ( β, J , h ) = 2 N O ( β, J , h ) . (53) VI. CONCLUSIONS
We have shown that the classical spin models asso-ciated to quantum topological color code states in thetwo dimensional lattices called 2-colexes are Ising modelswith 3-body interactions. We have studied this mappingin the triangular and the Union Jack lattices, which arethe duals of the two very represenative 2-colexes, namely,the hexagonal and the square-octogonal lattices, respec-tively. This is a genuine difference with respect to thecase with toric code states which yield the partition func-tion of the standard Ising model with two-body inter-actions. Ising models with different n-body interactionshave very different properties in general. Remarkbly, dif-ferent computational capabilities of the topological colorcodes depending on the chosen 2-colex correspond to dif-ferent universality classes of the associated classical 3-body Ising models.The tools employed to relate classical spin models withtopological color code states can be extended to studythe performance of such topological states when theyare considered as input in a measurement-based quan-tum computation. Then, the classical 3-body modelsthat arise involve arbitrary complex couplings and lat-tice shapes. The problem of evaluating their correspond-ing generalized partition functions cannot be performedwith the dimer covering technique that is so successfulin the case of the classical 2-body Ising model. Thesimilar technique in the 3-body case is a particular site-coloring problem that only in very specific instances hasbeen solved by means of the Bethe ansatz. The com-pleteness of the Bethe ansatz poses in turn more fun-damental problems in this regard. Therefore, the fact1that the 3-body Ising model is exactly solvable in certainconditions is not enough to conclude so far that the par-ent quantum topological color color states are efficientlyclassically simulated by MQC.Another interesting result is the construction of a clus-ter state from which we can construct the topologicalcolor code state. This turns out to be useful in order toobtain classical spin models in the presence of arbitraryexternal magnetic fields. Likewise, there are other two- dimensional multipartite states that arise in the study ofquantum antiferromagnets that may lead to a variety ofclassical spin models [25].
Acknowledgements
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