Hamiltonian Formulation of Quantum Error Correction and Correlated Noise: The Effects Of Syndrome Extraction in the Long Time Limit
aa r X i v : . [ qu a n t - ph ] J u l Hamiltonian Formulation of Quantum Error Correction and Correlated Noise:The Effects Of Syndrome Extraction in the Long Time Limit
E. Novais, Eduardo R. Mucciolo, and Harold U. Baranger Department of Physics, Duke University, Box 90305, Durham, North Carolina 27708-0305, USA Department of Physics, University of Central Florida, Box 162385, Orlando, Florida 32816-2385, USA (Dated: November 18, 2018)We analyze the long time behavior of a quantum computer running a quantum error correction (QEC) codein the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models,and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndromehistory and the associated residual decoherence encoded in the reduced density matrix. Systems with non-zerogate times (“long gates”) are included in our analysis by using an upper bound on the noise. In order to introducethe local error probability for a qubit, we assume that propagation of signals through the environment is slowerthan the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalizedspin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlationsdecay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem offault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, thetraditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities tocriteria that occur in the theory of quantum phase transitions.
PACS numbers: 03.67.Lx,03.67.Pp,03.65.Yz,73.21.-b
I. INTRODUCTION
Quantum computation provides a fundamentally new wayto process data; as a theory, it is complete and remarkably rich[1]. However, any real quantum computer is subject to an im-placable physical reality: components of a computer will al-ways be faulty due to environmental noise. Hence, the builderof a quantum computer faces the conundrum of having to iso-late the device from its surroundings and, simultaneously, ofneeding to act on it and read its output [2]. Many strategieshave been devised to address this problem [1, 3, 4, 5, 6, 7], themost general being quantum error correction [1, 8, 9, 10, 11].Quantum error correction (QEC) should be understood as aperturbative approach [12], where one can estimate the proba-bility of having an “error” in the wave function of the quantumcomputer after a certain time. It is naturally formulated as aperturbation expansion in powers of the coupling between thecomputer and the environment [12]. QEC cannot, in general,perfectly correct the quantum evolution, and the interferenceof the amplitudes for the various processes that occur impliesthat quantum information is always lost to the environment[12]. However, as we discuss below, QEC can very effec-tively slow down this loss. In fact, a central theoretical resultis the “threshold theorem”: it states that if the error probabilityis smaller than a critical value, quantum computation can besustained indefinitely [13, 14, 15, 16, 17, 18, 19, 20, 21]. Theword “indefinitely” deserves some clarification: For the prob-lems that we discuss, it means that given a calculation and adesired precision, it is always possible to construct a quantumcircuit that will provide the correct result with high enoughprobability.QEC has been largely developed using phenomenological“error models”. Rarely is a connection to a microscopic quan-tum dynamical system found in the literature (see, however,Refs. [22, 23, 24, 25]). In contrast, here we pursue exactlysuch a connection: We discuss the formal steps needed to link the theory of error correction with microscopic Hamiltonianmodels. Furthermore, because of the perturbative nature ofthe method, it is possible to draw a close parallel between the“threshold theorem” and the theory of quantum phase tran-sitions. We find that if a certain inequality holds, an errorthreshold always exists. When the inequality is not satisfied,either a new version of the threshold criterion is required orfault tolerant quantum computation is not possible at all. Forthe moment, we are not able to distinguish between these twopossibilities.Our analysis is based on the following assumptions. Firstand foremost, we assume that it is possible to perform thebuilding blocks of quantum error correction, namely, prepa-ration of states, quantum gates, and measurements. Second,we consider that the environment is described by a free fieldtheory in which thermal fluctuations can be effectively sup-pressed. Finally, the main simplifying assumption of our dis-cussion is that the qubits are sufficiently separated in spacefor an entire error correction procedure to be performed be-fore correlations between nearby qubits develop. The proba-bility of an error in an individual qubit within a QEC cycle is,therefore, independent of all other qubits. This does not im-ply that there are no spatial correlations; rather, they developon longer time scales, while the error correction procedure isdone faster than a certain characteristic time. We emphasizethat this hypothesis is not a limitation of the general theoreti-cal framework that we describe, but simply a way to connectto the traditional proofs of the “threshold theorem” in termsof stochastic error models.The paper is organized as follows. Because of the inter-disciplinary nature of the subject, this Introduction continueswith a discussion of two points. First, the difficulties in takinginto account correlations in the environment are explained inSec. I A from a perturbative point of view. Then, in Sec. I B,we discuss the QEC method from a physics viewpoint andpresent some results for the standard stochastic error model.We start the body of the paper by developing the relation be-tween error models and quantum codes (Sec. II). The keyissue of QEC in a correlated environment is treated in Sec. III.Our main results delineating when the perturbative treatmentis valid appear in Sec. IV. At the end of this Section, weprovide a brief comparison between our results and those ofRef. 26. Sec. V discusses parallels between the threshold the-orem of QEC and the theory of quantum phase transitions.Finally, in Sec. VI we summarize our results and comment onsome open problems.
A. The problem of correlated environments
In order to set the stage for the analysis in the presenceof QEC, we first look at the problem of errors created bya correlated environment in an unprotected system. In theSchr¨odinger equation governing the time evolution of a quan-tum system, the Hamiltonian H can usually be separated intoa single-particle term H and a many-particle interaction part V . A formal solution of this equation is given by the Dysonseries in the interaction picture. Solution by iteration showsthat the time evolution operator is U ( t,
0) = T t e − i ~ R t dt ′ V ( t ′ ) , (1)with T t denoting the time ordering operator and V ( t ) = e i ~ H t V e − i ~ H t . If V represents the interaction between thequantum computer and its surroundings, each insertion of V in Eq. (1) corresponds to an “error” in the computer evolution.Hence, Eq. (1) provides the natural framework to study the ef-fects of the environment on the state of the quantum computer.It is always possible to give an upper bound to the “errorprobability” [27]. The reason is that Dyson’s series is abso-lutely convergent for finite times and bounded operators (seeAppendix A). In short, the bounding is done by defining the“sup” operator norm and the evolution operator with at leastone “error” (one insertion of V ), E ( t ) = U ( t, − − i ~ Z t dt ′ V ( t ′ ) U ( t ′ , . (2)The norm of E is related to the probability of having errors inthe computer. The calculation is simple and yields ||E ( t ) || ≤ ~ Z t dt ′ || V ( t ′ ) || ≤ Λ t ~ , (3)where we used the triangular inequality, the unitarity of U , anddefined Λ as the largest eigenvalue of V (with correspondingeigenvector Ψ Λ ). One can understand this bound as simply arestatement of | sin x | ≤ | x | , as follows: E † ( t ) E ( t ) = (cid:2) − U † ( t ) − U ( t ) (cid:3) = 2 (cid:20) − T t cos 1 ~ Z t dt ′ V ( t ′ ) (cid:21) (4) so q h Ψ Λ | E † ( t ) E ( t ) | Ψ Λ i = s (cid:18) − cos Λ t ~ (cid:19) = (cid:12)(cid:12)(cid:12)(cid:12) sin Λ t ~ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Λ t ~ (cid:12)(cid:12)(cid:12)(cid:12) . (5)The norm ||E|| has been very useful in problems involvingnon-Markovian noise [26, 27, 28, 29, 30]. However, in QEC,an analysis based on the bound Eq. (3) only makes sense when ||E|| ≪ , while we are concerned with the long time limit, | Λ t | ≫ , for which this bound on the norm of the error di-verges. In this case, Dyson’s series is only asymptotically con-vergent and the “sup” norm is of no practical use. Hence, it isimportant to express the error probability differently.We must go back full circle and reexamine the Dyson seriesfor the time evolution of a particular state, instead of the worstcase scenario explored by the “sup” norm approach. Hence-forth, we will be mainly interested in an interaction Hamilto-nian with the general form V ( t ) = λ Z L d x f ( x , t ) , (6)where λ ≪ is a coupling constant, L is the size of the sys-tem, and f is some function of the degrees of freedom of afree theory whose Hamiltonian is H . Because we are inter-ested in correlated non-Markovian noise, we assume that thefree fields are such that the asymptotic expression for the two-point correlation function is a power law, h Ψ | f ( x , t ) f ( x , t ) | Ψ i ∼ F x ) δ , t ) δ/z ! , (7)where ∆ x = | x − x | and ∆ t = | t − t | [31]. Here, δ is thescaling dimension of f , z is the so-called dynamical exponent,and | Ψ i is a fixed eigenstate of H (which we will usually taketo be the ground state of the environment).The motivation for developing a perturbative expansion ofthe evolution operator (the Dyson series in the interaction pic-ture) is the hope that a few terms in the series or a summablefamily of them will capture most of the physics. It is thenassumed that small coupling can guarantee fast convergence.However, since ||E|| is not necessarily small, the number ofterms that contribute substantially to the series can grow fasterthat the smallness of consecutive terms. In order to see that, letus calculate the probability of an evolution with errors usingEq. (4), h Ψ | E † ( t ) E ( t ) | Ψ i − h Ψ | T t cos (cid:20) ~ Z t dt ′ V ( t ′ ) (cid:21) | Ψ i . (8)Since we are assuming a non-interacting free Hamiltonian, wecan use Wick’s theorem. It is then straightforward to showthat there is at least one term at each order m in the series thatcontributes “extensively” as ∼ λ m ( Lt ) m ( D + z − δ ) . A simpleexample is given by the series of “bubble” diagrams, wherethe m th order term is given by the contractions Z t dt ... Z t m − dt m h V ( t ) V ( t ) i ... h V ( t m − ) V ( t m ) i . (9)Disregarding numerical prefactors unimportant for our discus-sion, we sum the series as a geometric progression to obtain h Ψ | F † ( t ) F ( t ) | Ψ i ∼ λ ( Lt ) D + z − δ ) λ ( Lt ) D + z − δ ) . (10)Therefore, for D + z − δ > there is no guarantee that theperturbation series converges. Conversely, if D + z − δ < ,higher-order terms in the series should be increasingly lessimportant. Thus, for D + z − δ > the probability of an evo-lution with “errors” tends to one, whereas for D + z − δ < it will depend only on the “non-extensive” terms in the se-ries. The same analysis can be immediately transported to thestudy of the fidelity |h Ψ | U ( t ) | Ψ i| , where we see that for arelevant perturbation, D + z − δ > , the overlap betweenthe initial state and the evolving wave function tends to zero(an orthogonality catastrophe). This sort of “infrared” prob-lem provides a contact point with the theory of quantum phasetransitions, where the same kind of considerations also appearwhen calculating the partition function using the imaginarytime formalism (see Appendix B).In the body of this paper, our main goal is to transfer theseideas of relevance and irrelevance of a perturbation to the evo-lution of a quantum computer protected by QEC. B. Quantum error correction
