Topological quantum order: stability under local perturbations Sergey Bravyi, Matthew Hastings, and Spyridon Michalakis 0 ∗ † ‡ 1 0 January 3, 2010 2 n a J 3 Abstract ] We study zero-temperature stability of topological phases of matter under weak time- h p independentperturbations. OurresultsapplytoquantumspinHamiltoniansthatcanbewritten - t as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain n topological order conditions. Given such a Hamiltonian H we prove that there exists a con- a 0 u stant threshold ǫ > 0 such that for any perturbation V representable as a sum of short-range q bounded-norm interactions the perturbed Hamiltonian H = H +ǫV has well-defined spectral [ 0 bands originating from O(1) smallest eigenvalues of H . These bands are separated from the 0 1 rest of the spectrum and from each other by a constant gap. The band originating from the v 4 smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). 4 Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively 3 bounded operators, and the Lieb-Robinson bound. 0 . 1 0 0 1 : v i X r a ∗IBM Watson Research Center, Yorktown Heights NY 10594 (USA); [email protected] †Microsoft Research Station Q, CNSI Building, University of California, Santa Barbara, CA, 93106 (USA); [email protected] ‡T-4 and CNLS, LANL - Los Alamos, NM, 87544 (USA); [email protected] 1 Contents 1 Introduction 3 1.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Sketch of the stability proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Hamiltonians describing TQO 7 2.1 Frustration-free commuting Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Formal definition of TQO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Verification of TQO conditions for stabilizer Hamiltonians . . . . . . . . . . . . . 10 2.4 Unstable version of the toric code model . . . . . . . . . . . . . . . . . . . . . . . 11 3 Relatively bounded perturbations 12 3.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Stability of TQO under block-diagonal perturbations . . . . . . . . . . . . . . . . 13 4 Hamiltonian flow equations 16 4.1 Outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Local decompositions of Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Linearized block-diagonalization problem 23 5.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Finding the transformation S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.3 Local decomposition of the transformed Hamiltonian . . . . . . . . . . . . . . . . 28 6 Lieb-Robinson bounds 29 7 Adiabatic continuation of logical operators 34 7.1 Dressed Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 1 Introduction The traditional classification of different phases of matter due to Landau rests on symmetry breaking. GivenapairofgappedHamiltoniansH ,H withsomesymmetrygroupG,theground 1 2 statesofH andH wereconsideredtobeindifferentphasesiftheirsymmetrybreakingpatterns 1 2 are different. The discovery of topologically ordered phases, however, changes this paradigm. Models such as Kitaev’s toric code [1] have “topologically non-trivial” ground states despite lacking any symmetry breaking. Such states cannot be changed into a “topologically trivial” state such as a product state by any unitary locality-preserving operator [2]. One possible approach to classifying topological phases is to call a pair of gapped Hamilto- nians H ,H topologically equivalent iff it is possible to connect H and H by a continuous 1 2 1 2 path in the space of local gapped Hamiltonians. Using the idea of quasi-adiabatic continua- tion [3], one can describe the evolution of the ground state subspace along such a path by a unitary locality-preserving operator. In particular ground state degeneracy and the geometry of “logical operators” acting on the ground subspace is the same for H and H . 1 2 MostoftheHamiltoniansdescribingTQOmodelssuchasKitaev’squantumdoublemodel[1] orLevin-Wenstring-netmodel[4]arenotquitephysicalsincetheyinvolveinteractionsaffecting more than two spins at a time. One may hope however that such models emerge as low-energy effective Hamiltonians describing some simpler high-energy theories [5, 6, 7]. For example, the toric code model with four-spin interactions can be “implemented” as the fourth-order effective Hamiltonian describing low-energy limit of the honeycomb model [8] which involves only two- spin interactions. The higher-order corrections to the effective Hamiltonian must be regarded as a perturbation. Thus in order to show that the honeycomb model is topologically equivalent to the toric code (in the J J ,J phase) one has to prove that the spectral gap in the toric z x y ≫ code Hamiltonian does not close in a presence of weak perturbations V that can be represented as a sum of bounded-norm short-range (exponentially decaying) interactions. Even if one leaves aside the question of how multi-spin interactions can be implemented in a lab, one has to worry about precision up