On thermal stability of topological qubit in Kitaev's 4D model

On thermal stability of topologi al qubit in Kitaev's 4D model R. Ali ki1,2 , M. Horode ki1,2, P. Horode ki1,3 , and R. Horode ki1,2 1 2 arXiv:0811.0033v1 [quant-ph] 2 Nov 2008 3 National Quantum Information Centre of Gda«sk, Poland Institute of Theoreti al Physi s and Astrophysi s, University of Gda«sk, Poland Fa ulty of Applied Physi s and Mathemati s, Gda«sk University of Te hnology, Poland We analyse stability of the four-dimensional Kitaev model - a andidate for s alable quantum memory - in nite temperature within the weak oupling Markovian limit. It is shown that, below a riti al temperature, ertain topologi al qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their onstru tion involves polynomial in system's size algorithm whi h uses as an input the results of measurements performed on all individual spins. We also dis uss the drawba ks of su h andidate for quantum memory and mention the impli ations of the stability of qubit for statisti al me hani s. I. INTRODUCTION While quantum omputation oers algorithms whi h an outperform the lassi al ones, they are very fragile with respe t to external disturban e. Therefore, along with the dis overies of fast algorithms, the question of how to prote t quantum omputation against de oheren e was the subje t of extensive studies. As a result the whole domain was reated known as fault tolerant quantum omputation [1℄. The famous threshold theorems [2, 3℄, saying that arbitrary long quantum omputation is possible provided the error per gate is below ertain threshold has given the hope, that it is possible in priniple to over ome the de oheren e. However the initial theorems are based on phenomenologi al model of noise, and the problem, has not been solved within Hamiltonian dynami s [47℄. Even the problem of whether one an store qubits is open. There is, though a lass of andidates for quantum memories, whi h are in between realisti des ription and the phenomenologi al one: the Kitaev models of topologi al quantum memory [810℄. There is a heuristi reasoning, a ording to whi h su h memories are instable in two dimensions [8, 11℄, and stable in four dimensions (similarly like Ising model represent a stable lassi al memory in 2D, but not in 1D) [8℄. Behaviour of of Kitaev models in nite temperature was then investigated (see e.g. [1216℄). Quite re ently the thermal instability of 2D model has been rigorously proved in [17℄. In the present paper, we deal with the 4D Kitaev's model of Ref. [8℄ and prove rigorously, within Markovian weak oupling approximation, that the model provides thermally stable qubit. To this end we use the formalism of quantum semigroup theory [18℄, whi h has been su essfully applied to analysis of Kitaev 2D model in Ref. [17℄. As a byprodu t we obtain a very useful general upper bound for de ay rate. We perform our analysis in parallel for 3D and 4D ase. Indeed, though in 3D ase only one of the qubit observables is stable, as argued in [8℄, it is mu h more transparent and the reasoning is the same as in 4D ase. Sin e the very stability of qubit is not su ient for a good quantum memory, we also dis uss the open problems on erning existen e of self- orre ting quantum memory. Impli ations for des ription of thermodynami al limit are also dis ussed. The paper is organized as follows. In se tion II we provide some basi notions and results on erning Markovian weak oupling limit. We show, in parti ular, how the rate of de ay expressed in terms of noise generator is related to delity riterion. Finally we provide a general upper bound for de ay rate. In se tion III we show that analysis of noisy evolution of some parti ular topologi al observables is redu ed to the study of a lassi al model. Next (se . IV) we provide onditions for stability of these observables in terms of one-step auto orrelation fun tions. In se . V we nally prove the stability of the observables. In se . VI we provide polynomial algorithm to measure the observables. Finally (se . VII) we dis uss remaining open problems for existen e of self- orre ting quantum memory, as well as importan e of the result for des ription of systems in thermodynami al limit. II. MARKOVIAN APPROXIMATION IN WEAK COUPLING LIMIT Let us rst we briey sket h the general setup and properties of Davies generators. A quantum system with dis rete energy spe trum is oupled to a olle tion of heat baths leading to the global Hamiltonian X H = H sys +H bath +H int with H int = Sα ⊗fα , α (1) where the Sα are system operators and the fα bath operators. The main ingredients are the Fourier transforms ĥα of the auto orrelation fun tions of the fα . The fun tion ĥα des ribes the rate at whi h the oupling is able to transfer an energy ω from the bath to the system. Often a minimal oupling to the bath is hosen, minimal in the sense that the intera tion part of the Hamiltonian is as simple as possible but still addresses all energy levels of the system Hamiltonian in order to produ e nally an ergodi redu ed dynami s. The ne essary and su ient ondition for ergodi ity is [19, 20℄  ′ (2) Sα , H sys = C 1, 2 i.e. no system operator apart from the multiples of the identity ommutes with all the Sα and H sys . We begin by introdu ing the Fourier de ompositions of the Sα 's as they evolve in time under the system evolution dened by the equilibrium state. The spa e of observables equipped with the s alar produ t (3) is alled the Liouville spa e and the generator of the redu ed dynami s is a normal matrix on that spa e, i.e. the Hermitian and skew-Hermitian parts of the generator ommute. eitH sys Sα e−itH sys = X Sα (ω) e−iωt . ω Here the ω are the Bohr frequen ies of the system Hamiltonian. From self-adjointness we have the relation (4) Sα (−ω) = Sα (ω)† . The weak oupling limit pro edure then returns the following equation for the evolution of the spin system in Heisenberg pi ture dX = i[H sys , X] + Ldis (X) =: L(X) dt  1 XX Ldis (X) = ĥα (ω) Sα† (ω) [X, Sα (ω)]+ 2 α ω  + [Sα† (ω), X] Sα (ω) (5) (6) (7) For thermal baths one has moreover the relation whi h is a onsequen e of the KMS ondition [18℄. The operator L generates a semigroup of ompletely positive identity preserving transformations of the spin system. It des ribes the redu ed dynami s in the Markovian approximation and enjoys the following properties • The anoni al Gibbs state with density matrix sys e−βH   Tr e−βH sys (9) is a stationary state for the semigroup, i.e.  