Quantum error correction is arguably the most versatilemethod to protect quantum information from decoherence[32]. It is a clever use of two features of quantum mechanics:entanglement and (in its traditional form) wave packet reduc-tion due to measurement. Thus, before we start our discussionof QEC, it is important to carefully define what we mean byentanglement and decoherence.An entangled state of two quantum systems is a state thatcannot be described as a direct tensor product of states of in-dividual systems or probabilistic mixtures of tensor-productstates. As an example, consider two physical qubits (hereafterreferred to by the subscripts and ). Each qubit has a Hilbertspace isomorphic to a complex projective plane of dimensionone, CP (see Appendix C for details). However, the com-bined Hilbert space is not isomorphic to CP × CP , but tothe much larger CP . All states in CP outside CP × CP are said to be entangled. An important subtlety is the implicitnotion of a preferred “basis”. Although we can choose froman infinite number of CP × CP subspaces inside the same CP , nature gives us a natural choice, namely, CP × CP .In the working of a quantum computer, entanglement hastwo opposite roles. On the one hand, entanglement betweenqubits is the key element in a quantum computation that dis-tinguishes it from its classical counterpart [33]. On the otherhand, when the computer and the environment become en-tangled, precious quantum information is lost. Usually, the latter effect is referred to as decoherence. In the literature,there are two different definitions of decoherence. In a strictsense, decoherence is the decay in time of the coherences (off-diagonal elements of the reduced density matrix), while dissi-pation involves the exchange of energy with the environmentand changes in populations (the diagonal terms of the den-sity matrix). However, the word “decoherence” is also usedin a broader sense involving changes in both diagonal andoff-diagonal entries of the density matrix. In this paper wechoose the latter use of the word. The reason is that from aquantum error correction perspective changes in diagonal andoff-diagonal entries are “dual” to each other [1].There is a simple heuristic explanation for error correction:Usually, noise is regarded as a local phenomenon, thus itsdamaging effect in the computer should be less pronouncedif the information is delocalized among several qubits. This isprecisely how classical error correction codes work. A simpleexample of the latter is a majority vote, where the informationof a bit is copied into three physical bits, → and → .If the probability of an error in a given qubit is ǫ , the probabil-ity of having two independent errors, and consequently a totalinformation loss, is ǫ ≪ ǫ . Thus encoding increases the levelof protection of the information.It is tempting to start explaining QEC from this perspec-tive. However, the no-cloning theorem [1] states that it is im-possible to copy an unknown quantum state. The alternativeapproach is to use an entangled state involving two or morequbits to store the quantum information. This clearly delocal-izes the information, but it is at odds with the intuitive notionthat entangled states are in general more fragile to the effectsof the environment (this intuition is driven by the quantum-to-classical transition due to decoherence, see Appendix D for aconcrete example). Thus, delocalizing the information usingentanglement does not alone solve the problem. It is possibleto use unitary operations to transfer the entanglement betweenthe qubits and the environment to a constant fresh supply ofancilla qubits [1, 34]. However, it is more traditional in QECto use the partial measurements of some ancilla qubits to re-duce the quantum interference with the environment [1]. Mea-surements here have to be understood as the projection of thestate of one of the qubits (an ancilla) onto a certain basis orreference state. The outcome of this projection is a classicalbit (“zero” or “one”) and is called a syndrome. The partialwave packet reductions caused by syndrome extraction steerthe long-time evolution of the quantum computer. Recently,it has been shown that the duration of the measurement is notfundamental to the QEC procedure [35]. In fact, this processcan be quite long without jeopardizing the method.A simple example illustrates how QEC works [1, 9, 10].Suppose that we have an error model consisting of indepen-dent baths for each qubit which can cause only phase errors,and an initial qubit in the state | ψ i = α |↑i + β |↓i that wewant to protect. The 3-qubit code provides the simplest errorcorrection procedure for this problem. In Fig. 1, we define theencoding/decoding methods in a QEC cycle. At the end of acycle, the probability of measuring the syndrome of a phaseflip error in one of the three physical qubits is [39] p = 3 ǫ, (11) t t FIG. 1: A 3 qubit quantum error correction (QEC) code [1, 9,10, 36, 37]. The initial wave function, | ψ i ⊗ ( |↑i + |↓i ) / ⊗ ( |↑i + |↓i ) / , is encoded by two controlled-NOT (CNOT) gates, R CNOT = σ − i σ + i σ xj + σ + i σ − i , into an entangled state | ψ encode i = α ˛˛ ¯ ↑ ¸ + β ˛˛ ¯ ↓ ¸ with ˛˛ ¯ ↑ ¸ = ( |↑↑↑i + |↑↓↓i + |↓↑↓i + |↓↓↑i ) / and ˛˛ ¯ ↓ ¸ = ( |↓↓↓i + |↓↑↑i + |↑↓↑i + |↑↑↓i ) / . After some time, it isdecoded by a second pair of CNOT gates. An error in | ψ i is iden-tified by measuring the values of σ x and σ x (rectangle). The QECcycle ends with the correction of a possible phase-flip (arrow). and the probability of the syndrome indicating no error in thelogical qubit is p = 1 − p . (12)The residual decoherence that can not be corrected by theQEC procedure is closely related to these probabilities. In thecase of a cycle in which the syndrome indicates that one erroroccurred in any of the physical qubits, dephasing of the logi-cal qubit is given by the reduction of the off-diagonal densitymatrix element [39], ρ (1)¯ ↑ ¯ ↓ ≈ αβ ∗ (1 − ǫ ) , (13)while for a cycle with a syndrome indicating no error, the de-phasing is weaker, ρ (0)¯ ↑ ¯ ↓ ≈ αβ ∗ (cid:0) − ǫ (cid:1) . (14)After N of these cycles, the probability of having m uncorre-lated errors is P m = (cid:18) Nm (cid:19) p N − m p m , (15)with an associated residual decoherence of ρ ( m )¯ ↑ ¯ ↓ ≈ αβ ∗ (cid:0) − ǫ (cid:1) N − m (1 − ǫ ) m . (16)An elegant visualization of these events is given by a “syn-drome history diagram” of Fig. 2 (see for instance Ref. 28 fora similar discussion). An ordered set of syndromes labels aparticular evolution of the logical qubit. From the syndromehistory one can find the most likely evolution and the asso-ciated residual decoherence. For our 3-qubit code example,the most likely evolution is given by the mean value of m , ¯ m = N p . Thus, the residual decoherence of the logical qubitis given by ρ ¯ ↑ ¯ ↓ ≈ αβ ∗ e − Nǫ . (17)Therefore, as long the number of QEC cycles N ≪ ǫ − , theprobability of measuring the correct initial state of the logi-cal qubit is very high. We can quantify the amount of infor-mation that is lost by calculating the von Neumann entropy p p p p p p p p p p t FIG. 2: A syndrome history diagram. Each solid line represents theevolution of a logical qubit. At the end of a QEC cycle, a phase fliperror is detected or not with probabilities p and p , respectively. Apath provides the history of the logical qubit and is recorded as asequence of syndromes. S = − tr ( ρ ln ρ ) : lim N ≪ ǫ − S ≈ N | α | | β | ǫ (cid:2) − ln (cid:0) N | α | | β | ǫ (cid:1)(cid:3) (18) lim N ≫ ǫ − S ≈ −| α | ln | α | − | β | ln | β | . (19)Note that the loss of information can be substantial if the num-ber of cycles is so large that N ≫ ǫ − .If the information needs to be protected for a long periodof time, we have to modify the protection scheme. The moststraightforward approach is to consider a concatenated circuitwhere each qubit in Fig. 1 is a logical qubit itself and eachgate is a logical gate, resulting in an effective reduction of p .Layers and layers of protection can be added as needed [1, 29].A chief concern when applying this approach is whether thesteps required in the addition of more qubits and operations donot actually increase the chance of errors (since they increasethe combinatorial factors in the probability distribution). Thisquestion is addressed by fault-tolerant quantum computationtheory [13, 15, 16, 17, 19, 29], which has as its main resultthe so-called threshold theorem: If the “noise strength” ǫ issmaller than a certain critical value, then the introduction ofan additional layer of concatenation improves the protectionof the information.A key ingredient in the derivation of the noise thresholdis the assumption that a probabilistic structure similar to theone that we outlined above exists. Here rests the main con-cern of this paper. There are many physical situations wherean environment can induce strong memory effects and spatialcorrelations among qubits. Hence, it may not be obvious howto define the “error probabilities” of a qubit. This hinders thetraditional theory of QEC and threshold analysis, thus moti-vating a careful study of the dynamics of quantum computersprotected by QEC. II. ERROR MODELS AND QUANTUM CODES
