Classical spin systems and the quantum stabilizer formalism: general mappings and applications
aa r X i v : . [ qu a n t - ph ] A ug Classi al spin systems and the quantum stabilizer formalism: general mappings andappli ationsR. Hübener , M. Van den Nest , W. Dür , and H. J. Briegel , Institut für Theoretis he Physik, Universität Innsbru k, Te hnikerstraÿe 25, A-6020 Innsbru k, Austria Institut für Quantenoptik und Quanteninformation der Österrei his henAkademie der Wissens haften, Te hnikerstraÿe 21, A-6020 Innsbru k, Austria Max-Plan k-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Gar hing, Germany.We present general mappings between lassi al spin systems and quantum physi s. More pre- isely, we show how to express partition fun tions and orrelation fun tions of arbitrary lassi alspin models as inner produ ts between quantum stabilizer states and produ t states, thereby gen-eralizing mappings for some spe i(cid:28) models established in our previous work [Phys. Rev. Lett. 98,117207 (2007)℄. For Ising- and Potts-type models with and without external magneti (cid:28)eld, we showhow the entanglement features of the orresponding stabilizer states are related to the intera tionpattern of the lassi al model, while the hoi e of produ t states en odes the details of intera tion.These mappings establish a link between the (cid:28)elds of lassi al statisti al me hani s and quantuminformation theory, whi h we utilize to transfer te hniques and methods developed in one (cid:28)eld togain insight into the other. For example, we use quantum information te hniques to re over wellknown duality relations and lo al symmetries of lassi al models in a simple way, and provide new lassi al simulation methods to simulate ertain types of lassi al spin models. We show that in thisway all inhomogeneous models of q -dimensional spins with pairwise intera tion pattern spe i(cid:28)edby a graph of bounded tree-width an be simulated e(cid:30) iently. Finally, we show relations between lassi al spin models and measurement-based quantum omputation.PACS numbers: 03.67.-a,03.67.Lx,75.10.Hk,75.10.Pq,02.70.- I. INTRODUCTIONClassi al spin systems are widely studied in statisti alphysi s [1℄. They also play an important role in model-ing omplex behavior also in other dis iplines, su h ase onomi s and biology. In spite of their often simplede(cid:28)nition, spin models show a highly non-trivial behav-ior, as is, e.g., apparent from their phase stru ture and riti ality. Surprisingly, even the simple Ising model ofintera ting 2-state spins arranged on a 2D square latti e(with external magneti (cid:28)eld) is in general not solvable,and al ulating, e.g., the ground state energy or the par-tition fun tion is known to be a omputationally hardproblem [2℄.In quantum information theory (QIT), on the otherhand, (entanglement) properties of quantum systemsare systemati ally studied, and possible appli ations re-garding, e.g., quantum omputation are investigated.QIT has be ome a (cid:28)eld of interdis iplinary interest, and on epts and methods developed in QIT have found ap-pli ations also in other bran hes of physi s. In the on-text of QIT, methods to e(cid:30) iently ompute and simu-late ertain quantum systems and their properties havebeen developed [3, 4℄. In parti ular, so- alled quantum(cid:16)stabilizer states(cid:17) [5, 6, 7℄ and (cid:16)graph states(cid:17) [8, 9℄ havebeen introdu ed and studied in detail. Stabilizer statesare used for ertain types of quantum error orre tion [5℄and measurement-based quantum omputation [10℄, and an be des ribed e(cid:30) iently in terms of their stabilizingoperators. This allows to determine many of their (en-tanglement) properties, and to e(cid:30) iently simulate somepro esses lassi ally.In this paper, we present general mappings between lassi al spin systems and quantum physi s related toQIT. More pre isely, we show how to express the par-tition fun tion and orrelation fun tions of an arbitrary lassi al spin system as a quantum me hani al ampli-tude (s alar produ t) between a stabilizer state | ψ i en- oding the intera tion pattern, and a ertain produ tstate ⊗ j | α j i en oding the details of the intera tion (i.e.the oupling strengths) and the temperature: Z G = h ψ | O j | α j i . (1)With su h a mapping at hand, we an use methods andte hniques established in one (cid:28)eld to gain insight intothe other, thereby providing a novel approa h to theseproblems. We have initiated this approa h in a re entpubli ation [11℄, where su h mappings have been estab-lished for Ising and Potts-type models. Here we gener-alize this approa h, and dis uss the mappings and theirappli ations in more detail.We further note that onne tions between quantuminformation theory and statisti al me hani s have re- ently been studied by several other resear hers [12, 13℄.A. Mappings between lassi al spin systems andquantum physi sIn this se tion we brie(cid:29)y sket h the general form ofthe proposed mappings between lassi al and quantumsystems.We onsider lassi al q -state spin systems with anarbitrary pairwise intera tion pattern, des ribed by agraph G with vertex set V (position of the lassi alspins) and edge set E ( orresponding to intera tions).Su h systems are sometimes alled (cid:16)edge models(cid:17) (i.e.,the intera tions take pla e on the edges). Ea h spin s may assume q di(cid:27)erent states: s ∈ { , . . . , q − } .We will onsider models where the pairwise intera tions h ( s, s ′ ) between spins s and s ′ are of the following forms:(i) h ( s, s ′ ) only depends on the di(cid:27)eren e (modulo q )of the two involved spins, h ≡ h ( | s − s ′ | q ) ;(ii) h ( s, s ′ ) is of the form (i), but with additional lo almagneti (cid:28)elds;(iii) h ( s, s ′ ) is ompletely arbitrary.We will also onsider (iv) models with arbitrary k -bodyintera tions.The Ising- and Potts model without [with℄ magneti (cid:28)eld are of type (i) [(ii)℄ respe tively, while so- alled(cid:16)vertex models(cid:17) (i.e., the intera tions take pla e on theverti es) are a spe ial ase of type (iv).In ea h of the ases (i)-(iv), we show how one an ex-press the partition fun tion Z G as an overlap between aquantum stabilizer state and a omplete produ t state,(Eq. (1)). Depending on the di(cid:27)erent forms of the in-tera tion (as in (i)-(iv)), these quantum states will bede(cid:28)ned slightly di(cid:27)erently.(i) For models without lo al (cid:28)elds, the orrespond-ing quantum states onsists of | E | q -level quan-tum systems (one for ea h pairwise intera tionterm). We will denote the stabilizer state by | ψ G i .The produ t state has the form | α i = N e ∈ E | α e i ,where the oe(cid:30) ients of ea h | α e i en ode thestrengths of the pairwise ouplings, as well as thetemperature of the system.(ii) For models with lo al magneti (cid:28)elds, the orre-sponding quantum states onsist of | V | + | E | q -level quantum systems (one for ea h pairwise in-tera tion term and one for ea h lo al (cid:28)eld), withstabilizer states denoted by | ϕ G i and a produ tstate | α i = N e ∈ E | α e i N a ∈ V | α a i .(iii) For models with general pairwise intera tion (iii),we provide a mapping where the stabilizer stateis a tensor produ t of | V | entangled states, | φ i = N a ∈ V ( P q − j =0 | j i ⊗ n a ) . Here, n a is the degree ofvertex a , i.e. the number of neighbors in the graph,whi h also determines the number of asso iated q -level quantum systems. Correspondingly, we now onsider states | α ab i of dimension q for the over-lap, whi h are asso iated to one quantum parti lebelonging to vertex a and and one quantum parti- le belonging to vertex b . A similar pi ture holdsfor models with arbitrary k -body intera tions (iv),where the produ t states have now dimension q k ,and are asso iated with multiple verti es.We will investigate the entanglement properties of thestates | ψ G i and | ϕ G i and their relation to the underly-ing intera tion pattern spe i(cid:28)ed by the graph G , and provide a number of examples to illustrate this onne -tion.The mappings (ii)-(iv) an be extended, and will allowus to express also lassi al orrelation fun tions in aquantum language.B. Appli ations of the mappingsBased on these mappings, we will then illustrate someappli ations. Here we brie(cid:29)y sket h whi h appli ations an be obtained.(a) Using well established stabilizer methods [5, 6, 7,8℄, we show how one an re over the well knownhigh-low temperature duality relations [1℄ for las-si al spin models on arbitrary planar graphs.(b) Using the fa t that stabilizer states are stabilizedby ertain tensor produ t operators, we derive lo- al symmetry relations for lassi al models, i.e. weidentify models with di(cid:27)erent oupling strengthsthat lead to the same partition fun tion.( ) We show how one an use re ently established re-sults in QIT to lassi ally simulate ertain lassesof quantum systems e(cid:30) iently [3, 4, 14℄ and thusobtain novel simulation algorithms for lassi alspin system. More pre isely, by des ribing sta-bilizer states in terms of an optimal tree tensornetwork [3℄ of dimension d , one an ompute theoverlap with produ t states with an e(cid:27)ort that ispolynomial in d . This leads to an e(cid:30) ient algo-rithm to lassi ally simulate arbitrary (inhomoge-neous) lassi al q -state models on graphs with abounded (or logarithmi ally growing) tree width.We also extend these results to models with k -body intera tion.(d) Finally, we dis uss links between lassi al spinmodels and measurement based quantum ompu-tation. This allows us to relate the omputa-tional omplexity of omputing partition fun tionsof lassi al spin models with the quantum ompu-tational power of the asso iated graph states.We also note that (d) has re ently been used inRef. [15℄ to show a (cid:16) ompleteness(cid:17) property of the2D Ising model. That is, invoking the onne tionto measurement-based quantum omputation, it wasshown that the partition fun tion of any model withpairwise intera tion in arbitrary dimension an be ex-pressed as a spe ial instan e of the partition fun tionof a 2D Ising model on an (enlarged) 2D square latti e(with omplex oupling strengths).C. Guideline through the paperThe paper is organized as follows. We start in Se . IIby brie(cid:29)y reviewing lassi al spin models, and olle tsome relevant results on stabilizer and graph states inSe . III. We then introdu e di(cid:27)erent mappings between lassi al spin systems and quantum me hani al ampli-tudes, and dis uss the properties of the involved quan-tum states in Se s. IV and V. We illustrate a numberof appli ations of these mappings in Se . VI, and sum-marize and on lude in Se . VII.II. BACKGROUND ON SPIN MODELSIn this se tion we des ribe the lassi al models thatwe want to onsider. Sin e the various approa hes to bedes ribed later are related and an be viewed as deriva-tions from an original s heme, we will fo us on the orig-inal approa h (cid:28)rst.The typi al model to be onsidered by the originalapproa h is the thermal state of a lassi al spin modeldes ribed by a Hamiltonian fun tion with two-body in-tera tion, and this model will serve as an introdu toryguide to the general idea. These systems have the virtuethat they admit a des ription by means of a graph [16℄:the spins of the system orrespond to the verti es andthe two-body intera tion pattern between the spins isgiven by the edge set.We will des ribe a mapping of su h an intera tiongraph to a stabilizer state of a quantum system. Per-forming an overlap of this quantum stabilizer state withanother quantum produ t state, en oding the temper-ature and individual intera tion strengths, then yieldsthe properties of the thermal state of the lassi al sys-tem. We want to emphasize that this evaluation is notapproximate but exa t. Later on, extensions of this for-malism will be given as well, going beyond this parti -ular kind of graphi al des ription and at the same timegoing beyond the limitation to two-body intera tions.It is important to keep in mind that the intera tionpattern and the intera tion strengths are en oded at dif-ferent pla es: the graph en odes the intera tion pattern,not the strengths, hen e an edge onne ting two verti essimply denotes the fa t that there is an intera tion tak-ing pla e. The strength and nature of this intera tionis not en oded in the graph, but in a produ t state tobe spe i(cid:28)ed later. This en oding admits the strengthsof all edge terms and all vertex terms to be hosen in-dividually, hen e the intera tion strength may vary fordi(cid:27)erent pairs of spins and also the lo al (cid:28)eld may vary.More pre isely, let, for now, H be a Hamiltonian fun -tion with two-body-intera tion between lassi al spins s that an assume q possible values s ∈ { , ..., q − } . Inthe graphi al des ription of this Hamiltonian fun tion,we let G = ( V, E ) denote the graph asso iated with H ,where the sets V and E ontain the verti es and theedges of the graph respe tively. In this pi ture, any ver-tex v ∈ V orresponds to a lassi al spin site s v and anyedge e ∈ E between adja ent verti es v , v of the graph orresponds to an intera tion term between the respe -tive spins s v and s v . Additionally, we allow ea h spin s v to ontribute a lo al term to the Hamiltonian fun - tion, i.e. a term that that depends on the state of thesite s v alone, although this is not re(cid:29)e ted in the graph.We might think of the energy of the spin in a lo al (cid:28)eld.We hoose the graph to be a dire ted one, denoting theorientation by σ . The exa t hoi e of the dire tions anbe arbitrary but has to be (cid:28)xed. This way, the two ad-ja ent verti es of an edge e ∈ E an be distinguished as(cid:16)head(cid:17) v + e and (cid:16)tail(cid:17) v − e of the edge, respe tively.We will derive several di(cid:27)erent mappings for Hamil-tonian fun tions des ribed by these graphs. The (cid:28)rstmapping admits des riptions of systems with lassi alHamiltonian fun tions of the form H ( { s i } ) = X e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) , (2)with h e being an energy term that depends on the rel-ative state of two intera ting spins s v + e and s v − e modulo q . In the se ond mapping we extend the quantum de-s ription to be able to in lude also external (cid:28)elds H ( { s i } ) = X e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) + X v ∈ V b v (cid:0) s v (cid:1) , (3)where b v is an energy term ontributed by a lo al ex-ternal (cid:28)eld, a ting on the spin s v . To go beyond thelimitation to intera tion Hamiltonian fun tions that de-pend on the relative state of the spins only, we (cid:28)nallyprovide further approa hes to treat Hamiltonian fun -tions of the form H ( { s i } ) = X ( ij ) ∈ E h ( ij ) (cid:0) s i , s j (cid:1) as well as arbitrary Hamiltonian fun tions with n -bodyterms.The degrees of freedom in the de(cid:28)nitions of theseHamiltonian fun tions give rise to a large set of pos-sible lassi al spin systems to be des ribed (cid:21) even ifwe restri ted ourselves to the sets of Hamiltonian fun -tions spe i(cid:28)ed in Eqns. (2) and (3). Among those arethe Ising model, the Potts model and the lo k modelon arbitrary latti es, all equipped with (lo al) magneti (cid:28)elds, and generalizations thereof [1℄.A. Ising modelThe Ising model des ribes a set of lassi al spins (orsimply dipoles) that an point either up or down and arepla ed on a graph. All next neighbors have the same dis-tan e (hen e the intera tion strength is uniformly givenby the real number J ) and long-range for es are ne-gle ted. Moreover, there is a global external (cid:28)eld whosestrength is given by the real number B , whi h puts anenergeti bias on the possible on(cid:28)gurations. Thus the lassi al Ising model is des ribed by the Hamiltonianfun tion H Ising ( { s i } ) = J X(cid:10) i,j (cid:11)(cid:12)(cid:12) s i − s j (cid:12)(cid:12) + B X i (cid:0) s i − (cid:1) , (4)where the s i ∈ { , } and (cid:10) i, j (cid:11) denotes that i and j areadja ent spins on the graph. We note that it an berewritten as H Ising ( { σ i } ) = − J ′ X(cid:10) i,j (cid:11) σ i σ j + B ′ X i σ i , where σ i ∈ { +1 , − } . This is the more familiar formand an be obtained from Eq. (4) by a res aling ofparameters and an addition of a onstant. Althoughthis model is highly idealized, it features (in appropri-ate dimensions) many properties of realisti solids, su has phase transitions, spontaneous symmetry breakinget . As will be shown, our treatment allows(cid:22)withouta hange of omputational e(cid:27)ort(cid:22)the generalization tospin-glass Hamiltonian fun tions, where the fa tor J isa tually dependent on the spe i(cid:28) pairs of spins thatintera t: J → J ij .B. Potts and lo k modelsA generalization of the Ising model is given by thePotts- and the lo k model. Whereas the individualspins in the Ising model an take only one of two valuesand hen e for neighbors there are only the alternativesof being parallel or anti-parallel, it might be desirableto allow the individual dipoles to assume more posi-tions and hen e to obtain more relative on(cid:28)gurations ofneighbors that an be dis riminated energeti ally. A - ordingly we hoose spin states s i ∈ { , ..., q − } and aHamiltonian fun tion H ( { s i } ) = − X(cid:10) i,j (cid:11) J (Θ ij ) + b X i (cid:0) s i − q − (cid:1) , (5)where Θ ij is a fun tion that dis riminates the relativestates of neighboring spins. We an interpret it forinstan e as the angle between adja ent dipoles, pro-vided that they an only rotate in a (cid:28)xed plane, e.g., Θ ij = Θ i − Θ j with dis retised positions Θ i = 2 πs i /q .The fun tion J , whi h hara terizes the Hamiltonianfun tion, maps the relative angle (i.e., relative state) ofadja ent spins to an energy value: The Potts model isde(cid:28)ned by J Potts (Θ ij ) := − εδ (Θ ij ) with ε ∈ R and the lo k model by J lo k (Θ ij ) := − ε cos(Θ ij ) . C. Partition fun tionThe fo us of this paper will be on the thermal equi-librium of these lassi al systems. More pre isely, the entral quantities of interest that we want to obtain arethe partition fun tion Z ( β ) = X { s i } e − βH ( { s i } ) as well as the n-point orrelation fun tions, whose de(cid:28)-nition an be found, e.g., in Ref. [17℄ h s i , s i , ..., s i n i β = Z − X { s i } cos (Θ i ) cos (Θ i ) ... cos (Θ i n ) e − βH ( { s i } ) . The partition fun tion en odes the ma ros opi prop-erties of a thermal ensemble. The parameters that en-ter depend on the kind of ensemble we look at, e.g.,the anoni al (temperature), grand anoni al (temper-ature and hemi al potential) and others. In the presentframework we will deal with the anoni al ensemble, be- ause the number of spin sites is (cid:28)xed, but energy anbe drawn from an external bath.Let us brie(cid:29)y illustrate the importan e of the par-tition fun tion. The partition fun tion of a anoni alensemble is Z = X i e − βE i , where the index i is the index for the states with energy E i that the system an take and β = ( k B T ) − with theBoltzmann onstant k B . Moreover, p i = Z − i e − βE i isthe probability to (cid:28)nd the system in the state with en-ergy E i . Several relevant quantities an now be derivedfrom Z : We an extra t the expe tation value of theenergy h E i β = Z − X i E i e − βE i = − ∂ log Z∂β , the varian e of the expe ted energy D ( δE ) E β = ∂ log Z∂β , as well as the free energy F = h E i β − T S = − β − log Z, where the entropy is S = − k B P i p i log p i , and more.We refer the reader to standard text books on this topi .III. STABILIZER STATES AND GRAPHSTATESIn this se tion, we give the de(cid:28)nition and some prop-erties of stabilizer states [5, 6, 7℄ and graph states [8, 9℄.We will (cid:28)rst onsider spin-1/2 quantum systems, thenpro eed to higher dimensional systems.A. Graph statesHere we will brie(cid:29)y familiarize the reader with thegraph states. In the present ontext, a graph G = ( V, E ) is identi(cid:28)ed with a quantum system. Ea h vertex a rep-resents a quantum spin, and the adja ent verti es ( on-ne ted with a by edges in the graph) form the neigh-borhood N a of a . This way, the graph de(cid:28)nes a set ofoperators K a := σ ( a ) x Y b ∈ N a σ ( b ) z , where the sigma-matri es are de(cid:28)ned as usual σ = (cid:18) (cid:19) , σ x = (cid:18) (cid:19) , (6) σ y = (cid:18) − ii (cid:19) , σ z = (cid:18) − (cid:19) , and the notation O ( a ) of an operator O means the tensorprodu t of the operator O , a ting on the subspa e of site a , and 1 everywhere else. A graph state | G i asso iatedwith to the graph G , and hen e with the set { K a } , isthe unique non-trivial (cid:28)xed point of the operators K a , ∀ K a : K a | G i = | G i . Graph states are a subset of the stabilizer states,whi h play an important role in the ontext of one-wayquantum omputing. Conversely, every stabilizer state an be written, up to a lo al rotation, as a graph state.B. Stabilizer statesWe will now turn our attention to the slightly moregeneral set of stabilizer states. The on ept of de(cid:28)ning astate as a simultaneous (cid:28)xed point of a set of operators an be used in a slightly more general way than in the ase of graph states, where the operators K a take avery spe ial form. To onstru t more general sets ofoperators we onsider the sigma-matri es, see formula(6), and the group they generate G = {± σ , ± iσ , ± σ x , ± iσ x , ± σ y , ± iσ y , ± σ z , ± iσ z } . Tensor produ ts of G with itself form the Pauli groups G n := G ⊗ n . It is known that any Abelian subgroup S ⊂ G n of a Pauli group with (cid:12)(cid:12) S (cid:12)(cid:12) = 2 n that does not ontain − n has a unique (cid:28)xed point | ψ i in the Hilbertspa e H that it a ts upon. We then all S the stabilizerof | ψ i and | ψ i a stabilizer state. It should be notedthat ea h stabilizer an be identi(cid:28)ed with its generator,i.e., a set of operators that generate it. Generators arenot unique sets, but share the ne essary requirement to ontain n independent operators.For our purposes, the prefa tor ( ± , ± i ) of an elementof a Pauli group will not be important. Moreover, there is a mapping between the Pauli group G n / ∼ ( G n moduloprefa tors) and the group F n , whi h will be used later.Sin e σ y = iσ x σ z and σ = for all sigma-matri es,we an en ode the generators of G / ∼ as follows σ ∼ σ x σ z (00) σ x ∼ σ x σ z (10) σ y ∼ σ x σ z (11) σ z ∼ σ x σ z (01) . where ∼ denotes equality modulo prefa tor. Tensorprodu ts of these operators and hen e elements of thegroups G n / ∼ will be en oded by the mapping G n / ∼ ∋ n O i =1 σ ξ i x σ ζ i z ( ξ , ..., ξ n , ζ ..., ζ n ) ∈ F n . The generalization to q -dimensional quantum systemswith H = (cid:0) C (cid:1) ⊗ q is straightforward. We repla e σ x and σ z by the operators X and Z respe tively, where X | j i = | j + 1 mod q i , Z | j i = e πij/q | j i ,q = 2 being a spe ial ase that gives us ba k σ x and σ z .The higher-dimensional groups G qn / ∼ are thus generatedby tensor produ ts of X a Z b where a, b = 0 , ..., q − . Themapping is generalized to the group homomorphism ( G qn / ∼ , · ) ∋ n O i =1 X ξ i Z ζ i ( ξ , ..., ξ n , ζ ..., ζ n ) ∈ (cid:0) F nq , + (cid:1) . The number of elements in a stabilizer that stabilizesone single stabilizer state is q n , the number of elementsof its generator is n .Related to this onstru tion is a theorem that we willuse later. Note that we do not negle t the phase thistime.Lemma 1. Any two operators N ni =1 X ξ i Z ζ i and N ni =1 X ξ ′ i Z ζ ′ i ommute if and only if ξ ′ · ζ − ξ · ζ ′ = 0 modulo q .Proof. The omputation for the single spin site yields X ξ i Z ζ i X ξ ′ i Z ζ ′ i = X ξ i + ξ ′ i Z ζ i + ζ ′ i e πiξ ′ i ζ i /q = X ξ ′ i Z ζ ′ i X ξ i Z ζ i e πi ( ξ ′ i ζ i − ξ i ζ ′ i ) /q . Hen e for all sites together we obtain a phase fa tor e πi ( ξ ′ · ζ − ξ · ζ ′ ) /q .It is noteworthy that for q = 2 ea h stabilizer state isrelated to a graph state by some lo al unitary transfor-mations. This means that the two sets do not di(cid:27)er asfar as their non-lo al properties are on erned.The stabilizer states are interesting to us, be ause(cid:22)aswill be shown(cid:22)the intera tion patterns of the Hamil-tonian fun tions of the lassi al spin systems that welook at orrespond to su h states. Moreover, stabilizerstates are well investigated and elaborate te hniques areknown for their manipulation [5℄, allowing us to investi-gate relationships between di(cid:27)erent (intera tion) graphsand hen e di(cid:27)erent Hamiltonian fun tions.IV. ENCODING CLASSICAL SPIN SYSTEMSIN QUANTUM LANGUAGEIn this se tion we will investigate in detail the or-responden e of the lassi al and the quantum systemsthat were presented in the pre eding se tions.A. The basi prin ipleThe basi approa h, whi h was introdu ed inRef. [11℄, is su(cid:30) ient to des ribe systems with las-si al Hamiltonian fun tions of the form H ( { s i } ) = P e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) . The idea is to map the graph G , des ribing the intera tion pattern into a stabilizerstate, together with a supplementary produ t state thaten odes the intera tion strengths as well as the temper-ature.Let the lassi al spin system be de(cid:28)ned by the (arbi-trarily oriented) intera tion graph G σ = ( V, E ) over (cid:12)(cid:12) V (cid:12)(cid:12) lassi al spins of dimension q , where σ denotes the ori-entation. Let in the following M = (cid:12)(cid:12) V (cid:12)(cid:12) and N = (cid:12)(cid:12) E (cid:12)(cid:12) .Now onsider the in iden e matrix B σ of the intera -tion graph G σ . This matrix has one row for ea h vertexand one olumn for ea h edge. The entries are either or ± , where B σv,e = − if the index pair ( v, e ) orre-sponds to the tail vertex v of edge e , B σv,e = +1 for thehead vertex v of edge e and B σv,e = 0 otherwise. Consis-tent with our notation, we do not onsider graphs withedges that onne t one point with itself. The rows of B σ span the Z q -ve tor spa e C G ( q ) , whi h is a linearsubspa e of Z Nq . The ve tors ( B σ ) T s ∈ C G ( q ) , where T denotes transposition, orrespond to the ve tors thaten ode spin on(cid:28)gurations ( s v ) v ∈ V , as the linear map-ping (cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V ( B σ ) Te,v s v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q shows.Lemma 2. The kernel of the linear mapping ( B σ ) T has q κ elements, where κ is the number of onne tedsub-graphs of G (without isolated points).Proof. We re-arrange the rows of the matrix of ( B σ ) T so that the onne ted sub-graphs G i = ( E i , V i ) are de-s ribed by blo ks B i , i.e., ( B σ ) T B · · · B · · · B · · · ... ... ... . . . . Within ea h onne ted sub-graph G i , there is at leastone path from ea h vertex v to ea h other vertex v ′ : ( v, v , v , ..., v ′ ) , ea h edge ( v n , v n +1 ) in this path be-ing represented by one row in the orresponding matrix B i . Sin e a ve tor s to be in the kernel of B i implies (cid:12)(cid:12) s v n − s v n +1 (cid:12)(cid:12) q = 0 for ea h edge ( v n , v n +1 ) , we dedu eimmediately that | s v − s v ′ | q = 0 for any two verti es v and v ′ in V i . Hen e if s is in the kernel of B i , all spins in { s v n } v n ∈ V i take the same value. So there are q di(cid:27)erentve tors in the kernel of ea h matrix B i , of whi h thereare κ .We are now ready to de(cid:28)ne an non-normalized sta-bilizer state en oding G σ . We obtain it by (cid:28)rst in-terpreting ea h ve tor c = ( c , c , ..., c N ) ∈ C G ( q ) asa produ t state of a multipartite quantum spin sys-tem with spin-dimensionality q a ording to the formula | c i := | c i ⊗ | c i ⊗ ... ⊗ | c N i and by a subsequent sum-mation of all these states [29℄ | ψ G i := q κ X c ∈ C G ( q ) | c i = X s ∈ Z Nq (cid:12)(cid:12) ( B σ ) T s (cid:11) , (7)where the se ond equality and the fa tor q κ follow im-mediately from lemma (2). For an illustrative examplesee Fig. 1.Lemma 3. The state | ψ G i is a stabilizer state. Its sta-bilizer onsists of the q N operators X ( v ) Z ( u ) := O e ∈ E X v e Z u e , (8)where v ∈ C G ( q ) and u ∈ C G ( q ) ⊥ . Proof. From Lemma (1) we derive immediately that, byusing the given onstru tion rule for the operators, weobtain a ommuting set. Considering the equation X ξ i Z ζ i X ξ ′ i Z ζ ′ i = X ξ i + ξ ′ i Z ζ i + ζ ′ i e πiξ ′ i ζ i /q and hen e X ( v ) Z ( u ) X ( v ′ ) Z ( u ′ ) = X ( v + v ′ ) Z ( u + u ′ ) e πiu · v ′ /q , where u · v ′ = P i u i v ′ i = 0 for ea h admissible hoi e ofthese ve tors, we also see that these operators form agroup. Furthermore, these operators a tually stabilizethe (nontrivial) state | ψ G i , sin e for all v ∈ C G ( q ) andfor all u ∈ C G ( q ) ⊥ X ( v ) Z ( u ) | ψ G i = X ( v ) Z ( u ) q κ X c ∈ C G ( q ) | c i = q κ X c ∈ C G ( q ) X ( v ) Z ( u ) | c i = q κ X c ∈ C G ( q ) e πiu · c/q | c + v i = q κ X c ′ ∈ C G ( q ) | c ′ i = | ψ G i . FIG. 1: The basi onstru tion prin iple. This (cid:28)gure shows an example of an en oding of a lassi al intera tion pattern intoa stabilizer state. Thin graph: the lassi al intera tion graph G; thi k graph: the derived graph relating quantum sites in astabilizer state. The lassi al spin sites orrespond to verti es in a graph G. The intera ting pairs of sites are mapped to aquantum site, one for ea h edge ((cid:16)edge qudits(cid:17)). The quantum sites form, by onstru tion, a stabilizer state.From this, we an moreover dedu e that − q N . This part of the proof is given in AppendixA. 1. Thermal quantitiesNow we are able to formulate the entral theorem ofthis se tion.Theorem 4. The partition fun tion Z G ( q, { h e } ) of a lassi al spin system de(cid:28)ned on the graph G = ( V, E ) bythe Hamiltonian fun tion H ( { s i } ) = P e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) an be written as the overlap of a stabilizer stateand a produ t state Z G ( q, { h e } ) = ( O e ∈ E h α e | ) | ψ G i of a quantum me hani al spin-system, where | α e i = q − X j =0 e − βh e ( j ) | j i . (9)Proof. The state | ψ G i is a stabilizer state a ording tolemma (3), and we ompute, with an arbitrarily hosenorientation σ of the graph G , ( O e ∈ E h α e | ) | ψ G i ( ) = X s ∈ Z Nq ( O e ∈ E h α e | ) (cid:12)(cid:12)(cid:12) ( B σ ) T s E ( ) = X s ∈ Z Nq Y e ∈ E e − βh e ( (cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q ) = X s ∈ Z Nq e − βH ( { s i } ) whi h on ludes the proof. Let us give a brief interpretation of the method usedto en ode the partition fun tion. We observe that to al- ulate partition fun tions of systems with Hamiltonianfun tions of the form H ( { s i } ) = X e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) (zero external (cid:28)eld) it is already su(cid:30) ient to know therelative state of spins whose orresponding verti es are onne ted by an edge. A ordingly, we map ea h ve -tor ( s v ) v ∈ V of spin on(cid:28)gurations to the orrespondingone ( B σ ) T s = (cid:16)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:17) e ∈ E of di(cid:27)eren es alongedges using the in iden e matrix B σ . These ve tors areautomati ally onsistent with spin- on(cid:28)gurations, andmoreover, there an be no more of them than we havealready given.As shown, an intera tion pattern is en oded into agraph and this graph is en oded into a stabilizer state.Furthermore, the orresponding intera tion strengths(as well as a temperature) are en oded into a produ tstate. This way we en ode all the information aboutthe partition fun tion of a thermal state into two stateswith omparatively simple stru ture.Example 5. Here we onsider examples of states | α i ,whi h en ode the intera tion strengths of the Hamilto-nian fun tion. For the models we onsider, these areprodu t states | α i = N e ∈ E | α e i , whi h are derived im-mediately from the respe tive Hamiltonian fun tionsgiven in se tion (II).1. For the q -state Potts model the state | α i is de-rived from the Hamiltonian fun tion (5), with thefun tion J given by J Potts (Θ ij ) := − εδ (Θ ij ) . ThisHamiltonian fun tion is hara terized by two-body intera tions, whose strengths are en odedinto states | α e i whi h take the form | α e i = | α i Potts = e βε | i + q − X j =1 | j i .