to which an ideal model Hamiltonian can be approximated in a real life. If the presence or absence of the gap depends on tiny variations of the Hamitonian parameters that are beyond experimentalist’s control, the distinction between gapped and gapless Hamiltonians is meaningless. The best we can hope for is to approximate individual interactions of the ideal model with some constant precision ǫ independent of the systemsizeN. Accordingly, theidealHamiltoniancanbeapproximatedonlyuptoanextensive error O(ǫN). Proving stability of topological phases thus reduces to proving that the spectral gap of the ideal TQO models does not close in the presence of such extensive perturbations. Currently, the tools for proving lower bounds on the spectral gap are fairly limited. For example, one of the outstanding problems in mathematical physics is to prove the existence of a spectral gap for the spin-1 Heisenberg chain, making rigorous the arguments of Haldane [9]. Some progress toward this was obtained by Yarotsky [10], who showed the stability of the gap neartheAKLTpoint[11]. Yarotsky’stoolshoweverarelimitedtoperturbationsofHamiltonians which are topologically trivial. Thus, new methods are needed to analyze topologically ordered phases. Some partial results were recently obtained by Trebst et al [12] and Klich [13] who proved gap stability for the toric code under a special type of perturbations diagonal in the 3 z-basis as well as for anyon lattices on a sphere. In the present paper we succeed in proving gap stability under generic local perturbations. Our results are valid not just for the toric code, but more generally for any Hamiltonian which can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions that we define later. This includes models such as Kitaev’s quantum double model [1] and the Levin-Wen string-net model [4]. Furthermore, we prove stability of the spectral gaps separating sufficiently low-lying eigenvalues of the unper- turbed Hamiltonian. In the case of 2D models with anyonic excitations it allows us to define string-like operators that create particle excitations for the perturbed Hamiltonian and prove stability of invariants describing the braiding statistics of excitations. We explain how this may be used to adiabatically control a perturbed topological model to perform braiding operations to manipulate topologically protected quantum information. 1.1 Summary of results Consider a system composed of finite-dimensional quantum particles (qudits) occupying sites of a D-dimensional lattice Λ of linear size L. The corresponding Hilbert space is a tensor product of the local Hilbert spaces, = , dim( ) = O(1). Suppose the unperturbed H u∈ΛHu Hu Hamiltonian H0 can be written as a sumNof geometrically local pairwise commuting projectors, H = Q , 0 A AX⊆Λ where the sum runs over all subsets of the lattice of diameter O(1) and Q is a projector acting A non-trivially only on sites of A (one may have Q = 0 for some subsets A). The commutativity A assumption implies that all projectors Q can be diagonalized in the same basis. Accordingly, A all eigenvalues of H are non-negative integers. We assume that the smallest eigenvalue of H 0 0 is zero, that is, ground states of H are annihilated by every projector Q . Such states span 0 A the ground subspace P, P = ψ : Q ψ = 0 for all A Λ . A {| i ∈ H | i ⊆ } For any subset B Λ we shall also define a local ground subspace as ⊆ P = ψ : Q ψ = 0 for all A B . B A {| i ∈ H | i ⊆ } We shall use the notations P and P both for linear subspaces and for the corresponding B projectors. Note that the projector P acts non-trivially only on the subset B. B We shall impose two extra conditions on H and the ground subspace P that guarantee the 0 gapstability. Letusfirststatetheseconditionsinformally(seeSection2.2forformaldefinitions): TQO-1: The ground subspace P is a quantum code with a macroscopic distance1, TQO-2: Local ground subspaces are consistent with the global one 1 For our purposes it suffices that the distance grows as a positive power of the lattice size L 4 ConditionTQO-1isthetraditionaldefinitionofTQO.Itguaranteesthatalocaloperatorcannot induce transitions between orthogonal ground states or distinguish a pair of orthogonal ground states from each other. Thus a local perturbation can lift the ground state degeneracy only in the n-th order of perturbation theory, where n can be made arbitrarily large by increasing the lattice size, see [1]. Surprising, condition TQO-1 by itself is not sufficient for stability, see a simple counter-example in Section 2.4. ConditionTQO-2demandsthatalocalgroundsubspaceP andtheglobalgroundsubspace B P must be consistent, namely, the projectors P and P must have the same reductions on any B subset A B which is ”sufficiently far” from the boundary of B. We need to impose TQO-2 ⊂ only for regions with trivial topology such a cube or a ball. The consistency between the global and the local