The syndrome history used to describe the logical qubit his-tory can be converted into a more formal description of thecomputer dynamics. In our discussion, we will assume an en-vironment, H , described by a free field theory with an ultra-violet cutoff Λ , a characteristic wave velocity v , and a dynami-cal exponent z . Although simple, a free field theory faithfullyrepresents many physically relevant environments: the elec-tromagnetic field, a phonon field, spin waves, a bosonic bath,or, more generally, any two-body direct interactions betweenqubits that was split by a Hubbard-Stratanovich field. In addi-tion, we include in the Hamiltonian a term to account for thesequence of quantum gates performed on the qubits, H QC ( t ) .Hence, the total Hamiltonian is H ( t ) = H + H QC ( t ) + V. (20)The interaction term will be assumed to have the form of avector coupling between qubits and the environment, V = X x X α = { x,y,z } λ α f α ( x ) σ α ( x ) , (21)where ~σ ( x ) are Pauli matrices for the qubit located at x , λ α are the coupling strengths, and f α ( x ) are functions of the en-vironment operators [38]. Since [ H , H QC ] = 0 , we adopt aninteraction picture that follows not only the environment butalso the evolution of the computer (see Appendix E). In thisrotating frame, the evolution operator is U ( t,
0) = T t e − i ~ R t dt ′ V ( t ′ ) . (22)The interaction V ( t ) depends on the quantum code and itsimplementation. Nevertheless, there are two possible ways tokeep the discussion code independent:(i) In our previous work [39, 40], we assumed that quantumgates were performed faster than the environment responsetime (which is of order the inverse of the ultraviolet cutofffrequency Λ ). We call this approximation the “fast gate” limit.For this case, we have the evolution of the computer betweengates given by V ( t ) = X x X α = { x,y,z } λ α f α ( x , t ) σ α ( x ) , (23)with f α ( x , t ) = e i ~ H t f α ( x ) e − i ~ H t . Then, when a gate isperformed the action on the qubit is instantaneous and the sub-sequent evolution is once again governed by Eq. (23).(ii) A second possibility is to derive an upper bound on theeffects of correlations. In order to do that, we must first dis-cuss how slow gates, which are performed over time intervalslarger than τ c = 1 / Λ , change Eq. (23). Then, we can define aneffective interaction V eff that takes into account the slownessof the gates and serves as an upper bound to the exact (andcode-dependent) V . Clearly, the real experimental situationrests between the two limits (i) and (ii).Before we begin a detailed description of how to handlecase (ii), let us note that here the terminology “fast” and “slow” gates follows the QEC literature: Fast (slow) gateshave a duration much shorter (longer) than τ c . However, aswill become clear later, the relevant time scale that appears inthe study of correlation effects is the period or duration of theerror correction cycle, ∆ . Thus, in that context, short (“fast”)or long (“slow”) dynamical effects will be naturally definedwith respect to ∆ , and not to τ c .Any quantum computer code is just a rotation in the Hilbertspace of the qubits and can be described as a trajectoryon CP N − , where N is the total number of qubits. Inthe Schr¨odinger picture, the evolution is given by the natu-ral action on S N − by SU (2 N ) . The most general fault-tolerant quantum circuit is therefore defined by the Hamilto-nian H QC ( t ) = P b j ( t ) e j , where { e j } are the generators ofthe Lie algebra of SU (2 N ) . The evolution operator associatedwith this Hamiltonian satisfies the integral equation W ( t,
0) = 1 − i ~ Z t dt ′ H QC ( t ′ ) W ( t ′ , T t e − i ~ R t dt ′ H QC ( t ′ ) , (24)such that the computer state vector at time t is given by | ψ ( t ) i = W ( t, | ψ (0) i , where | ψ (0) i represents the initialstate of the computer. Therefore, in the interaction picture, theinteraction operator is given by V ( t ) = W † ( t ) e i ~ H t V e − i ~ H t W ( t ) (25) = X x X α = { x,y,z } λ α h e i ~ H t f α ( x ) e − i ~ H t i × W † ( t ) σ α ( x ) W ( t )= X x X α = { x,y,z } λ α f α ( x , t ) W † ( t ) σ α ( x ) W ( t ) . (26)Since W ( t ) is a SU (2 N ) matrix, then G α ( x , t ) = W † ( t ) σ α ( x ) W ( t ) (27)is another matrix of SU (2 N ) , and we can write V ( t ) = X x X α = { x,y,z } λ α f α ( x , t ) G α ( x , t ) . (28)Although the expression in Eq. (28) is general, it is not veryinstructive. Furthermore, it is very undesirable from an er-ror correction standpoint: since G ( t ) is an arbitrary matrix of SU (2 N ) , the V ( t ) in Eq. (28) in principle generates a highlycomplex correlated error that is nevertheless first order in thecoupling to the environment. The problem with the deriva-tion of Eq. (28) is that it is too general since we assumed thatarbitrary rotations are performed at each single step. How-ever, one of the cornerstones of quantum computation is thatsuch general rotations can be approximately decomposed intoa series of elementary gates [1]. Hence, our strategy will beto specialize the calculation to these elementary gates and as-sume that general rotations can be implemented by a finiteseries of such gates which are well resolved in time. A. Single-qubit operations
When only single-qubit operations are performed, we have H QC ( t ) = X x X α = { x,y,z } b α ( x , t ) σ α ( x ) . (29)In this case, W ( t ) is the product of SU (2) matrices acting ineach qubit’s Hilbert space. Thus, G α ( x , t ) simplifies to G α ( x , t ) = (cid:20) ρ e − iφ − ρ e iϕ ρ e − iϕ ρ e iφ (cid:21) σ α ( x ) (cid:20) ρ e iφ ρ e iϕ − ρ e − iϕ ρ e − iφ (cid:21) , (30)where ρ + ρ = 1 and { ρ , ρ , φ, ϕ } are functions of x and t . The single-qubit rotations yield an expression of the form G α ( x , t ) = X β = { ,x,y,z } g αβ ( x , t ) σ β ( x ) (31)for some g αβ ( x , t ) . By decomposing the operators f α andfunctions g αβ into their Fourier components, we can give amore formal meaning to “fast” and “slow” gates, f α ( x , t ) g αβ ( x , t ) = X | ω | < Λ ,ω e i ( ω + ω ) t f α ( x , ω ) g αβ ( x , ω ) . (32)Hence, if we define ν = ω + ω , we can rewrite the pertur-bation as V = X β (X ν e iνt "X ω X α f α ( ν − ω ) g αβ ( ω ) σ β . (33)In the limit of fast gates, | ω | > Λ , f and g are not convolved,since they have distinct frequency domains. Therefore, thenoise operators f α are unaltered by the rotation. However, if g has a significant weight at frequencies smaller than Λ (slowgates), one must convolve f with g , yielding a substantiallydifferent noise operator. B. Two-qubit operations
The general Hamiltonian for two-qubit gates is of the form H QC ( t ) = X x , y X α,β = { x,y,z } J αβ ( x , y , t ) σ α ( x ) σ β ( y ) . (34)However, one can also generate a full set of gates using insteada single type of interaction, H QC ( t ) = X x , y J ( x , y , t ) σ a ( x ) σ b ( y ) (35)where a and b are fixed for each gate ( x , y ) . In order to seethat this is sufficient we can for instance set a = b = z . Thisgenerates the liquid NMR Hamiltonian [41], where the Isinginteraction, Eq. (35), and single qubit rotations can be used togenerate a control σ z gate. We keep a and b arbitrary. However, for the sake of sim-plicity, we assume that only operations between disjoint pairsare allowed; that is, if J ( x , y , t ) = 0 , then J ( x , y , t ) = 0 for all y = y . It is then straightforward to write down W ( t ) in a compact form: The time-ordering [Eq. (24)] is automati-cally taken care of by the sequence of gates, while for a gateinvolving qubits x and y the contribution to W ( t ) is W ( x , y , t ) = cos [ θ ( x , y , t )] (36) + i sin [ θ ( x , y , t )] σ a ( x ) σ b ( y ) , where θ ( x , y , t ) = R t dt ′ J ( x , y , t ′ ) . Hence, a two-qubit rota-tion yields G α ( x , t ) = sin [2 θ ( x , y , t )] ǫ aαγ σ γ ( x ) σ b ( y )+ cos [2 θ ( x , y , t )] (1 − δ a,α ) σ α ( x )+ δ a,α σ α ( x ) , (37)where ǫ aαγ is the usual antisymmetric tensor.The first term on the r.h.s. of Eq. (37) tells us that the 2-qubitgate can propagate the error from the qubit at x to the qubit atposition y . However, it also tells us that it is possible to choosea particular gate where this propagation does not happen (bychoosing a = α , for instance). Unfortunately, propagatingerrors in the quantum circuit is in general unavoidable (sincethe only gate that commutes with all Pauli operators is theidentity).The second and third terms on the r.h.s. of Eq. (37) aremuch less dramatic. They simply describe a local noise that isnot propagated by the gate. C. Upper-bounds for the evolution
In Eqs. (31) and (37), we showed that one- and two-qubitgates can introduce what is seemingly a very complicatednoise structure. The expressions depend on how the gates areimplemented, thus hiding a general assessment. We can ad-vance the discussion by recalling that W is always an unitarymatrix. Hence, the coefficients in Eqs.(31) and (37) have mod-ulus equal or smaller than unity. A suitable upper bound onthe effects of slow gates is then provided by setting all thesecoefficients equal to one. Thus, the operators expressed inEqs. (31) and (37) gain the upper bounds ˜ G α ( x ) = X β = { x,y,z } σ β ( x ) , (38) ˜ G α ( x ) = σ α ( x ) + ǫ aαγ σ γ ( x ) σ b ( y ) . (39) ˜ G α still looks troublesome, since it tells us that an errorin qubit x is propagated to y . However, this is not a prob-lem of the finite gate time operation, since an instantaneousand perfect gate will also propagate the error in a similar fash-ion. In order to obtain an upper bound for the effects intro-duced by the two-qubit gates, we precisely follow this fact.We consider that all the qubit components are exposed to allthe noise channels all the time. Thus, we replace Eq. (39) by G α ( x ) = P β = { x,y,z } σ β ( x ) and assume that two-qubit gatesare performed instantaneously. In summary, we reduce theproblem of finite time operation of the two-qubit gate to theproblem of a noisier qubit environment and propagating errorsin the quantum code by perfect gates. Now we can rely on thetheory of fault-tolerance [1, 29], and simply assume that theerror propagation is handled by the quantum code.The final conclusion is that an upper bound estimate on theeffects of slow gates is obtained by the interaction Hamilto-nian V eff ( t ) = X x X α = { x,y,z } λ f eff ( x , t ) σ α ( x , t ) , (40)where f eff ( x , t ) = 1 λ X β = { x,y,z } λ β f β ( x , t ) (41)and λ = qP β = { x,y,z } λ β is the new coupling parameter.Although this is a brutal approximation, it will be sufficientfor our discussion. As we will argue later, for the purposeof determining the effect of long-wavelength correlations onthe threshold theorem, the only relevant aspect of the f α istheir scaling dimension. Since dim f eff is in general equal to min (dim f α ) , it is sufficient to use Eq. (40) as the worst casescenario. Thus, in both limiting cases, fast and slow gates, we ar-rive at the same functional form for the effective interaction.
Hence, both cases can be handled simultaneously, and we pro-ceed to the analysis of QEC in the presence of this interaction.In order to simplify the notation, we hereafter drop the sub-script “eff” from the slow-gate operators.