Tr ρβ etL (X) = Tr ρβ X . (10) • The semigroup is relaxing, i.e. for any initial state ρ of the system  
lim Tr ρ etL (X) = Tr ρβ X . (11) t→∞ • Furthermore, the generator satises the detailed balan e ondition, often alled reversibility. Writing δ(X) := [H sys , X], [δ, Ldis ] = 0 (12) and   
†  Tr ρβ Y † Ldis (X) = Tr ρβ Ldis (Y ) X . (13) The last equation expresses the self-adjointness of the generator with respe t to the s alar produ t (14) Finally it is known that −L is a positive operator, hen e it has nonnegative eigenvalues. Moreover L(I) = 0, and for ergodi systems eigenvalue 0 is nondegenerate. A. Auto orrelation fun tions, de ay rate and delity Suppose that for observable X satisfying hX, Xiβ = 1 hX, Iiβ = 0 we have −hX, L(X)iβ ≤ ǫ. (15) Then the auto orrelation fun tion of the observable satises hX, eLt Xiβ ≥ e−ǫt (8) ĥα (−ω) = e−βω ĥα (ω) ρβ = hX, Y iβ := Trρβ X † Y (16) One proves it easily, by de omposing X into normalized eigenve tors of L, and using onvexity of fun tion e−x . Thus to show that an observable X is stable, it is enough to estimate −hX, L(X)i, whi h an be therefore alled de ay rate for the observable X . If this quantity de reases exponentially with size of the system, we obtain stability. Let us now rephrase it in the language of delity. Namely, suppose we have observables X and Z satisfy ommutation rules of Pauli algebra. They generate subalgebra whi h denes a virtual qubit, the one to be prote ted. Let the indu ed tensor produ t on the total Hilbert spa e be H = HQ ⊗ Hanc . (17) Now, we x some state ρanc on the system Hanc . For any state ψ of qubit the initial state of the total system is ρQ,anc (0) = |ψihψ| ⊗ ρanc . Then the system evolves into state ρQ,anc (t), and nally, the an illa is tra ed out. Thus the delity is given by F (ψ) = hψ|ρout Q (t)|ψi (18) ρout Q (t) = Tranc (ρQ,anc (t)). (19) where Let us denote the delity averaged uniformly over the states of qubit by F . Proposition 1 With the above notation, suppose now that the Gibbs state is of the form ρβ = 1 IQ ⊗ ρanc , 2 (20) 3 where ρanc III. is a state on an illa. We then have 1 (21) (hX, eLt Xiβ + hZ, eLt Ziβ ) ≥ e−ǫt 2 upper bound for the rates −hX, L(X)iβ and FROM QUANTUM TO CLASSICAL IN KITAEV-TYPE MODELS F ≥ ǫ is −hZ, L(Z)iβ . where Proof. Let Fx be given by 1 Fx = (F (|0i) + F (|1i)) (22) 2 where |0i, |1i are eigenstates of X treated as observable on system Q. Similarly we dene Fz . Using results of [21℄ and [22℄ one nds that F ≥ Fx + Fz − 1 (23) Thus it is enough to estimate e.g. Fx . Using the property (20) and orthogonality X ⊥ I one nds that 1 Fx = (1 + hX, eLt Xiβ ). 2 Upper bound for de ay rate We now present a useful bound for de ay rate, whi h holds for operators X whi h are eigenve tors of [H, ·]. For su h operators, one omputes − hX, L(X)iβ = X  + e−ωβ h[Sα (−ω), X], [Sα (−ω), X]iβ ≤ X ĥ(ω)h[Sα (ω), X], [Sα (ω), X]iβ ≤ ≤2 (25) ω X h[Sα (ω), X], [Sα (ω), X]iβ . (26) ĥmax = sup ĥ(ω). (27) ω where ω≥0 Sin e X and Sα (ω) are eigenve tors of [H, ·] it follows that [Sα (ω), X] are eigenve tors of [H, ·] too, hen e they are mutually orthogonal. We thus an write X h[Sα (ω), X], [Sα (ω), X]iβ = ω X However from denition of Sα (ω) it follows that X Sα (ω) = Sα (28) (29) ω X Zc (31) c Hint = X σjx ⊗ fj + X σjz ⊗ f˜j . (32) j j Then Davies operators fall into two types: aα = σxj Pα bα = σzj Rα (33) (34) where Pα belong to algebra spanned by those operators Zc whose support ontains the j -th spin and Rα belongs to algebra spanned by operators Xs , whose support ontains j -th spin. (If the spin does not belong to support of any s, then Pα = I , and similarly for R. However, in Kitaev-type models this latter ase does not o ur). The dissipative generator has the following form L = Lx + Lz , (35) where Lx , Lz onsist of Davies operators of type a and b respe tively. The Davies operators des ribe the elementary noise pro esses. In 2D model, they are reation, anihilation and motion of two types of point-like anyons. In 4D model, ex itations are not point-like, and the proesses are reation, anihilation and two types of modiation of loops (see [8℄, se . X, and se s. V, III A of the present paper). Consider now observables of the form XS and ZT , where S, T are some subsets of spins. Let us assume that XS and ZT ommute with all Xs and Zc . Then XS ommutes with Davies operators