2. For the q -state lo k model the state | α i is derivedfrom the Hamiltonian fun tion (5), with the fun -tion J given by J lo k (Θ ij ) := − ε cos(Θ ij ) . Theindividual two-body intera tion strengths are thusen oded into states | α e i = | α i lo k = q − X j =0 e βε cos(2 πj/q ) | j i .
3. As a spe ial ase, for q = 2 we obtain, in an anal-ogous fashion, the states | α i and | α e i for the Isingmodel | α e i = | α i Ising = | i + e − βJ | i . In the following part we look at examples of states | ψ G i , whi h en ode the intera tion patterns of asso i-ated Hamiltonian fun tions, thereby investigating spe- ial ases of graphs and their orresponding stabilizerstates.1. Tree Graphs. Here we onsider models whoseintera tion patterns are hara terized by treegraphs, i.e., graphs ontaining no loops. Thestatement that (cid:16) n olumns { c i } i =1 ,...,n of the in i-den e matrix B σ of a graph G are linearly depen-dent(cid:17) means that there is a non-trivial linear om-bination su h that P ni =1 λ i c i = 0 . Hen e there isat least one ve tor that equals the negative sumof the remaining ones, say, c = − λ − P ni =2 λ i c i .Sin e the olumns des ribe the start and endpoints of the edges, this means that the graph ontains a loop. In turn, loop-less graphs (= treegraphs) have an in iden e matrix with N = | E | linearly independent olumns and hen e N lin-early independent rows. This means that the rowsspan the entire spa e Z Nq (= C G ( q )) and hen e | ψ G i = X v ∈ Z Nq | v i ∝ q − X j =0 | j i ⊗ N . (10)In on lusion, we observe that the states derivedfrom tree-graphs are produ t states.2. A y le. Here we onsider models whose inter-a tion patters are y les, i.e. a losed loop. If thegraph is a losed hain, the in iden e matrix looks(besides reordering of the edges) like this B σ = − − · · ·
00 0 − ... . . .
11 0 0 · · · − . We see that a ve tor v that is perpendi ular toall rows has the property v i = v j for all i and j , and hen e C G ( q ) ⊥ = span n (1 , , , , ..., T o .We hen e hoose n Z ⊗ N , X ( n ) (cid:0) X − (cid:1) ( n +1) | n = 1 , ..., N − o as generating set of the stabilizer. We an verifythat the state | ψ G i = P N − j =0 (cid:16) | j x i ⊗ N (cid:17) , where | j x i is an eigenstate of the X -operator, is an eigenstateof the generator of the stabilizer and hen e the sta-bilizer itself. This state is invariant under reorder-ing of the edges and hen e the proof is independentof the hoi e of B σ that was hosen in the begin-ning. Thus, the states derived from graphs thatare losed hains are (generalized) Greenberger-Horne-Zeilinger states (GHZ states.) In parti u-lar, for q = 2 one obtains the state | + i ⊗ N + |−i ⊗ N (where | + i and |−i are the eigenstates of the Paulimatrix σ x ).3. The Kitaev model. The Kitaev model of topo-logi ally prote ted quantum states is de(cid:28)ned asfollows. On ea h edge of a tori latti e with he kerboard stru ture we pla e one qubit, theedge qubit. The tori ode state (a tually a sub-spa e) is the ommon eigenstate of a set of opera-tors that are onstru ted using the neighborhoodrelations of the tori latti e. More pre isely, forea h but one of the smallest possible loops L i (theplaquettes) in the latti e, we de(cid:28)ne one operator B i := Y ( a,b ) ∈ L i Z ( a,b ) . We leave out one be ause it would not be inde-pendent from the others. Similarly, ea h vertex a (there is no qubit in the verti es) has a neighbor-hood N a of adja ent edges, forming a star. On thequbits of ea h but one of these stars we de(cid:28)ne theoperators A a := Y b ∈ N a X ( a,b ) . One has to be left out be ause it is not indepen-dent of the others, as in the ase of the plaque-ttes. All these operators mutually ommute, be- ause in ea h loop a vertex has either zero or twonearest neighbors. Hen e, these operators gener-ate a stabilizer, whose (cid:28)xed point is the tori odestate. We noti e that this stabilizer onsists of N − independent operators de(cid:28)ned on a N -sitequantum system and hen e the stabilized obje tis not a single state but a subspa e of dimension . We remark that the onne tion between the2D Ising model and planar (tori ) ode states was(cid:28)rst proven and utilized in Ref. [13℄.In view of the huge variety of lassi al spin mod-els and their intera tion graphs, we want to pointout that this state an be de(cid:28)ned more abstra tlyand more losely related to the B σ -matrix on-stru tion used in the other examples, as shown inthe following: We assume q = 2 and onsider anarbitrary graph G σ = ( V, E ) with the essentialproperty to ontain N − | V | − independentloops { L i } N − i =1 . The loops now naturally de(cid:28)ne aspe i(cid:28) neighborhood N a of ea h vertex a , namelythe union of the sets of nearest neighbors of a inea h loop N a = S L i { b ∈ V ; ( a, b ) ∈ L i } . Withthe loops and neighborhoods spe i(cid:28)ed, we de(cid:28)neas above the operators A a := Y b ∈ N a X ( a,b ) ; B i := Y ( a,b ) ∈ L i Z ( a,b ) . All of these operators mutually ommute, be ausein ea h loop a vertex has either zero or two near-est neighbors. There are N − independent oper-ators A a and N − independent operators B i ,whi h an be seen as follows. Considering theoperators B i , the statement Q i ∈ I B i = I implies that I ontains dependentloops, whi h is impossible for | I | < N . Similarly, onsidering the operators A a , for ea h set of ver-ti es V ′ the identity Q a ∈ V ′ A a = N a = { ( a, b ) , b ∈ N a } (when onsid-ered together) ontain ea h edge twi e. This isimpossible if | V ′ | < N by onstru tion of the in-tera tion graph. On the other hand Q a ∈ V A a = Q i B i = G is the periodi two-dimensional latti e, where the ommon (cid:28)xed point of the operators A a and B i de(cid:28)nes the tori ode state [18℄, as introdu ed inthe ontext of topologi al quantum omputation.We will use yet another generalization of this on-stru tion pro edure in the subsequent se tion, in orderto a ess orrelation fun tions and partition fun tionsof systems with external magneti (cid:28)elds.B. External (cid:28)elds and orrelation fun tionsThe en oding s heme dis ussed in the last se tion isneither suited to evaluate orrelation fun tions nor par-tition fun tions of systems with external (cid:28)elds, like thosedes ribed by H ( { s i } ) = X e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) + X v ∈ V b v (cid:0) s v (cid:1) . To over ome this limitation we (have to) use a di(cid:27)erenten oding s heme. Instead of the state | ψ G i we will nowuse the state | ϕ G i := X s ∈ Z Nq | s i (cid:12)(cid:12) ( B σ ) T s (cid:11) to en ode the intera tion pattern, where B σ is again thein iden e matrix of the intera tion graph G . Lemma 6. The state | ϕ G i is a stabilizer state. Its sta-bilizer is generated by the N operators K a = X ( a ) Y e : ∃ b ∈ V s.th. ( a,b )= e ∈ E (cid:16) X ( e ) (cid:17) σ K e = Z ( e ) (cid:16) Z ( a ) (cid:17) − σ (cid:16) Z ( b ) (cid:17) σ , for every a ∈ V and for every e = ( a, b ) ∈ E , where σ is either +1 or − , depending on the orientation ofthe edge ( σ := B σe,a = − B σe,b ). In our notation, anupper index in bra kets denotes the qudit a ted on bythe operator.Proof. We have to show that these operators have | ϕ G i as a (cid:28)xed point and, sin e the number of operatorsde(cid:28)ned this way is N = | V | + | E | and hen e equals thenumber of qudits in the quantum system in state | ϕ G i ,we have to show that they are independent. Under this ondition they generate the stabilizer of a single state.To see the stabilizing property of the operators K a , we ompute X ( a ) Y e : ∃ b ∈ V s.th. ( a,b )= e ∈ E (cid:16) X ( e ) (cid:17) σ | s i (cid:12)(cid:12) ( B σ ) T s (cid:11) = | s ′ i (cid:12)(cid:12) ( B σ ) T s ′ (cid:11) with s ′ = (cid:0) s , ..., s a + 1 mod q, ..., s | V | (cid:1) , be ause X ( a ) | s i = | s ′ i and Y b :( a,b )= e ∈ E (cid:16) X ( e ) (cid:17) σ (cid:12)(cid:12) ( B σ ) T s (cid:11) = (cid:12)(cid:12)(cid:12)(cid:16) c a + ( B σ ) T s (cid:17) mod q E = (cid:12)(cid:12)(cid:12) ( B σ ) T s ′ E , where c a is the a th olumn of B σ . Likewise, K e | ϕ G i = | ϕ G i , be ause (cid:16) Z ( a ) (cid:17) − σ (cid:16) Z ( b ) (cid:17) σ | s i = (cid:16) Z ( a ) (cid:17) − B σe,a (cid:16) Z ( b ) (cid:17) − B σe,b | s i = exp (cid:8) − πi (cid:0) B σe,a s a + B σe,b s b (cid:1) /q (cid:9) (cid:12)(cid:12) ( B σ ) T s (cid:11) , and Z ( e ) (cid:12)(cid:12) ( B σ ) T s (cid:11) = exp (cid:8) πi (cid:0) B σe,a s a + B σe,b s b (cid:1) /q (cid:9) (cid:12)(cid:12) ( B σ ) T s (cid:11) , so the phases an el.The set of n operators that were just de(cid:28)ned aremapped, by the isomorphism F nq , to a set of n ve torsthat an be arranged in the following matrix | V | B σ ) T − B σ | E | . s . Hen ethe operators generate the full stabilizer. (cid:3) In the ase q = 2 we re over a true graph state byan appli ation of Hadamard transformation on the edgequbits. For an illustrative example, see Fig. 2. As be-fore, the lassi al spin sites orrespond to verti es in theintera tion graph G of the lassi al model. The inter-a ting pairs of sites are then mapped to a quantum site,one for ea h edge (the edge qudits). What is di(cid:27)erentfrom the original s heme is that the individual lassi- al spin sites (cid:21)a ted on by lo al (cid:28)elds (cid:21) are mapped toquantum sites as well, one for ea h vertex (the vertexqudits). The resulting graph is alled a de orated graph.The resulting many body quantum states are again, by onstru tion, stabilizer states.1. Thermal quantitiesWe now ome to the entral result of this se tion.By means of the state | ϕ G i and appropriately hosenprodu t states, we an ompute the partition fun tionof systems with lo al external (cid:28)elds as well as n-pointfun tions.Theorem 7. The partition fun tion Z G ( { h e , b v } , β ) ofa lassi al spin system at inverse temperature β , de(cid:28)nedon the graph G = ( V, E ) by the Hamiltonian fun tion H ( { s i } ) = P e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) + P v ∈ V b v (cid:0) s v (cid:1) , anbe written as the overlap of a stabilizer state and a prod-u t state Z G ( { h e , b v } , β ) = ( O v ∈ V h α ′ v | O e ∈ E h α e | ) | ϕ G i , where | α e i = q − X j =0 e − βh e ( j ) | j i| α ′ v i = q − X j =0 e − βb v ( j ) | j i . Proof. The state | ϕ G i is a stabilizer state a ording tolemma (6), and we ompute, with an arbitrarily