III. QUANTUM ERROR CORRECTION INCORRELATED ENVIRONMENTS
A QEC code is defined as the combination of encoding,decoding, and recovery operations. Since we were able tomake our analysis code independent, the unitary componentof the QEC protocol is described by U (∆ , , Eq. (22), withthe appropriate V ( t ) discussed in Sec. II. The final ingredientin standard QEC is just the syndrome extraction P , which is aprojective measurement.In Ref. 39 it was demonstrated how to define P and its ef-fects on U for stabilizer error correction codes. It is importantto remark that an error which keeps the computer in the logi-cal Hilbert space can never be corrected by QEC. This is sim-ply a statement that for the general assumptions we make, theproblem of protecting quantum information never satisfies thesecond criterion of Lafflame-Knill [12] for perfect QEC. Insimple terms, the criteria states that all allowed errors must al-ways take the logical one and logical zero to orthogonal states[Eq. (20) of Ref. 12]. By construction, these errors are high-order events in the coupling with the environment. Neverthe-less, as we already know (I A), this fact per se is not enoughto ensure that such errors will not be relevant at long times. One of our goals is to find out when it is appropriate to safelyneglect such uncorrectable errors in the presence of correlatedenvironments.In hindsight, it is not hard to understand the benefits ofQEC. Thus, for the sake of readability, we present first a qual-itative argument that captures the overall discussion.As we defined in the introduction, there are two quanti-ties that we are interested in calculating: (i) the probabilityof a given evolution, and (ii) the reduced density matrix of thecomputer. Both quantities are written as a double series in thecoupling with the environment. On the one hand, the initialket of computer and the environment, | Ψ i , evolves in the timeinterval [0 , t ] by the time ordered series U ( t ) . On the otherhand, the bra h Ψ | evolves in time with the anti-time-orderedseries U † . It is only a subset of each series that enters in theevaluation of either the probability or the reduced density ma-trix, because of the measurements present in the traditionalformulation of QEC. Hence, it is usually a non-trivial task tocalculate the necessary expectation values.Because we are dealing with a double series, it is naturalto use a formalism analogous to a time-loop expansion [42].There are six (interrelated) Green functions in such an ex-pansion: The usual advanced and retarded functions for thetime-ordered series; the advanced and retarded functions forthe anti-time-ordered series; and the lesser and greater func-tions, which contract a term from the time-ordered series withanother one from the anti-time-ordered series. This formal-ism is often referred to as the Schwinger-Keldysh approach[43, 44]. It is usually represented graphically by a doublecontour in time (see Fig. 3). The upper leg stands for the time-ordered evolution for the time interval [0 , t ] , while the lowerleg stands for the anti-time-ordered evolution in the reversedinterval [ t, .Let us for the moment assume that a short-time expansion is t t t t t (a) t tt t t (b) t t t t t (c) t t t t t (d) FIG. 3: Graphical representation of a few fourth-order terms in a“time-loop” expansion for either the probability of a given evolutionor the reduced density matrix (spatial dimensions are suppressed forclarity). Points of interaction with the bath (circles) are connected bypropagation of the environmental modes (wiggly lines). In the topdiagrams, the time integrals are unconstrained, as would be the casefor unitary evolution. In the bottom diagrams, the detection of anerror by a QEC protocol forces the interactions with the bath to occurat the same times on both the forward and backward legs in orderthat U and U † correspond to the same syndrome. This additionalconstraint introduced by QEC is crucial in the long-time behavior. valid and focus on a single qubit. Then, the evolution operator for that particular qubit within a QEC cycle is given by U (∆ , ≈ − i ~ Z ∆0 dt X α = { x,y,z } λ α f α ( x , t ) σ α ( x , t ) − ~ Z ∆0 dt Z t dt ′ X α = { x,y,z } λ α λ β f α ( x , t ) f β ( x , t ′ ) σ α ( x , t ) σ β ( x , t ′ ) + O ( λ ) . (42)In Fig. 3 we represent graphically a few terms of order λ . Allof these terms are the product of a second-order term from U and a second-order term from U † [see Eq. (42)]. Hence, theycorrespond to two “errors” in the qubit evolution and involvethe expectation value h Ψ | f † α ( x , t ) f † β ( x , t ′ ) f α ( x , t ′′ ) f β ( x , t ′′′ ) | Ψ i . (43)Using Wick’s theorem, we can immediately write (43) as aproduct of the non-interacting Green functions. Each possibleset of contractions leads to the different “diagrams” in Fig. 3.We usually do not know when an “error” occurs; hence,each Green function is accompanied in the series by a doubleintegral in time. This is precisely the case in an unprotectedcomputer’s evolution or inside a QEC cycle [see Figs.3 (a) and(b)]. However, a dramatic change happens in a Green functionbetween terms for different cycles. When the syndrome showsthat a particular error occurred in a certain QEC cycle, we canre-write Eq. (43) to reflect this knowledge: h Ψ | f † α ( x , t ) f † β ( x , t ′ ) f α ( x , t + δt ) f β ( x , t ′ + δt ′ ) | Ψ i , (44)where δt and δt ′ are time variables with range smaller thanthe QEC period. After integrating the “high frequency” part(the δt and δt ′ variables), we end up reducing Eq. (44) to h Ψ | f † α ( x , t ) f † β ( x , t ′ ) f α ( x , t ) f β ( x , t ′ ) | Ψ i (45)with t and t ′ representing a coarse-grained time scale of orderthe QEC period [see Figs. 3 (c) and (d)]. Therefore, althoughwe are considering terms of the same order in λ , the numberof “time integrals” in the coarse-grained scale (low frequen-cies) is half that in the original microscopic calculation (highfrequencies).The simple dimensional analysis of Sec. I A tells us nowthat QEC has changed the criteria for the stability of the per-turbation series at long times. As we demonstrate now, it isless stringent than the naive expectation. A. Quantum evolution steered by QEC
It is reasonable to assume that at the beginning of the com-putation the computer’s state vector, ψ , and the environ-ment’s, ϕ , are not entangled, | Ψ ( t = 0) i = | ψ i ⊗ | ϕ i . (46) In a realistic situation, ψ would have some initialization errorand be entangled with the environment to some degree (bothof which would yield errors in ψ ). However, here we neglectthese effects in order to keep the discussion focused.Just as in the case of the 3-qubit code, by the end of a QECcycle the computer will have evolved according to the unitaryoperator U (∆ , . Then, the syndrome is extracted and thecomputer wave function is projected, P m U (∆ , | Ψ (0) i , (47)where m corresponds to a particular syndrome, with P m P m = I and P m = P m . In the case of many logical qubitsevolving together, then m denotes the set of all the syndromesextracted at time ∆ . The last step in the code is the appro-priate recovery operation, R m , depending on the syndromeoutcome, | Ψ (∆) i = R m (∆ + δ, ∆) P m U (∆ , | Ψ (0) i . (48)Since in a fault-tolerant error correction scheme the infor-mation is never decoded (in contrast to the 3-qubit code dis-cussed above), the quantum information always remains pro-tected. Therefore, we can deal with our two limiting cases(slow- and fast-gates) in two different ways. In the case of aslow-gate recovery, we formally include it as the initial stepof the next QEC period. Conversely, in the case of fast gates,we assume that the recovery is performed flawlessly in a veryshort time scale after the projection. For the sake of clarity,we choose the latter below. We emphasize that this does notrestrict our discussion, since it is known that the time of re-covery is irrelevant to the error correction. In fact, it can bepostponed all the way to the end of the calculation [35]. B. Probability of a syndrome history and the loss ofinformation
The first quantity to discuss is the probability of measuringa particular syndrome at the end of the first QEC step, P m = h Ψ(0) | U † (∆ , P m U (∆ , | Ψ(0) i . (49)The corresponding reduced density matrix is ρ m~r,~s (∆) = tr ε (cid:2) h ~r | P m U (∆ , | Ψ(0) i h
Ψ(0) | U † (∆ , P m | ~s i (cid:3) h Ψ(0) | U † (∆ , P m U (∆ , | Ψ(0) i = h ϕ | (cid:2) h ψ | U † (∆ , P m | ~s i h ~r | P m U (∆ , | ψ i (cid:3) | ϕ ih ϕ | h ψ | U † (∆ , P m U (∆ , | ψ i | ϕ i , (50)where ~r and ~s denote states in the computer Hilbert space andtr ε is the trace over the environment Hilbert space. It is possi-ble to quantify how much information was leaked to the envi-ronment by calculating the von Neumann entropy S (∆) = − tr c [ ρ m (∆) ln | ρ m (∆) | ] , (51)where tr c is the trace over the computer Hilbert space.In Eqs. (49) and (50), one clearly sees the important roleplayed by the projection operators in the quantum evolutionsteered by QEC. The careful construction of the encodedstates combined with the measurement (syndromes) reducesthe quantum interference between different history paths ofthe computer. By partially collapsing the wave function of thecomputer, this traditional form of QEC reduces decoherence.Equations (49) and (50) define the local components of thenoise. When spatial correlation between qubits can be ig-nored, they are related to the stochastic probabilities and den-sity matrix discussed in Sec. I B [see Eqs. (15) and (16)].The generalization to a sequence of QEC cycles is straight-forward [39], Υ w = υ w N (cid:0) N ∆ , ( N − (cid:1) ...υ w (∆ , , (52)where w is the particular history of syndromes for all thequbits and υ w j (cid:0) j ∆ , ( j − (cid:1) = R w j (cid:0) j (∆ + δ ) , j ∆ (cid:1) P w j U (cid:0) j ∆ , ( j − (cid:1) , (53)is the QEC evolution after each cycle. Each history comeswith the associated probability P (Υ w ) = h ϕ | h ψ | Υ † w Υ w | ψ i | ϕ i . (54)Finally, there is always some residual decoherence which canbe found from the reduced density matrix ρ ~r,~s (Υ w ) = h ϕ | (cid:2) h ψ | Υ † w | ~s i h ~r | Υ w | ψ i (cid:3) | ϕ ih ϕ | h ψ | Υ † w Υ w | ψ i | ϕ i , (55)with ~r and ~s being elements of the logical subspace. This inturn yields the entropy S (Υ w ) = − tr c (cid:2) ρ (Υ w ) ln | ρ (Υ w ) | (cid:3) . (56)In the following, we will show for Eqs. (54) and (55) howto separate the effect of correlations between different QECcycles from the contributions due to the local component ofthe noise, as defined by Eqs. (49) and (50). IV. PERTURBATION THEORY AND THEHYPERCUBE ASSUMPTION
There is one additional issue that we must deal with beforewe can move forward. In principle, even the first order termin Eq. (42) is already beyond the QEC approach that has beenoutlined so far. The reason is that when calculating P or ρ we generate pair contractions of the type h f α ( x , t ) f α ( y , t ′ ) i .Therefore, the probability of finding an error at a given qubitis conditional on what happens with all other qubits. Thisautomatically hinders the simple probabilistic interpretationof QEC that we used in Sec. I B.The fact that we do not want to deal with such conditionalprobabilities leads us to the single most important simplifyinghypothesis of our work: We assume that the qubits are sepa-rated by a minimum distance ξ = ( v ∆) /z , (57)where v is the excitation velocity and z is the dynamical ex-ponent of the theory describing the environment. Hence, forall x = y and | t − t ′ | < ∆ , we have h f α ( x , t ) f α ( y , t ′ ) i ≈ .It is then possible to assign a probability for the short-timeevolution of each qubit independently of all others.To further organize the analysis we order the qubits in a D -dimensional array that defines hypercubes of volume ∆ × ξ D (see Fig. 4). In summary, for times smaller than ∆ , each qubithas a dynamics independent from the other qubits, hence re-sembling a quantum impurity problem. However, for timescales larger than ∆ , spatial correlations among them arepresent, thus making the problem similar to a spin lattice.Ideally, we would like to decompose the evolution operatorin inter- and intra-hypercube components, U (∆ ,
0) = U < (∆ , U > (∆ , , (58)where < labels frequencies smaller than ∆ − and > frequen-cies in the interval (cid:2) ∆ − , Λ (cid:3) . Whenever this is possible, wecan integrate the intra-hypercube part in order to define a “lo-cal” evolution and, consequently, a local error probability. ξ∆ FIG. 4: Two neighboring hypercubes in space-time, each one con-taining a qubit.