of type a and ZT ommutes with Davies operators of type b. Therefore from (7) we get that L(XS ) = Lz (XS ), L(ZT ) = Lx (ZT ) (36) Consider now a modi ation of the model. Let the Hamiltonian be of the form H=− X Xs (37) s and the oupling with environment be of the form This gives −hX, L(X)iβ ≤ 2ĥmax Xs − and we assume that the sets s and the sets c are hosen in su h a way that the operators Xs and Zc ommute with ea h other. Consider also the following oupling to environment h[Sα (ω), X], [Sα (ω ′ ), X]iβ . ω,ω ′ X s  ĥ(ω) h[Sα (ω), X], [Sα (ω), X]iβ ω≥0 ≤ 2ĥmax H=− (24) Combining the last two formulas with (16) ends the proof. B. We onsider a system of N spin-1/2 systems. For any set S of spins let us denote XS = Πj∈S σjx , ZS = Πj∈S σjz . Consider now Hamiltonian of the form X h[Sα , X], [Sα , X]iβ (30) Hint = X σzj ⊗ f˜j . (38) α j The advantage of the formula is that the only pla e where the self-Hamiltonian appears is the Gibbs state in s alar produ t. Then dissipative generator for this model onsists of Davies operators (34) i.e. it is given just by Lz . We obtain 4 Xs = Πj∈s σjx , Zc = Πj∈c σjz where the sets s and c are hosen in su h way that Xs and Zc ommute for all s, c. Consider XS whi h ommutes with all Xs and Zc . Then Proposition 2 Let (39) L(XS ) = L′ (XS ) L is dissipative generator oming from X X X X σjz ⊗ f˜j . σjx ⊗fj + H =− Xs − Zc , Hint = of stability in se . V and will be then des ribed in se . VI. The full "dressed observable" is the produ t XC Fx . A ording to Proposition 2 it evolves a ording to lassi al model with Hamiltonian HX = − where s and L′ c is dissipative generator H′ = − j j X X ′ Hint = Xs , (40) oming from s σjz ⊗ f˜j . (41) j Moreover

Tr ρβ X † L(X) = Tr ρ′β X † L′ (X) , where ρβ = 1 −βH and Ze Analogous result holds for Xs and (42) ′ ρ′β = Z1′ e−βH respe tively. ZT , whi h ommutes with all oupled to environment via operators σjz . The model is known as Z2 gauge Ising model (the Ising variables are in our ase eigenve tors of σjx ) [23℄. One an dene analogous observable ZP Fz . However in 3D there will be no X-Z symmetry. The observable ZP is asso iated with plane, and atomi proje tor of algebra spanned by Zc is labeled by ongurations of points (i.e. the plaquettes) rather than by loops. Observable P ZP Fz is evolving a ording to the model with HZ = − c Zc oupled via σjx . It will not be stable (as pointed out in [8℄) and most likely, one an prove it by use of te hniques worked out in [17℄. 2. 4D Observables 1. 3D X and Z Kitaev model X Xs − s X Zc (43) c where ea h s denotes set of four plaquettes whi h share ommon link, and and ea h c is six plaquettes forming ube. We will now dene a lass of observables of interest. To this end we will use observable XC with C being set of parallel plaquettes forming a loop that winds around the torus (there are three homologi ally inequivalent hoi es, we will onsider a xed one of them). Su h observable is very unstable, hen e we may all it "bare qubit observable". One needs to "dress" it with another di hotomi observable whi h would store the error syndrome. The latter observable will then belong to the abelian algebra spanned by star observables Xs , hen e depending solely on atomi proje tors of the algebra whi h orrespond to ongurations K of ex ited links (stars an be labeled by the links - their enters). Let us all the proje tors PK . The needed observable will be thus of the form Fx = X λK PK , X Xs − s The Hamiltonian for 3D Kitaev model is given by [8℄ H =− Kitaev model In four dimensional model the spins again sit on plaquettes, and the Hamiltonian is similar as in 3D ase: H=− A. (44) K where λK = ±1. We shall not determine the values of λK at the moment. They will emerge from our analysis (45) Xs s Zc . Remark. Further in text, h·, ·iβ will denote s alar produ t with the Gibbs state of type ρ′β (with suitable H ′ , depending whether we talk about X or Z ). X X Zc (46) c The only dieren e is that the star s has six plaquettes, be ause there is six plaquettes ommon to a single link. Thanks to it there is symmetry: We x two planes p1 and p2 on the latti e and on the dual latti e, respe tively, obtaining bare qubit observables Xp1 and Zp2 . Then andidates for stable observables will be the dressed ones Xp1 Fx , Zp2 Fz . The latter will again evolve separately, and sin e 4D latti e