ground subspaces may be violated for regions with non-trivial topology. For example, if B has a hole, the local ground subspace P may include sectors with a non-trivial B topological charge inside the hole as opposed to the global ground subspace. Condition TQO-2 by itself is also not sufficient for stability, see a counter-example in Section 2.4. Let us emphasize that all our results apply also to the special case when H has non- 0 degenerate ground state. In this case TQO-1 is automatically satisfied since P is a one- dimensional subspace and thus condition TQO-2 alone guarantees the gap stability. We consider a perturbation V that can be written as a sum of bounded-norm interactions V = V , r,A Xr≥1 AX∈S(r) where (r) is a set of cubes of linear size r and V is an operator acting on sites of A. We r,A S assume that the magnitude of interactions decays exponentially for large r, max V Je−µr, r,A A∈S(r)k k ≤ where J,µ > 0 are some constants independent of L. Our main result is the following. Theorem 1. Suppose H obeys TQO-1,2. Then there exist constants J ,c ,c > 0 depending 0 0 1 2 only on µ and the spatial dimension D such that for all J J the spectrum of H + V is 0 0 ≤ contained (up to an overall energy shift) in the union of intervals I , where k runs over k≥0 k the spectrum of H0 and S I = λ R : k(1 c J) δ λ k(1+c J)+δ , k 1 1 { ∈ − − ≤ ≤ } and δ = poly(L)exp( c L3/8). 2 − In other words, the perturbation V changes positive eigenvalues of H at most by a constant 0 factor 1 c J (neglecting the exponentially small correction δ) while the smallest eigenvalue 1 ± k = 0 is transformed into a band I of exponentially small width 2δ, see Fig. 1. One can easily 0 5 check that for any fixed k the band I is separated from all other bands I , m = k, by a gap k m 6 at least 1/2 provided that J < J , where k 1 J = . k c (4k+2) 1 Thus the bands originating from eigenvalues 0,1,...,k of H are separated from each other and 0 from the rest of the spectrum by a gap at least 1/2 provided that J < J . k In the case when excitations of H are anyons, one can infer all topological invariants such 0 as S, R, and F-matrices by evaluating fusion and braiding diagrams with only a few particles (for example 4 particles suffice to compute all F matrices). Accordingly, any matrix element of, say, F-matrix, canberepresentedasanexpectationvalue ψ O ...O O ψ where ψ isthe 0 m 2 1 0 0 h | | i | i ground state and O are operators creating pairs of excitations from the ground state, moving, i fusing,andannihilatingthem. OurstabilityresultforexcitedstateswithO(1)excitationsallows us to construct quasi-adiabatic continuation of operators O thus explicitly demonstrating that i the perturbed Hamiltonian has the same S,R, and F matrices as the ideal one, see Section 7. Energy (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 2(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)I (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 2 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 1 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)I (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 1 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 0 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)I0 J Figure 1: Energy bands I describing the spectrum of a perturbed Hamiltonian H +V. k 0 1.2 Sketch of the stability proof Let us sketch the main steps of the proof of Theorem 1. We start from proving the theorem for a special class of perturbation V such that all individual interactions V preserve the ground r,A subspace P, that is, [P,V ] = 0. We call such perturbations block-diagonal. In Section 3 we r,A prove that block-diagonal perturbations are relatively bounded by H , that is, Vψ b H ψ 0 0 k k ≤ k k for any state ψ and for some coefficient b = O(J). Here for simplicity we ignore some | i ∈ H exponentially small corrections. A nice feature of relatively bounded perturbations is that the spectrum of a perturbed Hamiltonian H +V is contained in the union of intervals I where k 0 k runs over the spectrum of H and I = (k(1 b),k(1+b)), see Section 3.1. The proof of the 0 k − relative boundness is rather elementary and uses certain decomposition of the Hilbert space in terms of syndrome subspaces which is a standard tool in the theory of quantum error correcting codes. In order to get a strong enough bound on the coefficient b we use a novel technique of “coarse-graining” the syndrome subspaces, see Section 3.2 for details. 