A. Perturbation theory improved by RG
Our objective in this section is to define an effective evo-lution operator that can reasonably describe the evolution ofthe qubit within each hypercube. All terms consistent with thesame syndrome and having the same leading long-time prop- erties should be included. Within a hypercube, the environ-ment induces interaction of a qubit only with itself; commu-nication between qubits at longer times is treated in the nextsection.We use the renormalization group (RG) [45] to sum themost relevant families of terms in the perturbation series. Inorder to improve the lowest order terms in the perturbationtheory through RG, we need to introduce the next higher-orderterms in the perturbation series. However, as we discussedpreviously, we are not interested in the full unitary evolution,but rather the projected terms obtained after the extraction ofthe syndrome. Therefore, in order to apply RG to the first-order term, we need to consider υ α ( x , λ α ) ≈ − i ~ λ α Z ∆0 dt f α ( x , t ) − ~ | ǫ αβγ | λ β λ γ σ α (∆) T t Z ∆0 dt dt f β ( x , t ) f γ ( x , t ) σ β ( t ) σ γ ( t )+ i ~ X β λ α λ β σ α (∆) T t Z ∆0 dt dt dt f α ( x , t ) f β ( x , t ) f β ( x , t ) σ α ( t ) σ β ( t ) σ β ( t ) , (59)where ǫ αβγ is the antisymmetric tensor [46]. There is onlyone spatial index in (59) because of the hypercube assump-tion: we have included only terms in which contraction ofthe f ’s yields a non-zero value, as these will contribute to theeffective short time evolution. At long times, connections be-tween the qubits are, of course, essential, and this is treated inthe next section.The RG is naturally implemented in the case of ohmic baths(which leads to logarithmic singularities). However, suitablegeneralizations can be defined by dimensional regularizationor by summing series in the expansion. Thus, in general, it ispossible to write the following beta function for υ α x : dλ α dℓ = g βγ ( ℓ ) λ β λ γ + X β h αβ ( ℓ ) λ α λ β , (60)where g and h are functions specific to a particular environ-ment, ℓ = Λ / Λ ′ , and Λ ′ is the reduced (i.e. rescaled) cutofffrequency. By integrating the beta function from the bare cut-off, Λ , to ∆ − , we are summing the most relevant componentsof the noise inside a hypercube. If the renormalized value ofthe running coupling at frequency ∆ − , λ ∗ , is still a smallnumber, then it is a good approximation to consider υ α ( x , λ ∗ α ) ≈ − iλ ∗ α ~ Z ∆0 dt f α ( x , t ) (61)as the evolution operator of the qubit at position x which was diagnosed with an error α by the QEC procedure.We illustrate the renormalization group procedure with twosimple examples of ohmic baths: (i) marginally relevant and(ii) marginally irrelevant couplings.
1. The k-channel Kondo problem
The first example is a qubit exposed to a bosonic bath thatis modeled by a SU (2) k Kac-Moody algebra – the bosonizedHamiltonian of a k -channel Kondo problem. Here we closelyfollow the work of Affleck and Ludwig (see appendix B ofRef. 47). We define chiral bosonic currents : ~J L : obeying theoperator product expansion (OPE) : J aL ( t ) : : J bL ( t ′ ) : → f abc : J cL ( t ) : v ( t − t ′ ) − kδ ab v ( t − t ′ ) , (62)where f abc are the group structure constants and v is the ve-locity of excitations. In the interaction picture, the qubit cou-ples to the currents by the usual Kondo interaction, yieldingan evolution operator (or, equivalently, a scattering matrix) ofthe form U = T t e − iλ v ~ R ∞−∞ dt : ~J L ( t ): · ~σ . (63)Following our general discussion, we expand the evolutionoperator to lowest order in the coupling, U ≈ − iλ v ~ Z ∞−∞ dt : ~J L ( t ) : · ~σ − (cid:18) λ v ~ (cid:19) X a,b Z ∞−∞ dt Z t −∞ dt ′ : J aL ( t ) : : J bL ( t ′ ) : σ a σ b + i (cid:18) λ v ~ (cid:19) X a,b,c Z ∞−∞ dt Z t −∞ dt ′ Z t ′ −∞ dt ′′ : J aL ( t ) : : J bL ( t ′ ) : : J cL ( t ′′ ) : σ a σ b σ c . (64)Due to the QEC evolution, only some of these terms are kept after the syndrome is extracted [see Eq. (59)]. For clarity, let usassume that we know from the syndrome that a phase flip has occurred. Hence, we must truncate the evolution operator to reflectthis fact and apply the recovery operation (in this case multiply by σ z ), yielding v z ≈ − iλ v ~ Z ∞−∞ dt : J zL ( t ) : − i (cid:18) λ v ~ (cid:19) Z ∞−∞ dt Z t −∞ dt ′ [: J xL ( t ) : : J yL ( t ′ ) : − : J yL ( t ) : : J xL ( t ′ ) :]+ i (cid:18) λ v ~ (cid:19) X a Z ∞−∞ dt Z t −∞ dt ′ Z t ′ −∞ dt ′′ [: J aL ( t ) : : J aL ( t ′ ) : : J zL ( t ′′ ) : + : J zL ( t ) : : J aL ( t ′ ) : : J aL ( t ′′ ) : − : J aL ( t ) : : J zL ( t ′ ) : : J aL ( t ′′ ) :] . (65)Now, we integrate over a small frequency shell [Λ − δ Λ , Λ] and invoke the OPE. The result is a renormalization of thecoupling λ by an infinitesimal composed of quadratic and cu-bic terms, dλdℓ = λ − k λ . (66)The resulting running coupling λ ( ℓ ) can be used to improvethe results of our bare perturbation theory. For that purpose,we integrate the beta function from the bare cutoff until ∆ − .For the case of a small number of channels, we obtain a renor-malized coupling of the form λ ∗ ≈ λ − λ ln | Λ∆ | . (67)Although the RG flow goes toward the strong coupling limit,we do not integrate the beta function all the way to zero fre-quency. Thus, if the renormalized coupling λ ∗ is still a smallparameter, it replaces λ leading to the first-order renormalized evolution v z ≈ − iλ ∗ v ~ Z ∞−∞ dt : J zL ( t ) : σ z . (68)
2. Quantum frustrated system
Correlations are not necessarily malignant to the com-puter’s behavior. This is illustrated by our second example:a quantum frustrated environment [48, 49, 50]. Consider thecase of three independent Abelian ohmic baths coupled as inEq. (63), but with the OPE : J aL ( t ) : : J bL ( t ′ ) : → − δ ab v ( t − t ′ ) . (69)Following precisely the same methodology of the previous ex-ample, we obtain the beta function dλdℓ = − λ , (70) which leads to the renormalized coupling λ ∗ ≈ λ p λ ln | Λ∆ | . (71)A quantum frustrated system has the remarkable property ofasymptotic freedom. Hence, even very large bare couplingsflow towards a perturbative regime. The physical reason be-hind this is the lack of a pointer basis [51], thus effectivelydecoupling the qubit from its surroundings [49]. This phe-nomena can also be understood as self-inflicted π -pulse de-coupling working at the cutoff frequency Λ [7, 52].If the three coupling constants have different bare values,then the flow stops at some finite frequency since two of thecouplings will flow to zero before the third. In other words,there will be a pointer basis. In a quantum computer protectedby QEC, however, we are effectively stopping the flow at afinite frequency. Hence, the effect described in the previousparagraph is relevant even for large anisotropic couplings. B. Probability of a faulty path
Now that we have obtained a reasonable approximation tothe evolution operator at each QEC step, we can turn to theproblem of evaluating how much protection QEC yields atlong times. The simplest quantity to calculate is the proba-bility of finding a particular history of syndromes, Eq. (54).Using Eq. (61) and the known commutation relations of the f α operators, we in general can write that Υ † w Υ w = υ w N (cid:0) N ∆ , ( N −
1) ∆ (cid:1) ...υ w (cid:0) ∆ , (cid:1) , (72)and define υ w (cid:0) ∆ , (cid:1) ≈ X ij λ ∗ α i λ ∗ α j ~ Z ∆0 dt dt f † α i ( x i , t ) f α j ( x j , t ) . (73)We now can evoke Wick’s theorem once again to separatethe intra- and inter-hypercube contributions to the probabil-ity: The quantum average P (Υ w ) ≈ h ϕ | Υ † w Υ w | ϕ i can bewritten as a sum of all possible pair contractions. It is conve-nient to separate the sum into two distinct parts.2First, the sum of all pair contractions in the same hyper-cube gives the stochastic error probability of a qubit, that wedefined in Eq. (49), namely, ǫ α = h ϕ | υ α ( x , λ ∗ α ) | ϕ i = (cid:18) λ ∗ α ~ (cid:19) Z ∆0 dt dt h f † α ( x , t ) f α ( x , t ) i , (74)where we used again that for | x − y | > ξ and | t − t | < ∆ , we have (cid:10) f † α ( x , t ) f α ( y , t ) (cid:11) ≈ . Note that whenwe calculated λ ∗ we already summed intra-hypercube paircontractions; however, these were contractions on the sameKeldysh branch [see Fig. 3(a)] and therefore are related tothe wave function amplitude. Equation (74) corresponds topair contractions between two distinct Keldysh branches [seeFig. 3(b)], hence it gives the probability of that evolution.With this two-step procedure, we sum up the most relevantcontributions to the probability within a hypercube.Second, we sum contractions between hypercubes. Foreach possible syndrome outcome we define the operators F ( x ,
0) = 1 − P α ( λ ∗ α ∆ / ~ ) − P α ǫ α : | f α ( x , | : (75)and F α ( x ,
0) = 1 ǫ α (cid:18) λ ∗ α ∆2 ~ (cid:19) : | f α ( x , | : , (76)where : : stands for normal ordering with respect to the envi-ronment ground state (see Appendix F). We use these opera-tors to express the remaining pair contractions of each hyper-cube in the probabilities, namely, υ ( x , ∆ , ≈ − X α ǫ α ! F ( x , (77)and υ α ( x , ∆ , ≈ ǫ α [1 + F α ( x , . (78)Equations (77) and (78) are the final ingredients needed toevaluate the probability of a particular history of syndromes,Eq. (54). The remarkable aspect of these equations is that theyprovide a very elegant reorganization of the perturbation se-ries. They were tailored to separate the local contribution, ǫ α ,from the long-distance, long-time components of the noise, F α . The high-frequency part gives rise to the stochastic noise that is well discussed in the QEC literature. We rewrote therest of the series taking into account the unusual non-unitarydriven dynamics of QEC. The only remaining issue is to eval-uate the stability of the perturbation expansion in the renor-malized coupling λ ∗ .In Sec. I A we discussed how the scaling dimension of anoperator is important when studying a perturbative expan-sion. The same argument holds when evaluating the protec-tion yielded by QEC in a correlated environment. If the scal-ing dimension of f α is δ α , then dim F α = 2 δ α (see AppendixG). Hence, the original criterion for the validity of the pertur-bative expansion in λ , D + z − δ α < , becomes D + z − δ α < (79)once the expansion in λ ∗ is adopted. Note the factor of inthis equation caused by QEC.Whenever Eq. (79) is satisfied, the long-range correlationswill produce small corrections to the stochastic error proba-bility. Below, we illustrate this point with an example. Probability of a “flawless” evolution.