is self-dual, the evolutions are the same. We arrive at the 4D Z2 gauge Ising model. If we prove that e.g. observable of the form Xp1 Fx is stable, then also similar Zp2 Fz will be stable too, so that we will obtain stable qubit. IV. STABILITY CONDITIONS FOR KITAEV MODEL A. Bound for de ay rate for dressed observables The bound (30) applied to generator onsisting of Davies generators (33), (34) takes the form −hA, L(A)iβ ≤ 2ĥmax X h[σjx , A], [σjx , A]iβ + j X h[σjz , A], [σjz , A]iβ . + (47) j The quantity hmax given by (27) is a onstant independent of the size of the system. This is due to the fa t that 5 Kitaev models exhibits strong lo ality property, implying that there is a onstant number of frequen ies involved in the generator (e.g. just one positive frequen y in 2D model) whi h are independent of the number of spins N . Sin e the observables Z = ZP Fz , X = XC Fx (or analogous ones from 4D model) ommute with Hamilotnian, the bound is appli able. We obtain where δ = β − ln µ is positive and does not depend on the size of the system. We then evaluate probability of appearing a onguration that has a loop greater than L′ ′ P (l ≥ L ) ≤ poly(L) ∞ X e−δl = poly(L)e−δL ′ l=L′ −hX, L(X)iβ ≤ 4ĥmax X (1 − hX, σjz Xσjz iβ ) j −hZ, L(Z)iβ ≤ 4ĥmax X (1 − hZ, σjx Zσjx iβ ) (48) j where j runs over all spins. We see that the problem of de ay of time auto orrelation fun tion has been redu ed to the mu h simpler problem of "one step" auto orrelation fun tion. B. Gibbs state is on entrated on ongurations without long loops First we will estimate probability that a onguration has loop of length l. We shall use the Peierls argument following Dennis et al. [8℄ and Griths [24℄. To this end we rst estimate probability that a xed loop λ with length l emerges. Let C be the set of all ongurations whi h ontain loop λ. The probability is then given by P −βE(K) K∈C e P (λ) = P −βE(K) Ke (49) where in denominator we have sum over all ongurations. For any onguration K ontaining λ we ip spins on a hosen surfa e whose boundary is λ, obtaining new onguration K∗ whi h diers from K only in that the loop λ is not present anymore. Hen e E(K) = E(K∗ )e−βl (or the quantities here are taken to be dimensionless). Thus we write P (λ) = e−βl P −βE(K∗ ) K∗ ∈C e P −βE(K) Ke (50) Leaving in denominator only ongurations K∗ , we an only de rease it, so that P (λ) ≤ e−βl . Now, the probability P (l) of appearing a onguration whi h has a loop of length l is bounded by the number of all possible loops of length l times e−βl . A trivial bound for the number of loops in ube of linear size L in dimension d, that start from a xed node is 2d(2d − 1)l . This should be multiplied by the number of nodes, whi h is proportional to the volume i.e. polynomial in linear size L of the system. Finally, we obtain that P (l) ≤ poly(L)µl e−βl = poly(L)e−l(β−ln µ) (51) where µ is a onstant depending only on d. Thus below ertain riti al temperature Tcrit we have P (l) ≤ poly(L)e−δl (52) 1 1 − e−δ (53) Thus we see that below Tcrit the probability of obtaining e.g. a loop of length L/8 or greater is exponentially de aying in L. C. Stability of Kitaev 4D model In next se tion we shall prove that for ongurations having only loops shorter than L′ = L/8 a single ip does not hange observables X and Z for Kitaev 4D model. This implies that X j (1 − hZ, σjx Zσjx iβ ) ≤ X 2P (l ≥ L′ ) (54) j so that ′ −hZ, L(Z)i ≤ poly(L)e−δ L (55) where δ ′ = δ/8 is a onstant that is positive below some riti al temperature. The same happens for observable X , hen e due to proposition 1 the de ay time of delity is exponentially long in size of the system. V. STABILITY OF TOPOLOGICAL OBSERVABLES In previous se tion we have shown that below ertain riti al temperature Tcrit the Gibbs state is on entrated on ongurations with short loops. Thus if on su h ongurations an observable does not hange under single spin ip, it is stable within the lassi al model. If in addition it is of the spe ial form XC Fx , then it is also stable within the quantum model (see se tion V C). In this se tion we shall build su h observable. To this end we shall rst dene homology lasses of spin ongurations orresponding to ongurations of loops with short loops only. We will then show that, as expe ted, single spin ip does not hange those homology lasses. This implies that any observable whi h depends solely on the homology lasses does not hange under single spin ips (for ongurations ontaining only short loops). This result holds for torus of any dimension. Subsequently we shall show, that some observables of the form XC Fx share this property. 