hosenorientation σ of the graph G , ( O v ∈ V h α ′ v | O e ∈ E h α e | ) | ϕ G i = ( O v ∈ V h α ′ v | O e ∈ E h α e | ) X s ∈ Z Nq | s i (cid:12)(cid:12)(cid:12) ( B σ ) T s E = X s ∈ Z Nq Y v ∈ V e − βb v ( s v ) Y e ∈ E e − βh e ( (cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q ) = X s ∈ Z Nq e − βH ( { s i } ) . Likewise, we an write down a theorem for the n-point orrelation fun tions.Theorem 8. The n-point orrelation fun -tions h s i , s i , ..., s i n i β of a lassi al spin sys-tem at inverse temperature β , de(cid:28)ned on thegraph G σ = ( V, E ) by the Hamiltonian fun tion H ( { s i } ) = P e ∈ E h e (cid:0)(cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q (cid:1) + P v ∈ V b v (cid:0) s v (cid:1) , anbe written as an overlap of a stabilizer state and aprodu t state (up to a fa tor of Z , whi h is the partitionfun tion of the lassi al spin system). More pre isely, h s i , s i , ..., s i n i β = Z − ( O v ∈ V h α ′ v ( i , ..., i n ) | O e ∈ E h α e | ) || ϕ G i , where | α e i = q − X j =0 e − βh e ( j ) | j i| α ′ v ( i , ..., i n ) i = q − X j =0 cos (2 πj/q ) m ν e − βb v ( j ) | j i , and m ν is the number of o urren es of ν in the n-tuple ( i , ..., i n ) .Proof. The state | ϕ G i is a stabilizer state a ording tolemma (6), and we ompute ( O v ∈ V h α ′ v ( i , ..., i n ) | O e ∈ E h α e | ) | ϕ G i = ( O v ∈ V h α ′ v ( i , ..., i n ) | O e ∈ E h α e | ) X s ∈ Z Nq | s i (cid:12)(cid:12)(cid:12) ( B σ ) T s E = X s ∈ Z Nq Y v ∈ V cos (2 πs ν /q ) m ν e − βb v ( s v ) Y e ∈ E e − βh e ( (cid:12)(cid:12) s v + e − s v − e (cid:12)(cid:12) q ) = X s ∈ Z Nq cos (Θ i ) cos (Θ i ) ... cos (Θ i n ) e − βH ( { s i } ) . where Θ i = 2 πs i /q . We ompare this to the de(cid:28)nitionof the n-point orrelation fun tion given in se tion II.This on ludes the theorem.V. EXTENDING THE FORMALISMA. The most general frameworkSo far we have used produ t states of single edge-quditsites, namely the states | α i = N e ∈ E | α e i , in the overlapwith the states | ϕ G i and | ψ G i , to al ulate partitionfun tions and orrelation fun tions. Allowing for tensorprodu ts of entangled states, | α i = N ε ⊂ E | α ε i , wherethe ε are subsets of E with few elements, extends theset of possible en odings of lassi al spin systems. Thisis the ontent of this se tion.One short oming of the en oding of the intera tiongraph into the states | ϕ G i and | ψ G i is the inability of the1FIG. 2: The extended onstru tion prin iple. This (cid:28)gure shows an example of the extended en oding of a lassi al intera tionpattern into a stabilizer state. Thin graph on the left: the lassi al intera tion graph G; thi k graph on the right: the derivedgraph relating quantum sites in a stabilizer state. The lassi al spin sites orrespond to verti es in a graph G. The intera tingpairs of sites are mapped to a quantum site, one for ea h edge (edge qudits). The individual lassi al spin sites on whi h thelo al (cid:28)elds a t are then, too, mapped to quantum sites, one for ea h vertex (vertex qudits) (cid:21) this is di(cid:27)erent from the originals heme. The resulting graph is alled a de orated graph. The quantum sites are, by onstru tion, in a stabilizer state.intera tion to distinguish between lassi al spin statesthat have the same relative state | s i − s j | q = (cid:12)(cid:12) s ′ i − s ′ j (cid:12)(cid:12) q but have di(cid:27)erent values s i = s ′ i , s j = s ′ j . This in-ability stems from the fa t that an attempt to en odepairs of neighboring states ( s i , s j ) into one edge qudit ( s i , s j )
7→ | e ij i via the B -matrix formalism does notlead to a stabilizer state and hen e fails, if | e ij i takesmore states than ea h of the sites s i or s j . One way outof this dilemma is to en ode the pairs of neighboringspin sites in the graph of the lassi al model into morethan one qubit, while extending the overlap state | α i to states beyond produ t states. Although these statesare not produ t states anymore, we an still interpretthem as produ t states of omposite parti les, extend-ing over few sites as we restri t ourselves to subsets ε of E with few elements. The entangled states moreoverin lude neighboring sites only, whi h adds to the pi tureof omposite sites (quasi-lo al states).We des ribe this generalization now and investigatethe relationship to the more spe ialized ases. Under ertain assumptions on erning the lassi al Hamilto-nian fun tion, a formal mapping from the most general ase to the more spe ialized ones is possible. Taking thisstep, i.e. performing this formal transformation, givesus a mathemati al pi ture whi h is often mu h moreenlightening than the original one.B. En oding m -body intera tions, ea h siteappearing in maximally n terms of the Hamiltonianfun tion.The most general ase to onsider is the one wherewe • allow ea h lassi al spin to appear in as many as n terms of the Hamiltonian fun tion • allow ea h site to intera t with m − others(Hamiltonian fun tion with m -body terms) • allow all on(cid:28)gurations of the m intera ting spinsin ea h term to be di(cid:27)erentiated energeti ally.Note however, that a simulation of thermal states ofthese systems on a lassi al omputer s ale unfavorablyin m and n , as we will see in se tion VI C.The (cid:28)rst point in the list is addressed in the followingway. Sin e ea h site is allowed to take part in n in-tera tions, we need n instan es of it in the stabilizerstate. Of ourse, all instan es of the lo al quantumsystems have to be in the same state when measured.Hen e we map ea h site e i to an n -body GHZ state: e i P | s i, s i, ...s i,n i . To address the latter two pointsof this list, we onsider the following. To reate a quan-tum state | γ G i that enables us to di(cid:27)erentiate energet-i ally between all possible spin on(cid:28)gurations of an m -body intera tion, we map ea h site e i taking part in theintera tion to a single quantum spin state e i
7→ | s i i e i .The orresponding state | α i , whi h is used for the on-tra tion that yields the partition fun tion and whi h inthe pre eding se tions used to be a produ t state, onse-quently has to be an entangled state in this pi ture. Onthe sites { e i } mi =1 taking part in one m -body intera tion,the state | α i takes the form | α i = X ( s ,s ,...,s m ) e − βh ( s ,s ,...,s m ) | s s ...s m i . Note however, that a simulation of thermal states ofthese systems on a lassi al omputer s ales unfavor-ably in m and n , as we will see in se tion VI C. For2further details of this en oding, let us now have a lookat examples.1. En oding -body intera tions, ea h site appearing inmaximally n terms of the Hamiltonian fun tion: EdgemodelsA spe ial ase of the dis ussed generalization is theone where we (as in the pre eding se tions) sti k toHamiltonian fun tions with -body terms where ea hsite is involved in n intera tions. This kind of Hamilto-nian fun tion plays a role in higher dimensional latti esand spin glasses, for example. For their treatment wepropose, in the following, a way to dis riminate the las-si al spin on(cid:28)gurations beyond resolving relative states(as we did before).To reate a quantum state | γ G i that enables usto di(cid:27)erentiate energeti ally between all possible spin on(cid:28)gurations, we pro eed as follows. We identifytwo qudits with ea h edge e = ( ij ) ∈ E of thegraph G and provide them with a produ t basis n | s i i e i | s j i e j | s i , s j = 0 ...q − o , where e i is one of theedge qudits and e j is the other one. These qudits willbe alled the edge qudits orresponding to the edge. Wemap states of the lassi al spin sites to quantum stateof the whole quantum many body system of edge quditsvia Z Nq ∋ s = ( s , ..., s N ) O e =( ij ) ∈ E | s i i e i | s j i e j . This way, we atta h GHZ-states to the verti es, withthe number of parti les equaling the number of in i-dent edges. A graphi al representation of this en odingis given in Fig. 3 (a). Note that ea h lassi al spin ismapped to as many edge qudits as there are edges at-ta hed to the lassi al spin vertex.Lemma 9. The superposition of the quantum states be-longing to all possible lassi al states | γ G i := X s ∈ Z Nq O e =( ij ) ∈ E | s i i e i | s j i e j is a produ t of GHZ-states and hen e a stabilizer state.Proof. A reordering of the sites groups all edge qubitsbelonging to the same spin site i O e =( ij ) ∈ E | s i i e i | s j i e j = O i ∈ V O e =( ij ) | s i i e i and writing it this way we see that the state | γ G i hasthe stru ture | γ G i = X s ∈ Z Nq O i ∈ V O e =( ij ) | s i i e i reordering O i ∈ V X s i ∈ Z q O e =( ij ) | s i i e i , where P s i ∈ Z q N e =( ij ) | s i i e i is a GHZ state. (cid:3) The overlap to evaluate the partition fun tion or or-relation fun tions has now to be performed with onestate per edge qubit pair h α e | s i s j i . Sin e this overlap-ping state | α i allows us to adapt the energies h ij to ea hindividual spin, the possibility of evaluating partitionfun tions with lo al energy terms as well as orrelationfun tions is immediately given.To avoid the ne essity of en oding the orrelationfun tion dire tly into the states | α e i , we add one morequantum site to the GHZ state. This enables us to mea-sure the state of the lassi al site dire tly. Keep in mindthat this is te hni ally not ne essary, be ause the stateof the site is dire tly a essible already without the ex-tension.Theorem 10. The partition fun tion Z G ( { h e , b v } , β ) of a lassi al spin system at inverse temperature β , de-(cid:28)ned on the graph G = ( V, E ) by the Hamiltonian fun -tion H ( { s i } ) = P ( ij ) ∈ E h ( ij ) (cid:0) s i , s j (cid:1) , an be written asthe overlap of a stabilizer state and a produ t state (overedge qudit pairs) Z G ( { h ij , b v } , β ) = ( O ( ij ) ∈ E (cid:10) α ( ij ) (cid:12)(cid:12) ) | γ G i , where (cid:12)(cid:12) α ( ij ) (cid:11) = q X s i ,s j =1 e − βh ij ( s ei ,s ej ) | s i i e i | s j i e j Proof. The state | γ G i is a produ t of GHZ states andhen e a stabilizer state a ording to lemma 9, and we ompute, with an arbitrarily hosen orientation σ of thegraph G , ( O ( ij ) ∈ E (cid:10) α ( ij ) (cid:12)(cid:12) ) | γ G i = ( O ( ij ) ∈ E (cid:10) α ( ij ) (cid:12)(cid:12) ) X s ∈ Z Nq O e =( ij ) ∈ E | s i i e i h s j | e j = X s ∈ Z Nq Y ( ij ) ∈ E e − βh ij ( s ei ,s ej )= X s ∈ Z Nq e − βH ( { s i } ) .