Consider the case of a non-Markovian noise model withonly one type of error (phase flips, for instance). For sim-plicity, assume that no spatial correlations exist ( D = 0 ).Hence, we can consider each qubit separately and do not haveto worry about the spatial structure of the quantum computer.We also assume a two-point correlation function of the form h f ( x , t ) f ( y , t ) i = 12 (cid:18) τ | t − t | (cid:19) δ/z δ x , y , (80)where τ is a constant with the dimension of time. How dothese long-range correlations change the probability of a flaw-less evolution of a qubit after N ≫ QEC steps? To answerthis question, we evaluate P (Υ ) ≈ h ϕ | N − Y j =0 υ ( x i , j ∆) | ϕ i≈ (1 − ǫ ) N h ϕ | N − Y j =0 F ( x i , j ∆) | ϕ i . (81)Assuming ǫ, λ ∗ ≪ , we can rewrite the probability as P (Υ ) ≈ e − Nǫ h ϕ | T t exp ( − [ λ ∗ ∆ / (2 ~ )] − ǫ Z N ∆0 dt ∆ : | f ( t ) | : ) | ϕ i≈ e − Nǫ ( λ ∗ ∆ / (2 ~ )] (1 − ǫ ) Z N ∆0 dt ∆ Z t dt ∆ τ δ/z ( t − t ) δ/z + . . . ) ≈ e − Nǫ ( λ ∗ ∆ / ~ )] (1 − ǫ ) ( τ / ∆) δ/z N − δ/z ) − δ/z )(1 − δ/z ) + . . . ) , (82)where we have kept only the leading term. There are twosimple limits:(i) If z < δ , the corrections become increasingly irrele-vant as N grows. The stochastic probability in the limit oflarge N is given by P (Υ ) ≈ e − Nǫ and the correction due tocorrelations are small.(ii) The tipping point is z = 2 δ . By summing the subset ofdominant terms Z N ∆0 dt ∆ ... Z t j dt j +1 ∆ j Y i =1 D : | f ( t i − ) | : : | f ( t i ) | : E , (83)we obtain P (Υ ) ≈ e − Nǫ − ( λ ∗ ∆ / ~ ) (1 − ǫ ) ln N . (84)This signals a problem with the perturbative expansion when N ≈ exp (cid:0) ~ − ǫλ ∗ ∆ (cid:1) . For times larger than ∆ exp (cid:0) ~ − ǫλ ∗ ∆ (cid:1) ,correlations substantially change the probability. C. Residual decoherence
In addition to the probability of a given syndrome history,we also identified the residual decoherence, Eq. (55), as a fun-damental quantity to QEC. The reason is that the noise modelsthat we consider do not satisfy the Lafflame-Knill conditionfor perfect error correction [12], as is the case for most physi-cally relevant decoherence mechanisms. Hence, it may not besafe to ignore these high-order events in the coupling λ . It is straightforward to develop a calculation for the den-sity matrix along the same lines used for the syndrome historyprobability. After separating the intra- and inter-hypercubecontributions, the perturbative expansion is reorganized usingthe renormalized coupling λ ∗ . The result is exactly the sameas for the case of the probability: If D + z − δ < , the per-turbation theory in λ ∗ is stable and the analysis of the residualdecoherence done with the corresponding stochastic model isa good approximation of the true quantum result. We revisitthe example used in Sec. III B to make this point clear. Decoherence of a “flawless” evolution.
For this example, we assume an environment that can onlyintroduce phase flip errors in the computer. As we discussedin Sec. I, for this error model we can use the simple 3-qubitcode. However, unlike the calculation of the probability of aflawless evolution, we now make some assumptions about thespatial structure of the computer: We consider for simplicitythat each logical qubit is composed of three adjacent physicalqubits. The encoding and decoding are described in Fig. 1.Following Ref. 39, we write the evolution operator for aparticular logical qubit in a QEC cycle as w (0 , ¯ x ) = υ ( x , υ ( x , υ ( x , . (85)By expanding Eq. (85) in powers of λ , we obtain w (0 , ¯ x ) = 1 − (cid:18) λ ~ (cid:19) X j Z ∆0 dt Z t dt f ( x j , t ) f ( x j , t )+ i (cid:18) λ ~ (cid:19) Z ∆0 dt Z ∆0 dt Z ∆0 dt f ( x , t ) f ( x , t ) f ( x , t ) ¯ Z, (86)where ¯ Z is the logical phase flip for that particular logical qubit. Note that the third order term keeps the logical qubit inside thelogical Hilbert space [39] and therefore is not corrected by the QEC code.We choose to evaluate the most off-diagonal term of the reduced density matrix, ρ ~ ↑ ,~ ↓ (Υ ) = h ϕ | h h ψ | Q j = N − Q Mk =1 w † ( j ∆ , ¯ x k ) (cid:12)(cid:12)(cid:12) ~ ↓ E D ~ ↑ (cid:12)(cid:12)(cid:12) Q N − j =0 Q Mk =1 w ( j ∆ , ¯ x k ) | ψ i i | ϕ ih ϕ | h ψ | Q N − j =0 Q Mk =1 w (¯ x k , j ∆) | ψ i | ϕ i , (87)where ~ ↑ = |↑ ... ↑i and ~ ↓ = |↓ ... ↓i denote the state of the physical qubits, ¯ x k is labeling M logical qubits, and N is the totalnumber of QEC steps.After integrating all the modes inside a hypercube, we define a renormalized coupling λ ∗ and a local error probability ǫ .4Finally, we evoke again Wick’s theorem to write ρ ~ ↑ ,~ ↓ (Υ ) = D ψ | ~ ↓ E D ~ ↑| ψ E − A − N M ǫ − ǫ (cid:16) λ ∗ ∆2 ~ (cid:17) P ¯ x , ¯ y R N ∆0 dt R t dt (cid:10) : f (¯ x , t ) : : f (¯ y , t ) : (cid:11) + ... − A + N M ǫ + ǫ (cid:0) λ ∗ ∆2 ~ (cid:1) P ¯ x , ¯ y R N ∆0 dt R t dt h : f (¯ x , t ) : : f (¯ y , t ) : i + ... , (88)where A is a number proportional to ǫ and λ ∗ . Hence, for ǫ, λ ∗ ≪ , this simplifies to [53] ρ ~ ↑ ,~ ↓ (Υ ) ≈ D ψ | ~ ↓ E D ~ ↑| ψ E " − N M ǫ − ǫ (cid:18) λ ∗ ∆2 ~ (cid:19) Z d x Z d y Z N ∆0 dt Z t dt (cid:10) : f (¯ x , t ) : : f (¯ y , t ) : (cid:11) + . . . . (89)If we now recall the two-point correlation function of Eq. (7),it becomes clear that the corrections due to correlations arerelevant when D + z > δ . D. Relation to the work of Aharonov, Kitaev, and Preskill
The study of correlated noise has been a central problemfor quite some time. Among the most recent advances is apaper by Aharonov, Kitaev, and Preskill (AKP) [26]. Using amethod completely different from ours, AKP proved that: Fora computer where qubits are interacting through an instanta-neous interaction of the form λ / ∆ x δ , it is possible to proveresilience for λ < λ c and D − δ < . The key distinctionbetween the work of AKP and ours is the instantaneous na-ture of their interaction. Hence, while in our work each qubitis inside a distinct hypercube, for AKP they are all containedin a single hypercube. There is however a trade-off. Sincetheir interaction is instantaneous and perfect error correctionis assumed, there is no propagation of errors in time throughthe gauge field of the environment. Hence, effectively, AKPare considering a model with z = 0 . As a result, our Eq. (79)holds in the case they analyzed as well. V. THRESHOLD THEOREM AS A QUANTUM PHASETRANSITION
The main result of fault-tolerant quantum computation isthe threshold theorem. The theorem states that if a stochasticerror probability ǫ is smaller than a critical value ǫ c , then theintroduction of an additional layer of concatenation improvesthe protection of the information. Hence, for a fixed ǫ , it ispossible to sustain a quantum computation for any desire timeat the cost of some reasonable additional hardware overhead.Even though quantum computation and QEC are out-of-equilibrium problems, it is intuitive to talk about differentphases in the computer-environment parameter space. Alongthis line of thought, each phase corresponds to a distinctsteady state. A natural choice for an order parameter is thatgiven by the entanglement among the qubits and the environ-ment. We summarize our thinking in Fig. 5, where we presenta schematic phase diagram for a quantum computer runningQEC. For stochastic noise models, such an idea was explored byAharonov [18]. Following that work, we can separate the be-havior of the computer into two distinct regimes:(i) For ǫ < ǫ c , the computer components can maintainlarge entanglement through fault-tolerant procedures, whichin turns means that the computer and the environment areweakly entangled. Hence, due to this large internal entangle-ment, the quantum computer departs from the classical com-puter model. We can formalize these remarks by remember-ing that QEC tries to keep the system in the “steady state”described by the reduced density matrix. In order to keep thenotation simple, lets take the ideal computer state as a purestate, ρ ( t ) = | ψ i h ψ | , (90)with | ψ i = P i α i ( t ) | i i expressed in terms of the computa-tional basis {| i i} . As consequence, it has a reduced entropy S ≈ . In this case, if we look at the full Hilbert space (that is,before tracing out the environment), we find the tensor state | Ψ i ≈ | ψ i ⊗ (cid:12)(cid:12) ϕ environment (cid:11) . (91)(ii) For ǫ > ǫ c , the computer components are weakly en-tangled and, therefore, can be efficiently simulated by a Tur-ing machine. In other words, the computer density matrix nolonger represents a pure state, but rather a statistical mixture.Thus, the computer components are strongly entangled withthe environment. This corresponds to a steady state with alarge reduced entropy (in the limit of ǫ → , S ≈ N ln 2 , with N the number of qubits).In such a description, we see that ǫ plays a role analo-gous to an effective temperature [54]. Hence, the thresholdtheorem defines a phase transition from a high-temperaturephase, where qubits are independent from each other, to alow-temperature phase, where quantum coherence and entan-glement are possible [18]. This also sheds new light on therole of periodic measurements in QEC: They can be seen asa refrigeration that extracts entropy from the computer (verymuch like the Schulman-Vazirani initialization procedure [55]or the transfer of entanglement to fresh ancillas [34]). If theentropy production in the computer is below a certain level,then the computer can be kept in its “low-temperature” phase.Our analysis of correlated noise also fits perfectly into thisdescription. The dimension criterion provided by Eq. (79) isthe hallmark of a quantum phase transition [56]. For D + z < (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) "upper critical dimension""lower critical dimension" δ (efficiently simulated by a Turing machine) l o ca l e rr o r p r ob a b ilit y ( ε ) noisy quantum computer compute traditionalunconventionalthreshold theorem threshold theoremnot possible to FIG. 5: (Color online) Phase diagram of a quantum computer running QEC. The parameter δ is the scaling dimension of the environmentoperator, D is the dimensionality of the computer, and z is the dynamical exponent of the environment [see discussion preceding Eq. (79)]. Inthe red phase, qubits and environment are strongly entangled causing strong decoherence. In the light blue phase, QEC keeps the qubits andenvironment disentangled, making computation possible. δ , V can only produce small corrections to the stochastic er-ror model. The steady state of the system is therefore given byEq. (91). There is a clear separation of scales and the thresh-old theorem holds as it is. Conversely, for D + z > δ , there isno clear separation of scales. The computer and the environ-ment become increasingly entangled and the system is driventowards a different steady state. Such a state is probably dis-tinct from the “high temperature” one and it is likely that it ischaracterized by a smaller residual entropy.This does not mean that for D + z > δ it would not bepossible to perform quantum computation. It only means thatthe threshold theorem as we stated it does not hold. It is con-ceivable that some different derivation of the theorem existsin this case. In this sense, D + z = 2 δ defines what is usuallyreferred to as the upper critical dimension of the model (seeAppendix B). Below the upper critical dimension, there canbe substantial corrections to the steady state given by Eq. (91),but it may still be possible to prove resilience. The questionthat remains open is whether a lower critical dimension exists,namely, a criterion for V that would tell us when it is impos-sible to perform long-time quantum computation. VI. SUMMARY AND CONCLUSIONS