6 A. Observables depending only on homology lasses Let us introdu e some notation. By S we will denote onguration of spins on the latti e (in the form of ongurations of bits whose values en ode spin orientation). Given two spin ongurations S1 and S2 , we an add them to obtain new onguration S . We denote it by S = S1 ⊕ S2 , and the addition is bit-wise, modulo 2. I.e. if at given site the spins are the same, resulting spin is down, if they are dierent, resulting spin is up. We denote by S0 onguration of all spins down. By K we will denote set of ex ited links. A link is ex ited, if the parity of spins on adja ent plaquettes is odd (in 3D a link has four su h plaquettes, and in 4D  six ones). One nds that K is sum of disjoint loops lj (the loops an have self rossing at nodes): Lemma 1 For given onguration K onsider a onne ted set of links. It is sum of losed loops, whi h visit ea h link and ea h node at most one time. Equivalently, it is a losed walk, whi h visit one link only at most on e. Proof. The proof is by indu tion. We will all su h onne ted sets "loops". We will say that a loop is short, when its length is no greater than cL, where c is a xed onstant, whi h we an take e.g. 1/8. Any spin onguration S denes link onguration K. We will then write S(K). Of ourse for given K there are many spin ongurations leading to them. Sometimes for given S the orresponding K will be denoted by ∂S and alled boundary of S . Denition 1 By ontinuous deformation of spin ong- uration we mean operation, whi h an be omposed of the following elementary operations: ipping spins on all plaquettes belonging to an elementary d-dimensional ubes. Remark 1. Continuous deformation does not hange the onguration of links. For 3D easy to see: indeed, ipping spins on fa es of ube, hange at the same time spins on two plaquettes adja ent to a link from the ube. Fa t 1 All shortest ongurations S ∗ for given K are homologi ally equivalent, provided K ontains only short loops. Proof. have S1∗ ⊕ S2∗ = ⊕j [S1∗ (lj ) ⊕ S2∗ (lj )] Denition 3 We say that S1 and S2 whi h have the same boundary are homologi ally equivalent and denote it by S1 ∼ S2 , if S1 ⊕ S2 is homolgi ally trivial Denition 4 ("Shortest onguration") Consider given S K = j lj . For ea h loop lj x a shortest surfa e whose boundary is lj . Consider then S ∗ (lj ) whi h has spins up on this surfa e and all other spins down. The onguration S ∗ = ⊕j S ∗ (lj ) will be alled shortest onguration for L. (56) However, ea h onguration S1∗ (lj ) ⊕ S2∗ (lj ) is trivial. Indeed, sin e loop lj is short then |S1∗ (lj )| and |S2∗ (lj )| are small, and annot form homologi ally nontrivial surfa e. Denition 5 For K ontaining only short loops, with any S leading to K we an asso iate the homology lass of S ⊕ S ∗ (K). Denote it by h(S). Remark 2. For xed K obviously S1 ∼ S2 i h(S1 ) = h(S2 ). Thus the above denition allows to as ribe labels to homology lasses of spin onguration, by relating to distinguished lass i.e. the lass of S ∗ . But the homology lasses are now dened for any K. Thus we will be able to ask later, whether a spin ip (whi h of ourse hanges K) an preserve homology lass. For any K there are eight homology lasses in 3D ase, asso iated with three possible ways of winding around torus. In 4D there is 16 lasses. We have obvious fa t: Fa t 2 We have S1 ⊕ S2 = σj (S1 ) ⊕ σj (S2 ), where σj ips j -th spin. Now we will show that for short loops, single spin ip does not hange homology lass of S . To this end we rst prove the following lemma Lemma 2 For K ontaining only short loops we have σi (S ∗ (K)) ∼ S ∗ (σi (K)). (57) Here σi (K) is understood as the onguration of loops arising from onguration K by applying σi Proof. Divide K into two sets: K1 onsisting of loops that ontain some links from i-th plaquette, and K2 whi h does not ontain links from this plaquette. Then σi (K) = σi (K) ∪ K1 hen e Denition 2 We say that S1 and S2 with empty bound- ary are homologi ally equivalent if they an be transformed into one another by ontinuous deformation. S is alled homologi ally trivial, if it an be