2. En oding -body intera tions, ea h site appearing inmaximally terms of the Hamiltonian fun tion: VertexmodelsAn important lass of models are the vertex models.These models also (cid:28)t into our framework, as will beshown now. A prominent example of a vertex modelstems from a D regular latti e where ea h lassi al siteintera ts with two groups of three neighboring parti- les (ea h individually) (see Fig. 4). Hen e we have3FIG. 3: Alternative en oding s hemes I: Edge models. (The GHZ s heme.) This (cid:28)gure shows an example of an en oding of a lassi al intera tion pattern into a produ t of GHZ states. The lassi al intera tion graph, a square latti e in this example,is given by the underlying thin grid, the verti es symbolizing lassi al spin sites and the edges symbolizing their intera tions.Ea h edge holds a pair of edge qubits, as indi ated by the dots. The edge-qubits that belong to the same lassi al spin siteare onne ted by thi k lines, indi ating that they form a GHZ state. The ir le with dashed ir umferen e indi ates one pairof qubits | s i i | s j i ontra ted with one state | α ij i in the Hilbert spa e of the pair of edge qubits.the situation where -body intera tions take pla e, ea hsite appearing in maximally terms of the Hamiltonianfun tion. Consequently, we en ode ea h lassi al spinsite into a -body GHZ-state (Bell state), and entanglethe quartets of sites, orresponding to the intera tions,in the state | α i .This setting yields a vertex model in two dimensions,where the proje tions (of the subsets of the GHZ statestaking pla e at ea h vertex of the vertex model) are de-termined by the set of states | α i . Similar models inhigher dimensions an be obtained easily in an analo-gous fashion.C. Relations between the en oding s hemesAn interesting question is how the generalized modelsthat were just des ribed relate to the en oding s hemeen ompassing the states | ϕ G i and | ψ G i . We want to dis- uss this now and furthermore give additional relationsbetween the states | ϕ G i and | ψ G i , adding to what waspresented in the pre eding se tions. 1. Relations between | ϕ G i and | ψ H i There are instan es where for di(cid:27)erent graphs G and H the utspa es of | ϕ G i and | ψ H i are losely related.Two examples will now be demonstrated and give ussome more insight into the internals of the onstru tion.The (cid:28)rst way to look at the onstru tion of the utspa e of | ϕ G i is to modify the graph G by hang-ing the mapping of G to the quantum spin sites. Weremember that (in the ase of the onstru tion of | ψ G i ),the method was to map ea h edge to one quantum spinsite. As an alternative, we derive now from G an newgraph by pla ing on ea h edge one additional vertex.This new graph we all the de orated graph ˜ G , whi hpossesses N = | V | + | E | verti es. The ru ial point isnow to identify the verti es in ˜ G with the qudits thatwe hose as a produ t basis in the de(cid:28)nition of the state | ϕ G i . The original verti es (that appear in G and in ˜ G ) are alled vertex qudits and the qudits that wereadded at the edges are alled edge qudits. The in i-den e matrix of the de orated graph ˜ G is now ( | B σ ) with | ϕ G i = P s ∈ Z Nq | ( | B σ ) T s (cid:11) , where for ea h sum-mand | s i (cid:12)(cid:12) ( B σ ) T s (cid:11) the state | s i is a state of the vertexqudits and (cid:12)(cid:12) ( B σ ) T s (cid:11) is a state of the edge qudits. Wenote that the original method is a restri tion of the just4FIG. 4: Alternative en oding s hemes II: Vertex models. This (cid:28)gure shows an example of an en oding of a lassi al intera tionpattern into a vertex model, where ea h thi k line represents one Bell pair. In this example, ea h lassi al spin site entersin two four-site lassi al intera tions. A ordingly, four edge qubits ( ir le with dashed ir umferen e) form the smallestsubsystem of the Hilbert spa e used for the overlap states | α ε i .proposed mapping of verti es to the edge qudits.A se ond way of mapping the graph G to another onethat an be used to onstru t the utspa e of | ϕ G i is thefollowing. Let us add one vertex to the graph G that is onne ted to all other verti es. Let us all this vertex h and the new graph G + h . The in iden e matrix of G + h is B ( G + h ) T = B σ ( G ) T ... . The ve tor of lassi al spin sites s has to be extendedto in lude the site h , hen e we obtain a new ve tor s ′ =( s , s h ) . The anoni al way to onstru t | ψ G + h i now is | ψ G + h i = X s ′ (cid:12)(cid:12)(cid:12) B ( G + h ) T s ′ E = X s ,s h (cid:12)(cid:12)(cid:12) B σ ( G ) T s E | s + ( s h , s h , ...s h ) i = 2 X s (cid:12)(cid:12)(cid:12) B σ ( G ) T s E | s i , be ause for all values of s h , the equation (cid:12)(cid:12)(cid:12) B σ ( G ) T ( s + ( s h , s h , ...s h )) E = (cid:12)(cid:12)(cid:12) B σ ( G ) T s E holds. Hen e | ψ G + h i = 2 | ϕ G i .A on lusive remark seems appropriate. As has beenshown, the stabilizer of the states are derived from thein iden e matrix of their intera tion graph. In the aseof | ψ G i , the span of the rows of B σ forms the utspa edire tly. Following the arguments in the se tions above,the stabilizer of | ϕ G i is onstru ted analogously, butfrom the span of the rows of the matrix (cid:0) | V | | B σ (cid:1) or B ( G + h ) T instead. Although obviously being related,the di(cid:27)eren e in the onstru tion hanges the quantumstates qualitatively to a great extend. For instan e,when onstru ting | ϕ G i we do not obtain the same state lasses as in the examples (5). Instead of the state (cid:16)P N − j =0 | j x i (cid:17) ⊗ N (in ase of a tree graph), or the state P N − j =0 (cid:16) | j x i ⊗ N (cid:17) (in ase of a y le) or the tori odestate, we always obtain states that are lo ally equivalentto one-dimensional or two-dimensional luster states, re-spe tively.Finally, by means of measurements, we are able to ob-tain the state | ψ G i from the state | ϕ G i . By overlappingthe vertex qudits of | ϕ G i with the state (cid:16)P q − j =0 | j i (cid:17) ⊗| V | we immediately re over | ψ G i . On the one hand, this for-mally has the meaning of proje ting out the dimensionsof the state that are stabilized by operators orrespond-ing to the 1 | V | -part in the matrix (cid:0) | V | | B σ (cid:1) . On theother hand, it has the physi al interpretation of setting5the lo al external (cid:28)elds to zero.2. Going from the general pi ture to | ϕ G i and | ψ G i The states | ϕ G i and | ψ G i en ode two-body intera -tions. Hen e the s heme that is a dire t super-set of the | ϕ G i and | ψ G i en odings is the GHZ s heme. Let thisGHZ state be | GHZ i .Contra tions with states (cid:12)(cid:12) α ( ij ) (cid:11) that do not dis rimi-nate between quantum states | s i i e i | s j i e j with the samevalue of | s i − s j | q yield dire tly the appropriate quan-tum des ription for | ψ G i . To obtain the stabilizer de-s ription for this ase, ea h sub-state of | GHZ i on-sisting of a pair | s i i e i | s j i e j that is measured against atwo-qudit state (cid:12)(cid:12) α ( ij ) (cid:11) is identi(cid:28)ed with a new singlequdit. This qudit in turn orresponds to an edge inthe adja en y matrix of the graph G de(cid:28)ning the state | ψ G i . The verti es G orrespond to the GHZ sub-statesin the state | GHZ i . Hen e all the information aboutthe graph G an be re overed from the graphi al s heme orresponding to | GHZ i . The state | α ′ i that en odesthe intera tion strengths is not di(cid:30) ult to (cid:28)nd either.Sin e (cid:12)(cid:12) α ( ij ) (cid:11) does not dis riminate between quantumstates | s i i e i | s j i e j with the same value of | s i − s j | q , weobtain (cid:12)(cid:12)(cid:12) α ′ ( ij ) E = X s X | s i − s j | q = s e − βh ( s i ,s j ) | s i . Re overing a des ription of | GHZ i in terms of astate | ϕ G i an be performed similarly, provided thatthe Hamiltonian fun tion terms an be written as h ( s i , s j ) = ¯ h ij (cid:16) | s i − s j | q (cid:17) + h i ( s i ) + h j ( s j ) . The GHZ state in the general en oding is a produ tstate of smaller GHZ states | GHZ i = O k | GHZ k i . Ea h of the states | GHZ k i has to be extended by onesite by the mapping | GHZ k i = X s N k O i =1 | s i i X s N k +1 O i =1 | s i i =: | GHZ ′ k i . To obtain the stabilizer des ription for this ase, ea hsub-state of | GHZ i onsisting of a pair | s i i e i | s j i e j thatis ontra ted with a two-qudit state (cid:12)(cid:12) α ( ij ) (cid:11) is identi(cid:28)edwith a new single (edge) qudit of the de orated graph orresponding to | ϕ G i . This edge-qudit in turn orre-sponds to an edge in the adja en y matrix B of thegraph G = ( | B ) de(cid:28)ning the state | ϕ G i . The sub-states that are not measured this way are the ones thatwere added in the mapping above. These will be used to en ode the lo al (cid:28)elds and hen e will be mapped to thevertex qudits of the de orated graph de(cid:28)ning the state | ϕ G i . The part that is more ompli ated here than inthe ase of | ϕ G i is (cid:28)nding the new state | α ′ i en odingthe intera tion strengths. To do so, we have to (cid:28)nd, forea h term of the Hamiltonian fun tion h ( s i , s j ) , a or-responding form h ( s i , s j ) = ¯ h ij (cid:16) | s i − s j | q (cid:17) + h i ( s i ) + h j ( s j ) . The part ¯ h (cid:16) | s i − s j | q (cid:17) will be en oded in thepart of | α ′ i that is measured against the edge-qudits,e.g., (cid:12)(cid:12)(cid:12) α ′ ( ij ) E = X s X | s i − s j | q = s e − β ¯ h ij ( | s i − s j | q ) | s i . The lo al (cid:28)eld orresponding to the vertex qudit withstates | s k i N k +1 , belonging to the extended GHZ sub-state | GHZ k i , is found by a summation of all orre-sponding (cid:28)elds h N k +1 ( s k ) = N k X j =1 h j ( s k ) , where h j ( s j ) are the new terms of the Hamiltonian fun -tion gained from the original terms h ( s i , s j ) , that be-long to measurements on sites on the GHZ sub-state | GHZ k i . VI. APPLICATIONSThis se tion ontains appli ations of the frameworkgiven in the se tions above. The (cid:28)rst appli ation showshow to derive the relation between a lassi al spin modelon a graph and the orresponding model on the dualgraph. The se ond appli ation shows the impli ationsof quantum me hani al symmetries existing in our de-s ription of lassi al systems by means of a quantumsystem.Finally, we investigate the possibility to use simula-tions of the quantum system on a lassi al omputer inorder to obtain the statisti s of quantum measurementresults. This investigation yields some insight into the omplexity of the omputation of the partition fun -tion and orrelation fun tions of the lassi al system.We give a su(cid:30) ient riterion for the stru ture of theintera tion graph of the lassi al model, su h that the omputation of the partition fun tion and orrelationfun tions s ale polynomially with system size.A. Duality relations for planar graphsWe review Ref. [11℄. From graph theory it is knownthat for any planar graph G we an onstru t its dualgraph D . In this se tion we want to demonstrate thatthe partition fun tion Z G of a lassi al spin model de-(cid:28)ned on the graph G and the partition fun tion Z D of6the model derived on the orresponding dual graph D have a simple and meaningful relation.To show this, we note that any orientation σ of agraph G σ indu es an orientation of its dual graph D [16℄,whi h we also denote by σ (we refer to [16℄, page 168for details). Moreover the in iden e matri es B ( D σ ) and B ( G σ ) orresponding to the two graphs have theproperty B ( G σ ) B ( D σ ) T = 0 and the spa es gener-ated by the rows of these matri es are ea h others duals C G ( q ) ⊥ = C D ( q ) . Hen e, the stabilizer of | ψ D i an bewritten as S | ψ D i = { X ( v ) Z ( u ) | v ∈ C D ( q ) , u ∈ C G ( q ) } . The quantum Fourier transform, F := 1 √ q q − X j,k =0 e πikjq | j ih k | , has the property to map X and Z to ea h other un-der onjugation: F XF † = Z and F ZF † = X , and ana ordingly be used to map S | ψ D i to S | ψ G i , one-to-one,sin e F ⊗ N X ( v ) Z ( u ) (cid:0) F ⊗ N (cid:1) † = Z ( v ) X ( u ) . Consider-ing the identity ρ S = 1 q N X g ∈S g for the density matrix ρ S of a stabilizer state that isstabilized by the q N operators in S , we infer that | ψ D i = F ⊗ N | ψ G i . The orresponding partition fun tion Z G an thus berewritten as h ψ G | O e ∈ E | α e i ! = h ψ D | O e ∈ E | α ′ e i ! , where | α ′ e i = F † | α e i . This transformation arriesover to the energy terms in the Hamiltonian fun -tion of the model on the dual graph, where we (cid:28)nd Z G ( q, σ, { h e } ) = Z D ( q, σ, { h ′ e } ) with new energy terms h ′ e , whi h are derived from the old ones by e − βh ′ e ( j ) := 1 √ q q − X k =0 e − πikjq e − βh e ( k ) for every j = 0 , ..., q − .We now want to examine the relation of the the Pottsmodel on a graph G without external (cid:28)eld and its orre-sponding model on the dual graph D . The Potts model, hara terized by the Hamiltonian fun tion H ( { s i } ) = − X e = (cid:10) i,j (cid:11) J e δ ij , is en oded in two quantum states, | ψ G i and N e ∈ E | α e i with | α e i = | α i Potts = e βJ e | i + q − X j =1 | j i . The appli ation of F † on | α e i yields q / e − βh ′ e ( j ) = ( e βJ e + q − if j = 0 e βJ e − if j = 1 , ..., q − . Sin e the energies are again the same for all j = 1 , ..., q − , we have another Potts model (on the dual graph D )whose intera tion strength J ′ e ful(cid:28)lls the relation e βJ ′ e := e βJ e + q − e βJ e − . Equivalently, we write (cid:16) e βJ ′ e − (cid:17) (cid:0) e βJ e − (cid:1) = q, andhen e re over the well known high-low temperature du-ality relation for the Potts model partition fun tion [1℄.B. Lo al symmetriesSee Ref. [11℄. Lo al symmetries of stabilizer states an be used to show that several di(cid:27)erent models of lassi al spin systems a tually have the same partitionfun tions. More pre isely, any lo al unitary U = N e U e operator with eigenstate | ψ G i U | ψ G i = λ | ψ G i (11)generates a model with the same intera tion patternbut modi(cid:28)ed intera tion strengths. Using Eq. (11) weobtain the symmetry relation ( O e ∈ E h α e | ) | ψ G i = ( O e ∈ E h ˜ α e | ) | ψ G i where O e ∈ E | ˜ α e i = λ ∗ U O e ∈ E | α e i . The mapping q − X j =0 e − βh e ( j ) | j i = | α e i 7→ U e | α e i = q − X j =0 e − βh e ( j ) U e | j i implies another mapping of the energies de(cid:28)ning theprefa tors of the basis states | j i . This an lead to un-physi al intera tion strengths, e.g., imaginary ones.Similarly, a relation for the states | ϕ G i an be found,where the lo al symmetry is now orresponding to a hange of intera tion strengths and lo al (cid:28)eld strengths ( O v ∈ V h α ′ v | O e ∈ E h α e | ) | ϕ G i = ( O v ∈ V h ˜ α ′ v | O e ∈ E h ˜ α e | ) | ϕ G i , O v ∈ V | ˜ α ′ v i O e ∈ E | ˜ α e i = λ ∗ U O v ∈ V | α ′ v i O e ∈ E | α e i . The e(cid:27)e t on the orrelation fun tion is again similar,but generi ally di(cid:27)erent orrelation fun tions will, bythe same symmetry transformation, be mapped to the orresponding orrelation fun tions of di(cid:27)erent models.By de(cid:28)nition, the state | α i enabling us to read out thevalue h s i , s i , ..., s i n i β is N v ∈ V | α ′ v i N e ∈ E | α e i with | α e i = q − X j =0 e − βh e ( j ) | j i| α ′ v ( i , ..., i n ) i = q − X j =0 cos (2 πj/q ) m ν e − βb v ( j ) | j i , where m ν is the number of o urren es of ν in the n-tuple ( i , ..., i n ) . Now U ν | α ′ v ( i , ..., i n ) i = U ν q − X j =0 cos (2 πj/q ) m ν e − βb v ( j ) | j i , so in general not only h e ( j ) and b v ( j ) will be al-tered, but the prefa tors cos (2 πj/q ) m ν play the role ofweights. These are spe i(cid:28) for the orrelation fun tionin question and enter the al ulation of the energy terms b ν ( j ) belonging to the symmetry.The fa t that the states | ψ G i and | ϕ G i are stabi-lizer states is advantageous, be ause all elements fromthe stabilizer de(cid:28)ne su h a symmetry operation already,whi h we will use in the following examples.Example 11. We onsider now the hange of a lassi almodel with q = 