Most previous discussions of QEC have used the quantummaster equation and quantum dynamical semi-groups [57].This is a very natural approach: The computer is the objectof interest; hence, one starts the discussion by integrating outthe environmental degrees of freedom. However, the pricepaid in this approach is that some simplification is needed inorder to derive the quantum master equation [25, 57]. Theusual assumption is the Born-Markov approximation [25]. Inthat case, it is natural to define an error probability for a givenqubit, and a discussion in terms of error models naturally fol-lows [1, 29]. The situation is much less clear when the Born-Markov approximation cannot be justified [28, 58]. In thiscase, temporal and spatial correlations can build up and com- pletely destroy the notion of the probability of an error.A key characteristic of the discussion here is that we do nottry to use a quantum master equation. Rather, we follow theapproach put forward by Schwinger and Keldysh [42, 43, 44]to study out of equilibrium systems. The main conceptual dif-ference is that we trace the environmental degrees of freedomonly at the very last step of the calculation. Hence, we canmake the most of the unitary evolution of a quantum mechan-ical system.Following this “Schwinger-Keldysh” approach, we dis-cussed the evolution of a quantum computer operated withfast and slow gates. On the one hand, for fast gates the mi-croscopic Hamiltonian is the one relevant for the evolution ofthe computer, Eq. (23). On the other hand, for slow gates wedemonstrated that a suitable effective Hamiltonian, Eq. (40),can be used to provide an upper bound for the discussion ofdecoherence. With this effective Hamiltonian, the notationcan be unified, and both cases treated simultaneously. We de-rived two formal expressions that quantify the evolution ofthe computer under QEC in a correlated environment: (i) theprobability of a given syndrome history, Eq. (54), and (ii) thereduced density matrix of the computer, Eq. (55).In order to fully use standard QEC theory, we introducedthe important assumption of “hypercubes”, that is a minimumspatial distance between qubits, Eq. (57), in order to allow thedefinition of an error probability for a single qubit. With this“hypercube assumption”, it is straightforward to use Wick’stheorem to separate the environmental modes into intra- andinter-hypercube parts. The intra-hypercube component de-fines the error probability, while the inter-hypercube part istracked by an operator acting on the coarse-grained scale ofthe hypercubes. As examples, we treated a generalization ofthe spin-boson model and a quantum frustrated model.All the pieces are put together when we explicitly calculatethe probability of a syndrome history (Sec. IV B) and associ-ated residual decoherence (Sec. IV C). The main result is castas a dimensional criterion, Eq. (79). Finally, we discuss theparallels between the threshold theorem and a quantum phase6transition. A qualitative description of the possible fates of aquantum computer as a function of noise strength and degreeof correlation is given in Fig. 5.There are several clear directions in which our results couldbe extended or improved. First, it would obviously be desir-able to relax the hypercube assumption introduced in Sec. IV.There is nothing intrinsic to our approach which makes thisassumption necessary. Yet, progress without it seems muchmore difficult: The notion of a local error probability during asingle QEC cycle becomes problematic, making the connec-tion with analysis based on error models, such as the usualderivation of the threshold theorem, unclear.Second, non-instantaneous gate operation is clearly a deli-cate issue. By using a bound (Sec. II C), we are able to treatthis case in the same way as the fast-gate case. Thus we derivean upper bound for the local error probability together with thedimensional criterion. If a more accurate value for the errorprobability is desired, a specific error correction code as wellas the gates under consideration must be included in the analy-sis. However, the scaling argument and resulting dimensionalcriterion do not, in general, change.Note that it is possible to change the dimensional criterionfor the better (but not for the worse) by using the separationof scales introduced by QEC. Particular pulse sequences canreduce correlation at long times at the cost of increasing thelocal error probability. One example was given in our previouswork [39, 40].Finally, there may be a regime of parameters where, as indi-cated in Fig. 5, fault-tolerant quantum computation is possibleeven though the presently known derivations of the thresh-old theorem do not apply. By analogy with phase transi-tion phenomenology, there may be a lower critical dimen-sion such that a more sophisticated analysis than the one wepresent here shows that fault-tolerant computation is possiblefor δ < ( D + z ) / . It would be very interesting to show in anyexample that such is, or is not, the case.Quantum Error Correction is one of the most interestingframeworks which allows long quantum computations [59].Even though QEC is widely accepted, it has been argued thatit relies on a set of unphysical assumptions [22, 25, 60, 61],namely: (i) “fast” measurements, (ii) “fast” gates, and (iii) de-scribing decoherence by error models. Although these are le-gitimate concerns, it is now clear that they are not fundamen-tal: First, in Ref. 35 DiVincenzo and Aliferis demonstratedthat resilient circuits can be constructed with slow measure-ments. Second, in the current paper, we have demonstratedthat the fast gate assumption is not critical for fault tolerance.Finally, we have laid the groundwork here for a theoreticalframework that connects microscopic Hamiltonians with er-ror models in correlated environments. From our results forthe threshold theorem in conjunction with those of AKP [26],it is clear that a large class of correlated environments are al-ready properly treated within the QEC framework. Acknowledgments
We thank C. Kane, D. Khveshchenko, and R. Plesser foruseful discussions. This work was supported in part by NSFGrants No. CCF 0523509 and No. CCF 0523603. E.R.M. ac-knowledges partial support from the Interdisciplinary Infor-mation Science and Technology Laboratory (I Lab) at UCF.
APPENDIX A: ABSOLUTE CONVERGENCE OFDYSON’S SERIES
Dyson’s series is absolutely convergent for any bound op-erator evolving for any finite time [62]. This is particularlysimple to see using the sup operator norm [28], || A || = sup Ψ q h Ψ | A † A | Ψ i , (A1)where || Ψ || = 1 . If P = R t dt ′ || V ( t ′ ) || < ∞ , then thenorm of the m th -order term in Dyson’s series is bounded by P m /m ! . Thus, using the convergence of the exponential se-ries, we find that Dyson’s series is absolutely convergent. APPENDIX B: PERTURBATIVE EXPANSION IN φ THEORY
A classic example of a quantum phase transition is givenby the φ theory at criticality [63]. The model is compactlydescribed by the Euclidean action S = Z L d D r Z β dτ h ( ▽ r φ ) + ( ∂ τ φ ) + λφ i . (B1)The scaling dimension of the free field is usually defined as dim [ φ ] = ν/ . If we expand the partition function in pow-ers of λ , it is simple to see that each order in the pertur-bative expansion will have the power λ ( Lβ ) D +1 − ν . Hence, D + 1 − ν < is the criterion for the irrelevance of the pertur-bation. The simplest way to see that is to do power countingby rescaling space and time, r → br, τ → bτ, φ → b − ν φ, (B2)which immediately gives S = b D − − ν Z d D r Z dτ h ( ▽ r φ ) + ( ∂ τ φ ) i + λ b D +1 − ν Z d D r Z dτ φ . (B3)One finds the scaling λ → λb D +1 − ν , which is valid at eachorder of the perturbative expansion. The criterion for the ir-relevance of the perturbation is D + 1 − ν < There is one more important definition that this exampleprovides. Since the Gaussian action must be scale invariant,we automatically see that for this example ν = D − . Hence,the criteria for the irrelevance of λφ term as a perturbation7can be rewritten as − D < . This defines the upper criticaldimension for the model as d upper c = 4 (three spatial and onetemporal). When a system is above its upper critical dimen-sion, the physics is controlled by the Gaussian action. How-ever, when the system is below its upper critical dimension,there are substantial corrections to physical quantities whencompared with the Gaussian solution. APPENDIX C: HILBERT SPACE OF QUBITS