ontinuously transformed into S0 . Take two dierent shortest ongurations. We σi (S ∗ (K)) = σi (S ∗ (K1 )) ⊕ S ∗ (K2 ) (58) S ∗ (σi (K)) = S ∗ (σi (K1 )) ⊕ S ∗ (K2 ). (59) and Thus only K1 is in the game: σi (S ∗ (K)) ⊕ S ∗ (σi (K)) = σi (S ∗ (K1 )) ⊕ S ∗ (σi (K1 )) (60) and therefore we have to show that right-hand-side of the above formula is homologi ally trivial. Indeed, the set K1 ontains at most two loops independently of dimension. Now, sin e loops are short, both σi (S ∗ (K1 )) and S ∗ (σi (K1 )) are small, and added together must give a trivial surfa e. Now we are in position to prove the main result of this se tion 7 Consider onguration of spins S for whi h K has short loops only. Then single spin ip does not hange the homology lass of S . More expli itly, we have Proposition 3 sign on S1 and S2 . Thus it has the same sign also on S and S ′ . Now we are in position to build stable observable. Now let us assume that (65) (61) XT (S1 )XT (S2 ) = XT (S1 ⊕ S2 ). (62) We stress here that this assumption is easily seen to hold for parti ular observables onsidered in next subse tion. (One an a tually show, that it is true in general for observables satisfying assumptions of the above lemma). Using this we an write (63) XT (S) = XT (S ⊕ S ∗ ⊕ S ∗ ) = XT (S ⊕ S ∗ )XT (S ∗ ) (66) Combining the above two equations, we obtain the laim. Thus any observable T whi h for ongurations K ontaining only short loops depends only on homology lass, i.e. Sin e homology lass of S ∗ is always the same for short loops (independently on possible amiguity of S ∗ for given loop) then XT (S ∗ ) depends only on the loops onguration: XT (S ∗ ) = X ′′ (K), so that XT (S) = XT (S ⊕ S ∗ )X ′′ (K). Now, sin e for xed loops onguration XT depends only on homology lass and the loops onguration for S ⊕ S ∗ is always null (as S ⊕ S ∗ does not have a boundary), we get that XT (S ⊕ S ∗ ) depends only on homology lass of S ⊕ S ∗ . Therefore, a ording to definition 5, it depends only on homology h of S . Hen e XT (S ⊕ S ∗ ) = X ′ (h) and we have σi (S) ⊕ S ∗ (σi (K)) ∼ S ⊕ S ∗ (K) Proof. By lemma 2 we have σi (S ∗ (K)) ∼ S ∗ (σi (K)). By fa t 2 we have σi (S) ⊕ σi (S ∗ (K) = S ⊕ S ∗ (K) T (S) = T (h) (64) is dynami ally stable within the model of Proposition 2 below some riti al temperature. B. XT (S) = X ′ (h)X ′′ (K). Constru tion of stable topologi al observables Our bare observable will be XT = Πj∈T σxj where T is hosen in su h a way that XT is invariant under ipping spins on plaquettes from any ube (i.e. it is invariant under ontinuous transformations). Examples of su h observables exists, as will be shown later in next subse tion. We will show that one an nd di hotomi observable Fx whi h will depend on given onguration S only through K, su h that the dressed observable XT Fx depends only on homology of S (for short loops) i.e. it is stable within the lassi al model. We begin with the following lemma Lemma 3 The observable XT whi h is invariant under ipping spins on plaquettes of any ube is onstant on homology lasses for any xed link onguration K ontaining only short loops ( f. denition 5). Note that this does not mean that XT is stable. Indeed, for any xed link onguration, it is onstant on the whole homology lasses. However if the link onguration hanges, it may hange sign on the same homology lass. The stable observable des ribed in previous subse tion has the same value on a given homology lass independently of link ongurations, provided there are only short loops. Proof. Consider arbitrary spin ongurations S and S ′ whose boundary is K, and whi h are in the same lass of homology, i.e. S1 ≡ S ⊕ S ∗ is homologi ally equivalent to S2 ≡ S ′ ⊕ S ∗ . Therefore S1 an be transformed into S2 by ipping spins on a set of elementary ubes. This does not hange the sign of XT , so that XT has the same Remark. (67) Then the following observable T (S) = XT (S)X ′′ (K) (68) depends only on h. The above observable is dened unambiguously only for spin ongurations leading to short loops ongurations. This is be ause X ′′ is only well dened only on short loops ongurations. We then extend the denition of T to all spin ongurations, by letting X ′′ (K) = 1 for all other loops ongurations. Thus we shall take Fx = X ′′ and obtain that XT Fx depends only on homology of spin onguration, hen e is stable within lassi al