2 en oded into a state | ψ G i , aused bya symmetry operation. Let the lassi al graph have avertex a with a set of edges E a onne ting to it. One olumn c a of the in iden e matrix orresponds to thevertex a . The stabilizer element X ( c a ) Z (0) applied tothe state | α i (en oding the intera tion strengths) mapsall intera tions strengths J e , e ∈ E a to − J e and doesnot tou h the other ones. We hen e obtain the resultthat Z ( { J e } ) = Z (cid:16)n ˜ J e o(cid:17) , where ˜ J e = ( − J e e ∈ E a J e otherwise . Next, we onsider the hange of a lassi al modelwith q = 2 en oded into a state | ϕ G i , whi h is ausedby a symmetry operation. The matrix generating the utspa e is now C = ( | B ) T . The onstru tion of thelo al unitary symmetry operation using one olumn of C , like in the example above, yields now Z ( { b ν , J e } ) = Z (cid:16)n ˜ b ν , ˜ J e o(cid:17) , where ˜ J e = ( − J e e ∈ E a J e otherwiseand ˜ b a = − b a and ˜ b ν = b ν otherwise.C. Simulations on lassi al omputersAn interesting aspe t of the proposed mapping from lassi al to quantum systems is the established linkbetween two di(cid:27)erent mathemati al formalisms. Asshown, algorithms for the omputation of overlaps ofstabilizer states with produ t states an be used to om-pute partition sums and orrelation fun tions of lassi alspin systems (cid:21) and vi e versa. In both ases, hard and omputationally feasible instan es of these al ulationsare known, and we an now extend e(cid:30) ient algorithmsfrom one domain to the other. This onne tion allowsus to prove the followingTheorem 12. There exists an algorithm that allowsone to ompute the partition fun tion and the orrela-tion fun tions of lassi al spin models de(cid:28)ned on graphsexa tly and with an e(cid:27)ort that s ales polynomially in thenumber of spin sites, provided that the tree-width of thegraph used to de(cid:28)ne the lassi al model s ales logarith-mi ally in the number of spin sites.The proof is rather te hni al and is given in ap-pendix C. Thus, one (cid:28)nds that partition fun tions ongraphs whi h are su(cid:30) iently similar to a tree graph (aproperty made pre ise by the notion of tree-width) anbe e(cid:30) iently evaluated. Similar results have been ob-tained in, e.g., Refs. [14℄.D. Relations to measurement based quantum omputationIn this se tion we dis uss how the mappings between lassi al spin systems and the quantum stabilizer for-malism presented in this work, may provide insights inthe study of measurement-based (or (cid:16)one-way(cid:17)) quan-tum omputation (MQC).The one-way quantum omputer is a model of quan-tum omputation introdu ed in Ref. [10℄. In ontrast tothe quantum ir uit model, where quantum omputa-tions pro eed by unitary evolutions, in MQC any om-putation is realized via single-qubit measurements only.More pre isely, a one-way quantum omputation essen-tially onsists of two main steps: (cid:28)rst, a system of manyqubits is prepared in a highly entangled state, the (cid:16)2D luster state(cid:17) [19℄, whi h is an instan e of a stabilizerstate. Se ond, part (possibly all) of the qubits in thesystem are measured individually. The qubits are mea-sured one after the other in a spe i(cid:28) order, and ea h8qubit is measured in a ertain basis whi h may (and typ-i ally does) depend on the out omes of previous mea-surements. It is this (cid:16)measurement pattern(cid:17) whi h de-termines the quantum algorithm whi h is implemented.It was shown in Refs. [10, 20℄ that the one-way quan-tum omputer is a universal model for quantum om-puter, i.e., it is apable of (e(cid:30) iently) simulating everyquantum omputation performed within the quantum ir uit model. We refer to Ref. [20℄ for more detailsabout MQC.Note that the model of MQC exhibits a remarkablefeature, namely that the entire resour e of a quantum omputation is arried by the entangled luster state inwhi h the system is initially prepared. Indeed, as lo almeasurements an only destroy entanglement, all theentanglement present within a one-way quantum om-putation must be provided by the initial resour e state.Therefore, in order to understand the omputationalpower of quantum omputers, a study of the proper-ties of 2D luster states, and other resour e states, is alled for.Even though it is by now well-established that the2D luster states are universal resour e states for MQC(and several other states have also been found to beuniversal [21, 22℄, it is not yet fully understood whi hproperties of these states are responsible for their uni-versality. This issue has been the topi of re ent in-vestigations [21, 23℄ (see also [4, 13, 24, 25℄), where itwas studied under whi h onditions a given quantumstate may be a universal resour e for MQC, and underwhi h onditions it does not provide any omputationalspeed-up with respe t to lassi al omputation. Whilesigni(cid:28) ant progress has been made in these works, thisimportant problem is far from being fully understood.What an the present onne tions between lassi alspin systems and quantum stabilizer states tea h usabout MQC? To this end, onsider a one-way ompu-tation having one of the stabilizer states | ϕ G i or | ψ G i as a resour e, where G is some graph. One may thenask whi h omputational power an su h resour e statesprovide for MQC (cid:21) i.e., whi h states among the | ϕ G i and | ψ G i are universal resour e states, and whi h states arefully simulatable lassi ally. Next we will see how therelation between these quantum states and the asso i-ated lassi al spin systems, as established in this paper,provides insights in this issue.To do so, onsider Eq. (1), whi h identi(cid:28)es overlapsbetween a resour e state | η G i ( ≡ | ψ G i or | ϕ G i ) and aprodu t state | α i , as the partition fun tion Z G of the as-so iated lassi al spin model on the graph G . Now notethat su h overlaps (to be pre ise, their squared modu-lus) equal the probabilities of out omes of lo al measure-ments performed on the resour e state | η G i . Therefore,if it is possible to ompute su h overlaps (and thus the orresponding measurement probabilities) e(cid:30) iently, itbe omes possible to simulate lo al measurement pro- esses on su h a resour e, on a lassi al omputer. Re-sour es for whi h su h e(cid:30) ient lassi al simulation ispossible, by de(cid:28)nition annot o(cid:27)er any omputational speed-up as ompared to lassi al omputation. Us-ing Eq. (1), we now see that the problem of omputingmeasurement probabilities of lo al measurements boilsdown to the evaluation of the partition fun tion of theasso iated lassi al model. In parti ular, we (cid:28)nd that lassi al models whi h are (cid:16)solvable(cid:17)(cid:22)i.e., their parti-tion fun tion an be e(cid:30) iently evaluated(cid:22)give rise toresour e states for whi h the asso iated probabilities oflo al measurements an be omputed e(cid:30) iently. There-fore, the present mappings establish a relation betweenthe solvability of a lassi al spin systems and the om-putational power of the asso iated resour e state.Let us illustrate these relations with some examplesfor Ising models on di(cid:27)erent latti e types, with or with-out magneti (cid:28)elds (see also Figs. 1 and 2). Considere.g., the simple ase of a 1D Ising model with periodi boundary onditions, without external (cid:28)eld. This modelis known to be solvable: its partition fun tion an beevaluated in a time whi h s ales polynomially with thenumber of spins. Using our orresponden e, the asso i-ated quantum state | ψ G i is a GHZ state (see example5). This state is known to be an e(cid:30) iently lassi allysimulatable resour e state for MQC. A similar on lu-sion an be drawn for the 1D Ising model in the presen eof an external (cid:28)eld, whi h is solvable as well. Using ourmappings, the asso iated quantum state | ϕ G i is a 1D luster state, whi h is indeed also known to be simu-latable (see, e.g., [4℄). Finally, also the 2D Ising modelwithout (cid:28)eld is known to be solvable (cid:21) this is Onsager'sfamous result. The orresponding stabilizer state | ψ G i is the tori ode state. And indeed, this state is a sim-ulatable resour e (cid:21) in fa t, the latter property has beenshown in Ref. [13℄ by using the relation between thisstate and the solvable 2D Ising model.An Ising model whi h is not solvable is the 2D Isingmodel in the presen e of an external (cid:28)eld. In fa t, theevaluation of its partition fun tion is an NP-hard prob-lem. The orresponding stabilizer state is the 2D (de -orated) luster state. Interestingly, this state is a uni-versal resour e for MQC. Therefore, we (cid:28)nd that alsoin this ase the omputational di(cid:30) ulty of a lassi almodel is re(cid:29)e ted in the quantum omputational powerof the asso iated quantum state.VII. SUMMARY AND CONCLUSIONIn this work, we have displayed several mappings fromHamiltonian fun tions of lassi al spin systems to statesof quantum spin systems. We map the intera tion pat-tern given by the Hamiltonian fun tion of the lassi alsystem to quantum stabilizer states and the intera tionstrengths as well as lo al (cid:28)eld strengths to quantumprodu t states. The overlap of these states yields thema ros opi quantities of the thermal states of the las-si al spin system: the partition fun tion and orrelationfun tions at freely sele table temperatures (whi h arealso en oded into the produ t states).The des ribed mappings ir umfere di(cid:27)erent lasses9of admissible Hamiltonian fun tions. From the originaland exemplary approa h [11℄ suited for two-body inter-a tions without lo al (cid:28)elds, we derive a more generalizedmapping apable to yield orrelation fun tions as well asto in lude lo al (cid:28)elds. Finally, we introdu e a version apable to treat arbitrary Hamiltonian fun tions with n -body terms. Ea h of these mappings is interesting inits own right and o(cid:27)ers an individual viewpoint and in-dividual aspe ts in the formal approa h. The relationsbetween the di(cid:27)erent mappings were investigated.We moreover gave several appli ations of the pro-posed mappings, namely: a simple derivation of the du-ality relation of a graph and its dual; a simple derivationof the impa t of lo al symmetries of the stabilizer stateon the lassi al model des ribed by it; a onstru tiveproof of a su(cid:30) ient riterion for the possibility to e(cid:30)- iently evaluate of the thermal quantities of a lassi alspin system on a lassi al omputer; and we dis ussedthe relation of the omputational a essibility of a lassi- al spin system with the power of a quantum omputer.A knowledgmentsThis work was supported by the FWF and the Eu-ropean Union (QICS, OLAQUI, SCALA). MVDN a -knowledges support by the ex ellen e luster MAP.The authors thank G. Ortiz, M.A. Martín-Delgado andG. De las Cuevas for dis ussions.APPENDIX A: A PROOF FOR THE GIVENNUMBER OF STABILIZER ELEMENTSIn se tion IV A we onstru ted the set of opera-tors X ( v ) Z ( u ) [see Eq. (8)℄ with v ∈ C G ( q ) and u ∈ C G ( q ) ⊥ , where by onstru tion C G ( q ) is the Z q -sub-module of Z Nq that is generated by the rows of thein iden e matrix B σ . For this set to be a stabilizer ofthe single state | ψ G i it is ne essary that | ψ G i is a (cid:28)xedpoint of these operators (as already shown in the indi- ated se tion) and that is has ardinality q N . The latterpoint we show now.Lemma 13. The number of independent operators gen-erated by X ( v ) Z ( u ) [see Eq. (8)℄ with v ∈ C G ( q ) and u ∈ C G ( q ) ⊥ is q N .Proof. We note that the module Z Nq and hen e also allits sub-modules are free modules. A ordingly we an hose a basis, from whi h the modules or sub-modulesare generated respe tively. With the s alar produ t h·| · ·i we onstru t an orthonormal basis { c i } and withit the following mapping ϕ : Z Nq → Z Nq , w X c i ∈ C G ( q ) c i h c i | w i . This is a module-homomorphism, sin e for λ, µ ∈ Z q and a, b ∈ Z Nq ϕ ( λa + µb ) = X c i ∈ C G ( q ) c i h c i | λa + µb i = λ X c i ∈ C G ( q ) c i h c i | a i + µ X c i ∈ C G ( q ) c i h c i | b i = λϕ ( a )+ µϕ ( b ) , by the linearity of the s alar produ t. The kernelof ϕ , ker ( ϕ ) , is the set C G ( q ) ⊥ be ause being a(orthonormal) basis { c i } is independent. The rangeof ϕ , ran ( ϕ ) , is the set C G ( q ) , be ause for every w ∈ C G ( q ) we have w = P i λ i c i and ϕ ( w ) = P c i ∈ C G ( q ) P j λ j c i h c i | c j i = w . The homomorphism ϕ ,as any module-homomorphism indu es an isomorphism Z Nq / ker ( ϕ ) ˜ −→ ran ( ϕ ) , whi h provides us with the formula (cid:12)(cid:12) Z Nq (cid:12)(cid:12)(cid:12)(cid:12) C G ( q ) ⊥ (cid:12)(cid:12) = (cid:12)(cid:12) Z Nq (cid:12)(cid:12)(cid:12)(cid:12) ker ( ϕ ) (cid:12)(cid:12) = (cid:12)(cid:12) ran ( ϕ ) (cid:12)(cid:12) = (cid:12)(cid:12) C G ( q ) (cid:12)(cid:12) relating the number of elements in these sets. This im-plies q N = (cid:12)(cid:12) C G ( q ) ⊥ (cid:12)(cid:12)(cid:12)(cid:12) C G ( q ) (cid:12)(cid:12) . (A1)The number on the r.h.s. equals the number of the on-stru ted operators X ( v ) Z ( u ) , whi h are, as a set, iso-morphi to n ( c, s ) | c ∈ C G ( q ) , s ∈ C G ( q ) ⊥ o . This on ludes the proof. (cid:3)
APPENDIX B: TENSOR TREE NETWORKSAND TENSOR TREE STATESWe follow an approa h of Shi, Duan and Vidal and onsider the des ription of states in terms of a tensornetwork with tree stru ture [3, 14℄. We now want togive a short overview of fundamental de(cid:28)nitions andtheorems on erning these tensor tree states (TTS).1. Basi de(cid:28)nitionsThe building blo k of a tensor network are omplex d × d × ... × d n tensors with elements A i i ...i n . Thenumber n is alled the rank of the tensor A and the num-ber d k is alled the rank of the index i k . The maximalnumber d that the indi es an assume, d = max k d k , is alled the dimension of the tensor. A summation overtwo indi es i l and j l ′ of ommon rank of two tensors A [ r ] and A [ s ] , A [ r,s ] i i ... ˆ i l ...i n j j ... ˆ j l ′ ...j n ′ = X k A [ r ] i i ... ( i l = k ) ...i n A [ s ] j j ... ( j l ′ = k ) ...j n ′ , i l and j l ′ . A setof tensors together with pairs of indi es that are to be ontra ted is alled a tensor network. The maximal di-mension D of all tensors, D = max A d [ A ] , is alled thedimension of the network. Tensor networks an be rep-resented by graphs: the ea h vertex of the graph or-responding to one tensor of the network and ea h edge orresponding to one pair of ontra ted indi es. Theindi es to be ontra ted are referred to as internal in-di es and the other ones as open. The notation of graphtheory arry over to the tensor networks, e.g., we talkabout (cid:16)sub ubi (cid:17) tensor trees. A tree graph (network) is alled sub ubi if ea h vertex (tensor) has degree (rank)1 or 3. The verti es with rank 1 are alled leaves.It is possible to write the oe(cid:30) ients A s of a generi pure N -qudit state | ϕ i = P s A s | s i , where {| s i} is aprodu t basis, as a ontra tion of a (cid:28)xed set of ten-sors. Trivially, one tensor of rank N and a dimensionequal to the number of states of the qudits is su(cid:30) ient.In fa t, representations for any graph-stru ture an befound, provided the rank of the internal indi es beingsu(cid:30) iently large. Depending on the internal stru tureof the state to be represented, even representations withinternal indi es of omparatively small rank might befound, hen e redu ing the number of omplex parame-ters representing the network. This displays the prin i-ple that the more stru ture there is in the state, the lessinformation is (potentially) needed to settle the remain-ing degrees of freedom. Conversely, any tensor networkwith N open indi es an be used to de(cid:28)ne a pure N -qubit state.As an illustrative example, the tree depi ted inFig. 5 a) orresponds to the state | τ i = X s X i j A i i i A i s s A i s s A i s s | s i . Another well known example of sub ubi tensor treestates are the matrix produ t states (MPS) with openboundary onditions. They have the simple form | MPS i = X s X i j A i s s | s s i A i i s | s i A i i s | s i× ...A N − i N s N − s N | s N − s N i .