Due to the state vector normalization, the Hilbert space ofa qubit is isomorphic to a three-dimensional sphere S : For ageneral state | ψ i = α | i + β | i , we have the constraint ( Re α ) + ( Im α ) + ( Re β ) + ( Im β ) = 1 . (C1)However, an overall phase is physically irrelevant and the cor-rect mapping is to the complex projective plane of complexdimension , S /U (1) → CP . (C2)For the same reason, the Hilbert space of n qubits is isomor-phic to CP n − . For the discussion of entanglement, thereis a particularly important subspace of this space. It is com-posed by the direct product of each qubit Hilbert space minusan over all phase, n Y j =1 CP j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) modulus phase ⊂ CP n − , (C3)where j labels the j th qubit’s Hilbert space. The dimension ofthe subspace grows as n − while the dimension of the entireHilbert space grows as n − . Entangled states are defined asthe complementary set of this special subspace. APPENDIX D: DECOHERENCE IN THE SPIN-BOSONMODEL WITH OHMIC DISSIPATION
An example of a qubit coupled to an environment is thespin-boson model with ohmic dissipation [64, 65], which wasintensively studied in the context of quantum computation [2,66] even before quantum error correction was introduced. Inthis model, a qubit evolves according to the Hamiltonian H = Z dx h ( ∂ x φ ) + Π i + λ ∂ x φ (0) σ z , (D1)where φ is a chiral bosonic field, ~σ are Pauli matrices thatdescribe the qubit located at x = 0 , and λ is the environment-qubit coupling constant. If a qubit is prepared in an initialstate | ψ i = α |↑i + β |↓i , (D2)at large enough times, Λ − ≪ t ≪ ( k B T ) − , its densitymatrix evolves as ρ ( t ) = (cid:20) | α | αβ ∗ e − λ ln(1+Λ t ) α ∗ β e − λ ln(1+Λ t ) | β | (cid:21) , (D3) with Λ denoting the environment ultraviolet cutoff frequency.Since states with either α or β equal to zero do not experi-ence decoherence, they are called classical states. They de-fine a pointer basis. Conversely, any superposition state with α, β = 0 suffers decoherence and over a long time becomes astatistical mixture of the classical states.As one includes more qubits, the entries in the reduced den-sity matrix will decay faster as one moves away from the di-agonal. In the case where qubits are coupled to independentbaths, it is simple to see that the off-diagonal matrix elementsdecay as ρ ~p,~q (cid:0) t ≫ Λ − (cid:1) = ρ e − λ ( p − q ) ln(1+Λ t ) , (D4)where p and q are the total magnetization of the states ~p and ~q ,respectively [2]. The case of a common bath is also straight-forward, [2], and the result for qubits separated by a distancesmaller than Λ − is ρ ~p,~q (cid:0) t ≫ Λ − (cid:1) ≈ ρ e − λ ( p − q ) ln(1+Λ t ) . (D5)Some entangled states do not suffer decoherence (a sin-glet state, for example). However, these correspond to avery special and small decoherence-free subspace. In gen-eral, entangled states are made of quantum superpositions andtherefore have components in the off-diagonal entries of thedensity matrix. Hence, studying decoherence (the decay ofthe off-diagonal elements of the density matrix) is essentiallyequivalent to studying how entanglement between qubits isdestroyed by interaction with the environment. APPENDIX E: INTERACTION PICTURE
Since [ H , H QC ] = 0 , we can define the interaction picture O ( t ) = e i ~ H t R † ( t ) O R ( t ) e − i ~ H t , (E1) | Ψ( t ) i = e i ~ H t R † ( t ) ˜ U ( t ) | Ψ(0) i , (E2)where ˜ U ( t ) is the exact evolution operator, defined as U ( t ) = T t e − i ~ R t dt ′ H ( t ′ ) , (E3)and | Ψ i is the total state vector (computer plus environment).Now, let us consider the time evolution of | Ψ i , ddt | Ψ( t ) i = ddt e i ~ H t R † ( t ) ˜ U ( t ) | Ψ(0) i = − i ~ V ( t ) | Ψ( t ) i . (E4)Thus, we obtain the usual definition for the evolution operatorin the interaction picture ˜ U ( t ) = e i ~ H t R † ( t ) ˜ U ( t ) = T t e − i ~ R t dt ′ V ( t ′ ) . (E5)8 APPENDIX F: LOW FREQUENCY CONTRIBUTION TOTHE ERROR PROBABILITY
The simplest way to understand F α is to write f in its fre-quency representation υ α ( x , λ ∗ α ) ≈ λ ∗ α Z ∆0 dt f α ( x , t ) ≈ λ ∗ α Z ∆0 dt Z Λ0 dω e iωt f α ( x , ω ) ≈ λ ∗ α Z ∆0 dt Z ∆ − dω + Z Λ∆ − dω ! e iωt f α ( x , ω ) ≈ λ ∗ α Z ∆0 dt (cid:2) f >α ( x , t ) + f <α ( x , (cid:3) , (F1)where < stands for frequencies smaller than ∆ − and > for the frequencies between ∆ − and Λ . Thus, using that h f <α f >α i = 0 , we obtain υ α ( x , λ ∗ α ) ≈ ( λ ∗ α ) Z ∆0 dt dt f > † α ( x , t ) f >α ( x , t )+ ( λ ∗ α ∆) f < † α ( x , f <α ( x , . (F2) APPENDIX G: SCALING DIMENSION OF F α If the two-point correlation function of f α can be expressedas h f α ( x , t ) f α ( x , t ) i ∼ F (cid:18) x ) δ , t ) δ/z (cid:19) , (G1)the scaling dimension of f α is defined as dim f α = δ . UsingWick’s theorem, D : | f α ( x , t ) | : : | f α ( x , t ) | : E = (cid:10) f † α ( x , t ) f † α ( x , t ) (cid:11) h f α ( x , t ) f α ( x , t ) i + (cid:10) f † α ( x , t ) f α ( x , t ) (cid:11) (cid:10) f α ( x , t ) f † α ( x , t ) (cid:11) = 2 " F x ) δ , t ) δ/z ! . (G2)Therefore, dim F α = 2 δ . [1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press, Cam-bridge UK, 2000).[2] W. G. Unruh, Phys. Rev. A , 992 (1995).[3] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. , 2594 (1998).[4] L. Viola and S. Lloyd, Phys. Rev. A , 2733 (1998).[5] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. , 3520(2000).[6] D. A. Lidar and S. Schneider, Quant. Inf. Comp. , 350 (2005).[7] L. F. Santos and L. Viola, Phys. Rev. Lett. , 150501 (2006).[8] A. M. Steane, Phys. Rev. Lett. , 793 (1996).[9] A. Steane, Proc. R. Soc. Lond. A , 2551 (1996).[10] A. R. Calderbank and P. W. Shor, Phys. Rev. A , 1098 (1996).[11] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane,IEEE Trans. Inf. Theory , 1369 (1998).[12] E. Knill and R. Laflamme, Phys. Rev. A , 900 (1997).[13] D. Gottesman, Phys. Rev. A , 127 (1998).[14] J. Preskill, Introduction to Quantum computation and informa-tion (World Scientific Publishing Co., Singapore, 1998), chap.Fault-Tolerant Quantum Computation, pp. 213–269.[15] E. Knill, R. Laflamme, and W. H. Zurek, Science , 342(1998). [16] D. Aharonov and M. Ben-Or, arXiv:quant-ph/9906129 (1999).[17] D. Gottesman, Lect. Notes Comput. Sci. , 302 (1999).[18] D. Aharonov, Phys. Rev. A , 062311 (2000).[19] E. Knill, R. Laflamme, and W. H. Zurek, Science , 2395(2001).[20] A. M. Steane, Phys. Rev. A ,042322 (2003).[21] E. Knill, Nature , 39 (2005).[22] R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, Phys.Rev. A , 062101 (2002).[23] J. P. Clemens, S. Siddiqui, and J. Gea-Banacloche, Phys. Rev.A , 062313 (2004).[24] R. Klesse and S. Frank, Phys. Rev. Lett. , 230503 (2005).[25] R. Alicki, D. A. Lidar, and P. Zanardi, Phys. Rev. A , 052311(2006).[26] D. Aharonov, A. Kitaev, and J. Preskill, Phys. Rev. Lett. ,050504 (2006).[27] E. Knill, R. Laflamme, and L. Viola, Phys. Rev. Lett. , 2525(2000).[28] B. M. Terhal and G. Burkard, Phys. Rev. A , 012336 (2005).[29] P. Aliferis, D. Gottesman, and J. Preskill, Quant. Inf. Comp. ,97 (2006).[30] B. W. Reichardt, Lecture Notes in Computer Science (Springer,New York, 2006),Vol. 4051, pp.50-61. [31] As usual in field theories, short distances always need regular-ization.[32] W. H. Zurek, Rev. Mod. Phys. , 715 (2003).[33] N. Linden and S. Popescu, Phys. Rev. Lett. , 47901 (2001).[34] J. Preskill, lecture notes, URL .[35] D. P. DiVincenzo and P. Aliferis, Phys. Rev. Lett. , 020501(2007).[36] A. M. Steane, Phys. Rev. A , 4741 (1996).[37] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane,Phys. Rev. Lett. , 405 (1997).[38] Static imperfections can be dealt using randomization tech-niques [67].[39] E. Novais and H. U. Baranger, Phys. Rev. Lett. , 040501(2006).[40] E. Novais, E. R. Mucciolo, and H. U. Baranger, Phys. Rev. Lett. , 040501 (2007).[41] L. M. K. Vandersypen, Ph.D. thesis, Stanford University(2001), arXiv:quant-ph/0205193v1.[42] G. D. Mahan, Many-Particle Physics , (Kluwer Academic /Plenum Publishers, New York, 2000), 3rd ed.[43] J. Schwinger, J. Math. Phys. , 407 (1961).[44] L. V. Keldysh, Sov. Phys. JEPT , 1018 (1965).[45] R. Shankar, Rev. Mod. Phys. , 129 (1994).[46] Though the qubit operators σ α ( x ) have no explicit time de-pendence, a time argument is needed here to keep track of theproper order after application of T t .[47] I. Affleck and A. W. W. Ludwig, Nucl. Phys. B , 641(1991).[48] A. H. Castro Neto, E. Novais, L. Borda, G. Zar´and, and I. Af-fleck, Phys. Rev. Lett. , 096401 (2003).[49] E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, andG. Zar´and, Phys. Rev. B , 014417 (2005).[50] E. Novais, F. Guinea, and A. H. Castro Neto, Phys. Rev. Lett. , 170401 (2005).[51] W. H. Zurek, Phys. Rev. D , 1516 (1981).[52] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. , 2417(1999).[53] The fact that the corrections in Eq. (89) start at order ǫ is adirect demonstration of the efficacy of QEC. [54] A spin lattice model is quite appropriate to illustrate the anal-ogy. Suppose that we are looking at the 3-D Heisenberg spinHamiltonian, H = P h ij i ~S i · ~S j , where h ij i denotes nearestneighbors spins. The ground state of this Hamiltonian is a sin-glet (anti-ferromagnetic phase), hence it exhibits a very strongentanglement among the spins. At low temperatures, this fea-ture dominates the spin dynamics and the system is in its quan-tum phase. Conversely, at very high temperatures, all spin con-figurations are equally probable. The Hamiltonian cannot driveany coherent motion and the spins effectively decouple (i.e, thesystem is in a paramagnetic phase).[55] L. J. Schulman and U. V. Vazirani, in Proc. 31’st STOC (ACMSymp. Theory Comp.) (ACM Press, 1999).[56] S. Sachdev,
Quantum Phase Transitions (Cambridge UniversityPress, Cambridge UK, 1999).[57] H.-P. Breuer and F. Petruccione,
The Theory of Open QuantumSystems (Oxforf University Press Inc., New York, NY, 2002).[58] D.P DiVincenzo and D. Loss, Phys. Rev. B , 035318 (2005).[59] The other one is topological quantum computation [68, 69].[60] R. Alicki, M. L. Horodecki, P. L. Horodecki, and R. Horodecki,Open Sys. Inf. Dyn. , 205 (2004).[61] R. Alicki (2007), arXiv:quant-ph/0702050.[62] J. A. Oteo and J. Ros, J. Math. Phys. , 3268 (2000).[63] J. W. Negele and H. Orland, Quantum Many-Particle Systems (Perseus Books, Reading, Massahusetts, 1998).[64] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher,A. Garg, and W. Zwerger, Rev. Mod. Phys. , 1 (1987).[65] M. Grifoni, E. Paladino, and U. Weiss, Eur. Phys. J. B , 719(1999).[66] J. H. Reina, L. Quiroga, and N. F. Johnson, Phys. Rev. A ,032326 (2002).[67] O. Kern, G. Alber, and D. Shepelyansky, Euro. Phys. J. , 153(2005).[68] A. Yu. Kitaev (1997), arXiv:quant-ph/9707021.[69] A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classicaland Quantum Computation , vol. 47 of