model. C. Observable stable within quantum model The observable onstru ted in the previous subse tion is stable within lassi al model, be ause it depends only on homology lass. However, we know that only spe ial observables from the lassi al model evolve in the same way in quantum model. E.g. the observables of the form XT Fx , where Fx is from algebra generated by star operators Xs ,a and XT ommutes with Zc . Here we shall fo us on onstru tion of XT sin e it determines Fx via onsiderations of the previous se tion. Now, let us note that the rst ondition means simply that Fx depends only on loops. The se ond ondition means that XT does not hange under ips on all plaquettes of an elementary ube. Thus the observable (68) is of the above form, hen e it evolves in the same way 8 both in quantum and lassi al model, hen e it is stable also within quantum model. The last thing is to assure that the observable T is nontrivial, i.e it is not identity. To this end we have hoose the set T in a spe ial way, su h that on spin ongurations without boundary, XT an take dierent sign. For 3D it will be nontrivial loop in dual latti e, i.e. straight line onsisting of parallel plaquettes. The fa t that it is loop in dual latti e, implies that it XT is invariant under ontinuous transformations. Sin e it is nontrivial, then XT have value −1 for spin onguration onsisting of plane of ipped spins perpendi ular to T , while it takes value 1 on homologi ally trivial spin ongurations. Sin e there are three possible hoi es of inequivalent nontrivial loops, we an onstru t three independent observables. In 4D we take T to be plane in dual latti e, i.e. the value observable XT is dened as a produ t of values of all plaquettes belonging to the plane T . Again, XT does not hange under ipping spins on ube be ause arbitrary ube has exa tly two plaquettes in ommon with su h a plane. For this reason it will be also 1 on homologi ally trivial spin ongurations. However it will take value −1 on the onguration onsisting of ipped spins on a plane T ′ whose interse tion with T is a single plaquette. Note that sin e there are six homologi ally nontrivial planes, we an onstru t six independent observables of this sort. Now, sin e the torus in 4D is selfdual, we an onsider dual observable i.e. Tz = ZT′ Fz , and Fz depends only on onguration of three dimensional ubes (su h ubes are dual to link). Sin e Fz and Fx ommute, and planes T and T ′ interse t only in a single plaquette, we obtain that Tz and Tx anti ommute, so that they form a qubit. VI. POLYNOMIAL ALGORITHM FOR MEASURING THE TOPOLOGICAL OBSERVABLES The observables are symmetri , so it is enough to show algorithm for one of them, say Tz . The algorithm is the following. 1. Measure all spins. 2. Multiply out omes on a xed plane in dual latti e, this gives "raw value" of the observable. 3. Identify the loops. 4. For "short" loops we identify asso iated surfa es (the ones homologi ally equivalent to shortest ones). 5. If an odd number of surfa es rosses a xed plane in dual latti e, multiply the "raw value" with −1. The step 2 orresponds to measuring the bare observable XT , while the steps 3-5 dene observable Fx . The multipli ation in last step produ es the stable, dressed A) B) 3 2 1 FIG. 1: E ient algorithm for determining surfa e losing the loop. observable Tx = XT Fx . The only nontrivial problem here is to argue that the step 4 is polynomial. It is a tually enough to show that for a xed loop, one an nd e iently a surfa e whi h is ontained in the smallest ube ontaining the loop. Moreover, it is enough to nd a proto ol whi h in ef ient way allows to nd spins whi h, if ipped, redu e length of the loop by some amount (in our proto ol, it will be redu ed by two). The proto ol is the following. We rst hoose a Cartesian frame. We start with a link of the loop, and move along the loop. If there is ambiguity (the loop rosses itself) the priority is set by the hosen frame: if only we an we go in positive dire tion of the axis with the smallest number. If not, then we go in negative dire tion of the axis with the smallest number. The same rule governs hoi e of the starting link. The walk is stopped, if we are for ed at some point to go in opposite dire tion to any of the previous steps (see gure). When the walk is stopped, the link at whi h we stopped and the last "opposite" link, determine uniquely the set of plaquettes. This is be ause all the links of the walk lying between two "opposite links" are perpendi ular to them. Now, after ipping spins on the set of plaquettes, the two opposite links are removed from the urve. Note that this ipping may further diminish the length of loop, if by a han e, the hose plaquettes have some other links ommon with the loop. It may also divide the loop into smaller ones, however their joint length is not longer than l − 2. VII. CONCLUDING REMARKS We have shown that within Markovian weak oupling approximation, there exist a stable quantum subsystem in four dimensional Kitaev model of [8℄. While the qubit 9 is indeed stable, there are several other drawba ks, whi h makes the question of existen e of self orre ting quantum memory still open. Minimal requirement for good quantum memory is that it should allow to en ode arbitrary state of qubit (en oding), then to store it for long time (storage) and nally perform a measurement in arbitrary basis (readout). It would be also good if the measurement is repeatable. The present result shows that storage is possible, but does not tou h the problem of preparation and measurement. A tually, the algorithm for measuring topologi al observables is highly destru tive, hen e non-repeatable. The en oding and read-out one usually performs by preparing qubit in a standard state, and also measure standard observable, the rest being done by gates. Also repeatability an be then assured, if one an perform -not gates on the prote ted qubits. However the problem with the Kitaev's model is that it does not support universal omputation. A possible solution of this problem is to use the version of topologi al quantum memory developed by Bombin and Delgado [25℄ whi h supports universal omputation (we shall present the dynami al analysis of these models elsewhere). How- [1℄ M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press,Cambridge, 2000). [2℄ D. Aharonov and M. Ben-Or (1996), quant-ph/9611025. [3℄ E. Knill, R. Laamme, and W. H. Zurek, S ien e 279, 342 (1998). [4℄ R. Ali ki, M. Horode ki, P. Horode ki, and R. Horode ki, Phys. Rev. A 65, 062101 (2002), quant-ph/0105115. [5℄ B. M. Terhal and G. Burkard, Phys. Rev. A 71, 012336 (2005), quant-ph/0402104. [6℄ D. Aharonov, A. Kitaev, and J. Preskill, Phys. Rev. Lett. 96, 050504 (2006), quant-ph/0510231. [7℄ R. Ali ki (2004), quant-ph/0402139. [8℄ E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002), quant-ph/0110143. [9℄ A. Y. Kitaev, Annals Phys. 303, 2 (2003), quantph/9707021. [10℄ H. Bombin and M. A. Martin-Delgado, Physi al Review B 75, 075103 (2007), ond-mat/0607736. [11℄ R. Ali ki and M. Horode ki (2006), quant-ph/0603260. [12℄ R. Ali ki, M. Fannes, and M. Horode ki, 40, 6451 (2007), ever, still there is a separate problem of preparation of the qubit in standard state. Finally, let us mention, that the present result has separate impli ations in statisti al physi s. In the standard approa h to large quantum systems the metastable states of en oded qubits like those found for Kitaev models disappear in the thermodynami limit merging into a lassi al simplex of equilibrium (KMS) states [11℄. On the other hand they arry an interesting topologi al stru ture whi h might be physi ally relevant. In this ontext it is interesting to ask for a new des ription of innite system, whi h would take into a ount su h new metastable states. In parti ular, phase transitions whi h lead to su h urious states require further investigations. A knowledgements We are grateful to John Preskill for drawing our attention to results of Ref. [8℄ on 4D model and for numerous dis ussions. We thank Mark Fannes for stimulating disussions. M.H. would like also to thank He tor Bombin and Miguel Martin-Delgado for helpful dis ussions. This work is supported by the EU Proje t QAP-IST ontra t 015848 and EC IP SCALA. [13℄ [14℄ [15℄ [16℄ [17℄ [18℄ [19℄ [20℄ [21℄ [22℄ [23℄ [24℄ [25℄ arXiv:quant-ph/0702102. Z. Nussinov and G. Ortiz (2007), ond-mat/0702377. Z. Nussinov and G. Ortiz (2007), arXiv:0709.2717. A. Kay (2008), arXiv:0807.0287. S. Iblisdir, D. Perez-Gar ia, M. Aguado, and J. Pa hos (2007), arXive:0806.1853. R. Ali ki, M. Fannes, and M. Horode ki (2008), arXiv:0810.4584. R. Ali ki and L. 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