2. E(cid:30) ient s alingA sub ubi TTN with N open indi es (representinga sub ubi TTS of N qudits) and dimension D dependson at most O (cid:0) N D (cid:1) omplex parameters. Thus, a fam-ily of states over N qudits whose TTN-des ription hasa dimension s aling polynomially in N allows for a de-s ription with a number of parameters s aling polyno-mially in N . Con erning the ontra tions of TTS withprodu t states we obtain the following result, (see alsoRefs. [3, 14℄) Lemma 14. Cal ulating the overlap of a omplete prod-u t state of N qudits with a sub ubi TTS of dimension D over N qudits has a omplexity of at most O (cid:0) N D (cid:1) .Proof. Let the produ t state be | α i = O l ∈ leaves | α l i and the TTS be | τ i . The al ulation of h α | τ i is a on-tra tion of a sub ubi tensor network where the leavesare tensors with values h α l | l i . In a sub ubi tree, thereis at least one tensor with at least two leaves atta hed.A ontra tion of this tensor with its atta hed leaves re-quires an e(cid:27)ort of order D . If this tensor has threeleaves atta hed we are done. If not, this tensor will nowbe a leaf tensor atta hed to one other tensor and thetree will still be sub ubi . As before, the tree will nowhave at least one tensor whi h has at least two leavesatta hed. We ontinue this pro edure and be ause thereare N − tensors in the tree, we end up with an e(cid:27)ortof the order N D . (cid:3) Hen e the ontra tion of a family of states over N qudits, whose TTN-des ription has a dimension s alingpolynomially in N , with produ t states of the appropri-ate Hilbert spa es s ales polynomially in N .3. Entanglement in TTSWe want to state one more important result on ern-ing the entanglement ontent of TTS with dimension D . Sin e the rank of the index orresponding to anedge onne ting two tensors is limited by this number D , only D linearly independent ombinations of states orresponding to the sub-trees atta hed to this edge arepossible. Hen e we haveLemma 15. The number of S hmidt oe(cid:30) ients of aTTS with dimension D in a bipartition of the qudits that orresponds to utting exa tly one edge in the graph ofthe orresponding TTN is limited by D . This S hmidtnumber an be rea hed. (cid:3) APPENDIX C: A PROOF OF THEOREM 12We will prove theorem 12 using the ma hinery devel-oped in the se tions IV and V. The underlying idea ofthe proof is to map the lassi al spin problem (of (cid:28)ndingthe partition fun tion) to the orresponding quantumproblem (of (cid:28)nding an overlap), whi h is then solvedby a simulation on a lassi al omputer. To treat thesimulation aspe t, we need some results from the the-ory of tree tensor networks [3, 14℄. We will use the treetensor networks to en ode the stabilizer states whi hare the images of the intera tion patterns of the lassi- al spin systems. The ne essary notation and theoremshave been summarized in the appendix B. With thelanguage developed there, we reformulate theorem 12.1FIG. 5: A sub ubi tensor network. The verti es orrespond to tensors ( ir les) or physi al sites (squares, (cid:16)leaves(cid:17)) respe tively.Edges indi ate ontra tions over ommon indi es. The bipartition of a state orresponding to the ut of a single vertex annothave any S hmidt-rank higher than the rank of the onne ting indi es. The lass of states generated by tensor networks overs all possible pure states provided that the dimension of the network is su(cid:30) iently large.1. Simulation omplexity for the states | ϕ G i and | ψ G i Theorem 16. For the states | ψ G i and | ϕ G i (as de(cid:28)nedabove) a tree tensor network des ription an be om-puted with an e(cid:27)ort growing polynomially in the numberof lassi al spin sites N , provided that the tree-width ofthe graph G grows logarithmi ally in N . This tree ten-sor network des ription allows to ompute the overlaps h α | ψ G i and h α ′ | ϕ G i of these states with produ t stateswith an e(cid:27)ort that grows polynomially in N .Proof. The proof onsists of three parts. i) In apreparatory step, we will summarize the ties betweenthe tree-width of G and the bran h-width of its y lematroid. ii) We will then use this result to derive abound for the S hmidt-rank of a TTN-des ription of thestates | ψ G i and | ϕ G i and hen e derive an upper boundfor the omputational e(cid:27)ort to ompute the overlaps h α | ψ G i and h α ′ | ϕ G i . iii) Finally we give an algorithmto (cid:28)nd the (tensor-) oe(cid:30) ients in the TTN-des ription.Parts ii) and iii) have been given in a similar form for q = 2 already in Ref. [4℄.i) Let us (cid:28)rst (cid:28)x some notation, whi h an be found inmore detail, together with missing proofs, for examplein the referen es [16, 26, 28℄.De(cid:28)nition 17. (matroid) A matroid is a set Ω togetherwith a rank fun tion rk on its subsets. A rank fun tionful(cid:28)lls the following properties • If A and B are subsets of Ω and A ⊂ B , then rk ( A ) ≤ rk ( B ) . • For all subsets A and B of Ω , rk ( A ∩ B ) + rk ( A ∪ B ) ≤ rk ( A ) + rk ( B ) . • If A ⊂ Ω , then rk ( A ) ≤ | A | . A spe ial instan e of a matroid is the set of olumnsof the in iden e matrix B σ of a graph G = ( V, E ) , alledthe y le matroid M ( G ) of the graph G .A natural hoi e of a rank fun tion on a y le matroidis the dimension of the span of the olumn ve tors. Fol-lowing the ideas of lemma 2 we dedu e that with this hoi e of rank fun tion and for a subset of olumn ve -tors A we have the relation rk ( A ) = | V | − c , where c is the number of onne ted omponents in the graph G A = ( V, A ) .De(cid:28)nition 18. ( onne tivity fun tion) With the rankfun tion rk of the y le matroid M ( G ) of the graph G = ( V, E ) we de(cid:28)ne the onne tivity fun tion λ on asubset of edges A ⊂ E by λ ( A ) := rk ( A ) + rk ( E − A ) − rk ( E ) + 1 . It is a symmetri fun tion with respe t to A ↔ E − A .An important observation is that with rk ( A ) = | V | − c ( A ) follows the equality λ ( A ) := | V | + c ( E ) − c ( A ) − c ( E − A ) + 1 , (C1)where c ( E ) , c ( A ) and c ( E − A ) are the numbers ofthe onne ted omponents in the respe tive subsets ofedges.De(cid:28)nition 19. (bran h de omposition) Let T be a sub- ubi tree (see appendix B and Fig. 5) with edges E .The deletion of an edge e ∈ E of the tree orrespondsto a bipartition of the set of leaves of the tree, be ausethe deletion divides the tree into two ( onne ted) om-ponents. The set of bipartitions of leaves indu ed by atree is alled a bran h de omposition of the leaves.In the following we will identify the edges of our (de -orated) intera tion graph G des ribing the lassi al spinsystem with the leaves of a suitable tree. The set of pos-sible trees then orresponds to a set of di(cid:27)erent bran hde ompositions. With the onne tivity fun tion we ande(cid:28)ne the bran h width of a bran h de omposition.De(cid:28)nition 20. (bran h-width) The bran h-width b T ( λ ) asso iated with the bran h de omposition in-du ed by a tree T with edges E T is the value b T ( λ ) := max e ∈ E T λ ( A e ) , where A e ⊂ E ( T ) is the subset of edges belonging toone of the remaining ontiguous sub-trees of tree T ob-tained by deleting edge e ∈ E ( T ) from tree T [note the2symmetry λ ( A e ) = λ ( T − A e ) .℄ The bran h-width of the y le matroid of G is de(cid:28)ned as b ( λ ) := min T b T ( λ ) . With this notation, we formulate the following theo-rems, to be found, together with the proofs, in Ref. [26℄.Lemma 21. (Theorem 3.2 in Ref. [26℄) Let G be agraph with at least one edge, and let M ( G ) be the y lematroid of G . Then the tree-width of G equals the tree-width of M ( G ) . (cid:3) andLemma 22. (Theorem 4.2 in Ref. [26℄) Let M be amatroid of tree-width t and bran h-width b . Then b − ≤ t ≤ max (2 b − , . (cid:3) In parti ular this result tells us that the tree-widthis an upper bound to the bran h-width. The next the-orem, to be found in Ref. [27℄, now states that we analgorithmi ally ompute a sub ubi tree that at least omes lose to the optimal tree.Lemma 23. (Theorem 2.12 in Ref. [27℄) For given k , there is an algorithm as follows. It takes as inputa (cid:28)nite set E G with | E G | ≥ [and the onne tivityfun tion λ ℄. It either on ludes that b ( λ ) > k or out-puts a tree with b T ( λ ) ≤ k + 1 . Its running time is O (cid:16) δ | E G | log | E G | (cid:17) , where δ is the time to ompute λ . (cid:3) Be ause e(cid:30) ient algorithms to ompute the tree-width of a graph G (and hen e with the lemma above,upper bounds for the bran h-width) are known, we anassume to be able to input a k > b ( λ ) . This way we al-ways end up with a tree T with at most b T ( λ ) = 3 k + 1 .At the end of this part we know, given an intera tiongraph G with tree-width t , that we an e(cid:30) iently om-pute a bran h de omposition over the set of edges su hthat the bran h-width asso iated with this de omposi-tion is smaller than or equal to t .ii) We now want to establish a link between the χ − width asso iated with bipartitions of the states | ψ G i and | ϕ G i and the bran h-width of the a tree indu ingthese bipartitions. To (cid:28)x some notation, we de(cid:28)ne thematrix M := ( B σ for | ψ G i (cid:0) | V | | B σ (cid:1) for | ϕ G i . (C2)We re ognize that M is used to de(cid:28)ne the stabilizer ofthe respe tive states, be ause C G = n M T s , s ∈ Z | V | q o . Lemma 24. Let P ∪ Q = E with P ∩ Q = Ø be a biparti-tion of the edges in the intera tion graph G = ( V, E ) de-s ribing the intera tion pattern of the lassi al spin sys-tem. Let | η G i denote the quantum state whose stabilizeris onstru ted via M (e.g. | ϕ G i or | ψ G i ). The S hmidtrank χ of the bipartition | η G i = P χi =1 λ i (cid:12)(cid:12) η Pi (cid:11) (cid:12)(cid:12)(cid:12) η Qi E ,where (cid:12)(cid:12) η Pi (cid:11) and (cid:12)(cid:12)(cid:12) η Qi E are quantum states of the qudits orresponding to the edges in P and Q respe tively, sat-is(cid:28)es the equality χ = q | V | + c ( E ) − c ( P ) − c ( Q ) , where c ( E ) , c ( P ) , c ( Q ) are the number of onne ted omponents in the graphs ( V, E ) , ( V, P ) and ( V, Q ) re-spe tively.Proof. Corresponding to the bipartition P ∪ Q = E we have a bipartition of the olumns of the matrix M . After performing some (unimportant) permuta-tion of the olumns, the matrix M takes the form M = ( M P | M Q ) . Let c denote the number of olumns of M and p and q the number of olumns of M P and M Q respe tively.Now let S be the stabilizer of the state | η G i and S P ⊂S the subset of operators g that a t trivially on thequdits belonging to the labels in Q . We de(cid:28)ne ¯ S P := { Tr Q [ g ] , g ∈ S P } . From the theory of stabilizer statesit is known that | η G ih η G | = q − c P g ∈S g and hen e ρ P = Tr Q [ | η G ih η G | ] = Tr Q q − c X g ∈S g = q q − c X g ∈ ¯ S P g = q − p X g ∈ ¯ S P g. The fa tor q q omes in be ause the tra e over all oper-ators but 1 in the Pauli group is zero and Tr Q [ ] = q q .Furthermore, the stabilizer is a group, so we have theidentity ( ρ P ) = q − p X g ∈ ¯ S P g X h ∈ ¯ S P h = q − p X g ∈ ¯ S P X h ∈ ¯ S P h = (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) q p X h ∈ ¯ S P h = (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) q p ρ P . We de(cid:28)ne r := q p / (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) and obtain ( rρ P ) = rρ P .Hen e rρ P is a proje tor and has (after a possiblyne essary hange of basis and reordering of rows) theform rρ P = diag (1 , ..., , , ..., , or equivalently, ρ P = diag (cid:0) r − , ..., r − , , ..., (cid:1) . Sin e Tr [ ρ P ] = 1 , we have r − rank ( ρ P ) = 1 and hen e r equals the number ofS hmidt oe(cid:30) ients in the bipartition of the state | η G i a ording to the sets of edges P and Q . Thus χ = r = q p / (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) .3To obtain the number (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) , we have now a look atthe matrix M = ( M P | M Q ) , whi h we will from now oninterpret as a linear mapping M T : Z | V | q → Z c q . Here, M P is a | V | × p -matrix belonging to the olumns in P and M Q is a | V | × q -matrix belonging to the olumns in Q . Re all that the stabilizer is isomorphi to the set ofoperators X ( v ) Z ( u ) := O c ∈ olumns of M X v c Z u c , where v ∈ C G ( q ) and u ∈ C G ( q ) ⊥ . Hen e (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) is de-termined by the number of ve tors v ′ ⊂ C G ( q ) and u ′ ∈ C ⊥ G ( q ) whose elements are in the last q pla es(e.g. v ′ = (cid:0) v ′ , ..., v ′ p , , ..., (cid:1) ). Let this number for theset C G ( q ) be z C = | C P ( q ) | , where (cid:0) v ′ , ..., v ′ p (cid:1) ∈ C P ( q ) ,and the orresponding number for the set C G ( q ) ⊥ be z C ⊥ = (cid:12)(cid:12) C ⊥ P ( q ) (cid:12)(cid:12) , where (cid:0) u ′ , ..., u ′ p (cid:1) ∈ C ⊥ P ( q ) . Then (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) = z C z C ⊥ .Let us now al ulate z C . The elements of C G are theimage ve tors of M T . Furthermore, if s ∈ ker (cid:0) M TQ (cid:1) ,then the image of s has the desired form M T s = v ′ = (cid:0) v ′ , ..., v ′ p , , ..., (cid:1) . Considering that we an add anyve tor from the kernel of M T to s without hanging theimage v , it is z C = (cid:12)(cid:12) ker (cid:0) M TQ (cid:1)(cid:12)(cid:12) / (cid:12)(cid:12) ker (cid:0) M T (cid:1)(cid:12)(cid:12) .Similarly, z C ⊥ equals the number of elements in theset C P ( q ) ⊥ where C P ( q ) = ran (cid:0) M TP (cid:1) . As shown aspart of Appendix A, this number is equal to z C ⊥ = q p / (cid:12)(cid:12) ran (cid:0) M TP (cid:1)(cid:12)(cid:12) as the target spa e of the mapping M TP is Z p q .Another basi onsideration about the linear mapping M TP : Z | V | q → ran (cid:0) M TP (cid:1) (note: a mapping between (cid:28)nitespa es) tells us that q | V | = (cid:12)(cid:12) ran (cid:0) M TP (cid:1)(cid:12)(cid:12) (cid:12)(cid:12) ker (cid:0) M TP (cid:1)(cid:12)(cid:12) , hen e z C z C ⊥ = q p q | V | (cid:12)(cid:12) ker (cid:0) M TP (cid:1)(cid:12)(cid:12) (cid:12)(cid:12) ker (cid:0) M TQ (cid:1)(cid:12)(cid:12) | ker ( M T ) | and χ = q | V | (cid:12)(cid:12) ker (cid:0) M T (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ker (cid:0) M TP (cid:1)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ker (cid:16) M TQ (cid:17)(cid:12)(cid:12)(cid:12) . From lemma 2 we now derive that (cid:12)(cid:12) ker (cid:0) M T (cid:1)(cid:12)(cid:12) = q c ( E ) (with analogous results for M TP and M TQ ). (cid:3) Remark 25. As a side remark we note the identities r = q p / | C P ( q ) | (cid:12)(cid:12) C ⊥ P ( q ) (cid:12)(cid:12) (C3)and (cid:12)(cid:12) ran (cid:0) M TP (cid:1)(cid:12)(cid:12) = | C P ( q ) | = q | V | / (cid:12)(cid:12) ker (cid:0) M TP (cid:1)(cid:12)(cid:12) = q | V |− c ( P ) , whi h an be obtained from the proof above. Corollary 26. Considering identity (C1), we dedu ethat the S hmidt-rank χ of a bipartition of edge-qudits E = A ∪ ( E − A ) and the onne tivity fun tion λ de(cid:28)nedon the graph G satisfy the following equation χ = q λ ( A ) − . Considering that the matrix M de(cid:28)ned in Eq. (C2) isthe y le matroid linked to the states | ψ G i and | ϕ G i we an now state the following important result, on ludingthe se ond part of the proof.Corollary 27. Using the result of lemma 23 to (cid:28)nd,by means of the matrix M , a bran h de omposition ofthe qudits in the states | ψ G i and | ϕ G i , we an e(cid:30) iently(cid:28)nd a sub ubi TTN des ription su h that the S hmidtnumber of all bipartitions following this bran h de om-position satis(cid:28)es χ ≤ q t − . A ording to lemma 15, the dimension D of this TTSis limited by t − and hen e, following lemma 14, thee(cid:27)ort to ompute the overlaps h α | ψ G i and h α ′ | ϕ G i growswith at most O (cid:0) | E G | t (cid:1) . (cid:3) iii) In this part we want to dis uss how to omputethe tensor entries in the TTS des ription of the states | ψ G i and | ϕ G i , whi h we will again denote generi allyas | η G i where no distin tion is ne essary. The ansatzfor the al ulation of all tensor elements is the bran hde omposition of the edge qudits ( on erning the edgesof the graph G = ( V, E G ) ) indu ed by the tree tensornetwork T with edges E T des ribing the state. We sele tan arbitrary edge e ∈ E T of the tree to obtain an initialbipartition E G = P ∪ Q with Q = ( E G − P ) of the edgesin E G , indu ing a bipartition of the set of qudits of thestate | η G i . We will use the notation P and Q for theedges and the orresponding qudits alike.Let us onsider the S hmidt de omposition belong-ing to the bipartition. Re alling the proof of lemma24, the S hmidt oe(cid:30) ients of a de omposition | η G i = P i λ i | p i i | q i i , where the states | p i i live on the Hilbertspa e of the edge qudits in a part P ⊂ E G and thestates | q i i live on the part Q = E G − P ⊂ E G an beobtained immediately. They are all equal and have thevalue λ i = r − = (cid:12)(cid:12) ¯ S P (cid:12)(cid:12) /q p . We remember also thatthere are exa tly r of these oe(cid:30) ients. Con erning theS hmidt ve tors, we onsider the following lemma.Lemma 28. A S hmidt basis for a bipartition of theedge qudits E G = P ∪ Q , Q = E G − P of the state | η G i is given by the set of states {| p i i | q i i} ri =1 where | p i i := q ( p −| V | ) / X c P ∈ C P | c P + ˜ p i i| q i i := q ( q −| V | ) / X c Q ∈ C Q | c Q + ˜ q i i . Here ˜ p i ∈ (cid:0) C ⊥ P (cid:1) ⊥ , su h that the osets ˜ p i + C P are dis-tin t for di(cid:27)erent values of i , and C P is the ut spa e of4the subspa e belonging to the edges belonging to the edgequdits in P . ( { ˜ q i } ⊂ (cid:0) C ⊥ Q (cid:1) ⊥ is de(cid:28)ned analogously; alladditions in the kets are modulo q ).Proof. We look at the states | p i i (cid:28)rst; the states | q i i are treated analogously. The set of states {| p i i} has to be an orthonormal set whi h at the same timeis a set of eigenstates of the redu ed density operator ρ P = Tr Q [ | η G ih η G | ] . We de(cid:28)ne ¯ S P := { Tr Q [ g ] , g ∈ S P } and re all from the proof of lemma 24 that | η G ih η G | = q − c P g ∈S g and hen e ρ P = q − p P g ∈ ¯ S P g . Now ea h g ∈ ¯ S P an be written as g = X ( v ) Z ( u ) where v ∈ C P and u ∈ C ⊥ P . Applying su h an operator to | p i i yields q − ( p −| V | ) / g | p i i = X ( v ) Z ( u ) X c P ∈ C P | c P + ˜ p i i = X c P ∈ C P | c P + ˜ p i + v i e πiu · ˜ p i /q = X c ′ p ∈ C P | c ′ P + ˜ p i i , sin e ˜ p i ∈ (cid:0) c ⊥ P (cid:1) ⊥ and c P is a group. The perpendi -ularity property of the states | p i i stems from the fa tthat the ve tors ˜ p i are from distin t osets for di(cid:27)erentvalues of i . We furthermore al ulate h p i | p i i = q ( p −| V | ) X c P ,c ′ P ∈ C P δ c P ,c ′ P = 1 following from remark 25. The number of S hmidt ve -tors is indeed r , be ause the number of distin t osets is,with a slight generalization of the results of AppendixA, espe ially Eq. (A1), to C ⊥ P and (cid:0) C ⊥ P (cid:1) ⊥ , (cid:12)(cid:12)(cid:12)(cid:0) C ⊥ P (cid:1) ⊥ /C P (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) C ⊥ P (cid:1) ⊥ (cid:12)(cid:12)(cid:12) / | C P | = Eq. (A1) q | P | (cid:12)(cid:12) C ⊥ P (cid:12)(cid:12) | C P | = remark 25 r. Having proven that the individual states | ˜ p i i and | ˜ q i i have the given form, we note that the pairing (˜ p i , ˜ q i ) forea h i is not arbitrary and has to be found out. In thefollowing we give an algorithm to (cid:28)nd these pairs. Weassume in this ontext that joining the edges of P and Q results in a mere on atenation of the orrespondingve tors to simplify the notation. This an always bea hieved by a reordering of the edges. The algorithmthat we use is as follows1. Find the set { ˜ p i } , an orthonormal basis { ˜ c Q } ofthe spa e C Q and a ve tor ( c P | ∈ C , where c P ∈ C P .2. For ea h ˜ p i (cid:28)nd one ve tor ( c P + ˜ p i | a i ) ∈ C wherethe hoi e of a i is in prin iple arbitrary and justlimited by the set of ve tors in C . Keep the ve tors a i .3. For ea h ve tor a i al ulate the orresponding ve -tor ˜ q i := a i − P ˜ c Q ˜ c Q (˜ c Q · a i ) . By onstru tion, the ve tors ˜ q i are all elements of (cid:0) C ⊥ Q (cid:1) ⊥ . Furthermore we noti e that there are e(cid:30) ientalgorithms for all these steps. (cid:3) This bipartition enables us to ompute all tensor en-tries e(cid:30) iently.Consider that using the TTN des ription of the state | η G i the states | p i i and | q i i an be written as | p i i = X jk A [ P ] ijk | j i P | k i P , | p i i = X jk A [ Q ] ilm | l i Q | m i Q with suitable tensors A [ P ] and A [ Q ] and states | j i P , | k i P . The states | j i P are living on the Hilbertspa e P belonging to the leaves (and hen e to the orresponding qudits) that are part of the sub-treeof T atta hed to the tensor A [ P ] by its index i .Also the states | j i P an be written as | j i P = P rs A [ P ] jrs | r i P | s i P (analogous arguments apply tothe states | k i P , | l i Q , | m i Q .) To be able to omputethe entries of the tensors we hen e need the states be-longing to sub-trees whi h an be derived from the ini-tial S hmidt de omposition.Lemma 29. Let | i i E be a state on the qudits or-responding to a set of edges E , de(cid:28)ned as | i i E = P c E ∈ C E | c E + d ( i ) i , where C E is the ut spa e of thein iden e matrix of the graph G = ( V, E ) belonging tothe qudits as de(cid:28)ned above. Let E = P ∪ Q , Q = E − P be a bipartition of the qudits. Then | i i E = | P i i | Q i i , where | P i i = X c P ∈ C P | c P + d ( i ) P i| Q i i = X c Q ∈ C Q (cid:12)(cid:12)(cid:12) c Q + d ( i ) Q E . The states | P i i and | Q i i live on the Hilbert spa es of P and Q respe tively and the ve tors d ( i ) P and d ( i ) Q arethe parts of the ve tor d ( i ) belonging to the respe tivequdits.Proof. A reordering of the position of the qudits in | i i E , so that the merging of the ve tors c P , c Q , d ( i ) P and d ( i ) Q be omes a on atenation yields | i i E = X c P ∈ C P ,c Q ∈ C Q (cid:12)(cid:12)(cid:12) ( c P | c Q ) + (cid:16) d ( i ) P | d ( i ) Q (cid:17)E = X c P ∈ C P ,c Q ∈ C Q | ( c P | c Q ) + d ( i ) i . The sets of ve tors { c E } , { c P } and { c Q } are the im-ages of the matri es M TE , M TP and M TQ of identity C2respe tively, where the index denotes the edges that the olumns orrespond to. Sin e M E = ( M P | M Q ) , we ob-tain the result that for ea h c E there is exa tly one pair ( c P , c Q ) immediately. (cid:3) C P , C Q and theve tors d ( i ) P , d ( i ) Q an be found e(cid:30) iently. Now wewrite | i i E = P ijk A ijk | P j i | Q k i and dedu e that A ijk = δ ij δ ik , ex ept for A [0] and A [1] whi h have to absorb the squareroot of the S hmidt oe(cid:30) ients also and hen e A [0 , ijk = r − / δ ij δ ik .This on ludes the proof of theorem 16. (cid:3)
2. The bipartite entanglement of the generalen oding s hemes (e.g. GHZ-produ t state and thevertex model state)So far we have only onsidered the omputational omplexity using an en oding into the states | ψ G i and | ϕ G i . In this se tion we want to extend the e(cid:30) ien ystatement to the alternative en oding s hemes dis ussedin se tion V.The major modi(cid:28) ation leading to these s hemes and ompli ating the entanglement aspe t is the extensionof measurements from one qudit to two or more. In abran h de omposition, the sites being involved in thesemeasurements have to be pla ed in their own sub-trees,whi h we will refer to as ontra tion sites. The on-tra tion of the highly entangled states | α e i with these ontra tion sites will in general not be e(cid:30) ient, but sin ethe size of the ontra tion sites is limited, this only leadsto a onstant omputational overhead. In a bran h de- omposition of a state of the extended en oding s hemeswe an represent the ontra tion sites as leaves. The remaining question is (cid:16)What is the entanglementof bi-partitions in a bran h de omposition where the ontra tion sites are leaves(cid:17)? By onstru tion, we im-mediately (cid:28)nd that this question an be answered bylooking at the number of states (in our s hemes thoseare either q-dimensional Bell pairs or GHZ states) thatare shared by di(cid:27)erent ontra tion sites and ut by thebran h de omposition. See also Figs. 3 and 4.On e we ontra t the ontra tion sites in the graph-i al representation of the general pi ture (like given inFig. 3) to single verti es, we obtain a new graph wherethe edges represent Bell pairs shared by ontra tionsites. Graph theory immediately tells us that also in this ase the tree width is the de isive quantity of the ( on-tra ted) graph that governs the minimum number ofstates (and hen e ebits) that have to be ut in a bran hde omposition. The tree width of the ontra ted graphis arried over from the underlying graph of the lassi alintera tion graph. Thus we an on lude that theorem16 applies for the alternative en oding s hemes as well,and the de isive parameters an be derived immediatelyfrom the respe tive en oding patterns.We also emphasize that non-planar graphs of loga-rithmi ally bounded tree-width, as well as non-lo al in-tera tions are overed by this result. Results regardinge(cid:30) ient omputation of homogeneous Potts model par-tition fun tions on graphs of logarithmi ally boundedtree-width have been obtained before, though with en-tirely di(cid:27)erent methods. We emphasize that our ap-proa h, in ontrast to previous approa hes, an handlewithout di(cid:30) ulty also inhomogeneous models. 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