6 In the second part of the proof we reduce generic perturbations V to block-diagonal pertur- bations. Specifically, we construct a unitary operator U such that U(H +V)U† H +W, 0 0 ≈ where W is a block-diagonal perturbation. Since U does not change eigenvalues, we can use the techniques described above to analyze the spectrum of H + V. The operator U is con- 0 structed using a discrete version of Hamiltonian flow equations developed by Glazek, Wilson, and Wegner [14]. Specifically, we define a hierarchy of Hamiltonians H(n) = H +V(n)+W(n) 0 labeled by an integer level n 0, such that W(n) is a block-diagonal perturbation while V(n) ≥ is a generic perturbation. We start at the level n = 0 with the perturbed H + V, that is, 0 V(0) = V and W(0) = 0. As we go to higher levels, the Hamiltonian H(n) becomes more close to a block-diagonal form. The transformation from H(n) to H(n+1) is described by a unitary operator U(n) that block-diagonalizes H(n) up to errors of order V(n)2. These errors are dealt with at the next level of the hierarchy, see Section 4 for details. We construct U(n) by solving a linearized block-diagonalization problem, see Section 5. The solution can be easily constructed intermsoftheserieswhileconvergenceoftheseriesfollowsfromthefactthatW(n)isrelatively bounded by H . 0 We prove that the strength of V(n) decays doubly-exponentially as a function of n, while W(n) does not change essentially after the first few levels. We then choose the desired unitary operator U as U = U(n ) U(1)U(0) where the highest level n log(L) is chosen to make f f ··· ∼ thenormofV(n )exponentiallysmall(asafunctionofL). Themosttechnicalpartoftheproof f is to show that the unitary operators U(n) are locality preserving such that all Hamiltonians V(n) and W(n) remain sufficiently local. To this end we first prove that U(n) can be generated by a quasi-local Hamiltonian, see Section 5, and then employ the Lieb-Robinson bound, see Section 6. 2 Hamiltonians describing TQO 2.1 Frustration-free commuting Hamiltonians TosimplifynotationsweshallrestrictourselvestothespatialdimensionD = 2. Ageneralization to an arbitrary D is straightforward. Let Λ = Z Z be a two-dimensional square lattice of L L × linear size L with periodic boundary conditions. We assume that every site u Λ is occupied ∈ by a finite-dimensional quantum particle (qudit) such that the Hilbert space describing Λ is a tensor product = , dim = O(1). u u H H H Ou∈Λ Let (r) be a set of all square blocks A Λ of size r r, where r is a positive integer. Note that S ⊆ × (r) contains L2 translations of some elementary square of size r r for all r < L, (L) = Λ, S × S and (r) = for r > L. We can always assume that the unperturbed Hamiltonian H involves 0 S ∅ only 2 2 interactions (otherwise consider a coarse-grained lattice): × H = G . (2.1) 0 A X A∈S(2) 7 There will be three essential restrictions on the form of interactions G . Firstly, we require that A G are pairwise commuting operators, that is, A G G = G G for all A,B (2). A B B A ∈ S Thus all interactions G can be diagonalized in the same basis. Secondly, we require that H A 0 is a frustration free Hamiltonian, that is, the ground state of H minimizes energy of every 0 individual term G . Performing an overall energy shift we can always assume that all G are A A positive-semidefinite operators, G 0. A ≥ Then the condition of being frustration-free demands that ground states of H are common zero 0 eigenvectors of every term G . Thus the ground subspace of H is A 0 P = ψ : G ψ = 0 for all A (2) . (2.2) A {| i ∈ H | i ∈ S } Thirdly,weshallassumethateveryoperatorG hasaconstant spectral gap,thatis,thesmallest A positive eigenvalue of G is bounded from below by a constant independent of the lattice size A L. We can always normalize the Hamiltonian H such that the spectral gap of any G is at 0 A least 1. This is equivalent to a condition G2 G . A ≥ A Let P be the projector onto the zero subspace of G and Q = I P . Note that all the A A A A − projectors P ,Q are pairwise commuting. For any square B (r), r 2 define a projector A A ∈ S ≥ onto the local ground subspace P = P (2.3) B A Y A∈S(2) A⊆B and Q = I P . Note that P and Q have support on B. We shall often use the same B B B B − notation for a subspace and for the corresponding projector. 2.2 Formal definition of TQO We shall need two extra property of H and the ground subspace P that guarantee the gap 0 stability and robustness of the ground state degeneracy. We shall assume that there exists a constant c > 0 such that the following conditions hold for some integer L∗ cL for all ≥ sufficiently large L: 8 TQO-1: Let A (r) be any square of size r L∗. Let O be any operator acting on A. A ∈ S ≤ Then PO P = cP A for some complex number c. TQO-2: Let A (r) be any square of size r L∗ and let B (r +2) be the square ∈ S ≤ ∈ S that contains A and all nearest neighbors of A. Define reduced density matrices (B) ρA = TrAc(P) and ρA = TrAc(PB). Then the kernel of ρA coincides with the (B) kernel of ρ . A Remark1. Usingthelanguageofquantumerrorcorrectingcodesonecandefinetheminimum distance of P as the smallest integer d such that erasure of any subset of d particles can be corrected for any encoded state ψ P, see [15] for details. Note that TQO-1 holds for | i ∈ L∗ = √d since the reduced state of any square A (L∗) does not depend on the encoded ⌊ ⌋ ∈ S state. What is less trivial, TQO-1 holds also for L∗ = Ω(d), see [15]. Thus L∗ coincides with the distance of the code P up to a constant coefficient (as far as condition TQO-1 is concerned). Remark 2. Condition TQO-2 can be easily ‘proved’ if the excitations of H are anyons 0 (since the latter assumption lacks a rigorous formulation, the argument given below is not completely rigorous either). Indeed, in this case we can choose a complete basis of the excited subspace Q such that the basis vectors correspond to various configurations of anyons. For non-abelian theories one may have several basis vectors for a fixed configuration of anyons that describe different fusion channels, see [8]. Note that any state ψ P is a superposition B | i ∈ of configurations with no anyons inside B. Since A is a topological trivial region, any such configuration can be prepared from the vacuum P by some unitary operator UAc acting on complementary region Ac = Λ A. Thus ψ = UAc ψ0 for some ground state ψ0 P. Since \ | i | i | i ∈ all ground states ψ have the same reduced matrix on A, it means that ψ and P have the 0 | i | i same reduced matrix on A. This implies TQO-2. The above arguments suggest that TQO-2 holds for all 2D models of TQO that can be described by commuting frustration-free such as quantum double models [1] and Levin-Wen string-net models [4]. Remark 3. As was already mentioned, the consistency between the global and the local ground subspaces may be violated for regions with non-trivial topology. For example, if A has a hole, the local ground subspace P may include sectors with a non-trivial topological charge A inside the hole as opposed to the global ground subspace. We shall need the following corollary of TQO-2. Corollary 2.1. Let A (r) be any square of size r L∗ and O be any operator acting on A ∈ S ≤ A such that O P = 0. Let B (r+2) be the square that contains A and all nearest neighbors A ∈ S of A. Then O P = 0. A B † † Proof. Let ρA = TrAc(P). The assumption OAPOA = 0 implies that OAρAOA = 0, that is, OA annihilates any state in the range of ρA. From TQO-2, the range of TrAc(PB) coincides with † the range of ρ , and thus Tr(O P O ) = 0. It implies O P = 0. A A B A A B 9 2.3 Verification of TQO conditions for stabilizer Hamiltonians Conditions TQO-1,2 can be easily checked for those models of TQO that can be described using the stabilizer formalism such as the toric code model [1] or topological color codes [16]. For such models each site of the lattice Λ represents one or several qubits, while the ground state subspace P is a stabilizer code, i.e., the invariant subspace of some abelian stabilizer group Pauli(Λ). Here Pauli(Λ) is a group generated by single-qubit Pauli operators σx,σy,σz. G ⊆ i i i The stabilizer group must have a set of geometrically local generators, that is, = S ,...,S 1 M G h i whereanygeneratorS Pauli(Λ)actsnon-triviallyonlyonO(1)qubitslocatedwithindistance a ∈ O(1) from each other. Note that the generators need not to be independent. We choose the corresponding stabilizer Hamiltonian H as 0 H = (I S )/2 0 a − Xa such that states invariant under action of stabilizers have zero energy. The minimal distance of the code is the smallest integer d such that there exists a Pauli operator O that commutes with all elements of but does not belong to . Such an operator O can be regarded as a logical G G Pauli operator acting on encoded states. It follows from results of [15] that condition TQO-1 holds if we choose L∗ = Ω(d). Assume that the set of qubits is coarse-grained into sites of the lattice Λ such that the support of any generator S is contained in at least one 2 2 square. One can bring this a × Hamiltonian into the form Eq. (2.1) by distributing the generators over 2 2 squares in an × arbitrary way. For any square B (r) one can define two subgroups of : (i) a subgroup B ∈ S G G generated by generators S whose support is contained in B, and (ii) a subgroup (B) that a G includes all stabilizers S whose support is contained in B. By definition, (B), but B ∈ G G ⊆ G in general = (B). B G 6 G Lemma 2.1. The stabilizer Hamiltonian H satisfies condition TQO-2 iff for any square A 0 ∈ (r), r L∗, one has (A) , where B = b (A). B 1 S ≤ G ⊆ G Thus TQO-2 demands that any element of the stabilizer group whose support is contained in a square A can be written as a product of generators whose support is contained in A and a small neighborhood of A. We leave verification of this condition for the toric code model as an exercise for the reader. Proof. Indeed, the reduced density matrix ρ computed using the global ground subspace P is A proportionaltotheprojectorontothecodespaceofthestabilizercode (A). Thereduceddensity G matrix ρ computed using the local ground subspace P is proportional onto the codespace of A B the stabilizer code (A), where (A) includes all elements of whose support is contained B B B G G G inA. ThusTQO-2holdsiff (A) = (A). Thisisequivalenttotheconditionofthelemma. B G G 10