Detecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators
Tommaso F. Demarie, Trond Linjordet, Nicolas C. Menicucci, Gavin K. Brennen
DDetecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators
Tommaso F. Demarie, Trond Linjordet, Nicolas C. Menicucci, and Gavin K. Brennen Centre for Engineered Quantum Systems, Department of Physics and Astronomy,Macquarie University, North Ryde, NSW 2109, Australia School of Physics, The University of Sydney, Sydney, NSW 2006, Australia (Dated: October 31, 2018)The Kitaev surface-code model is the most studied example of a topologically ordered phase and typicallyinvolves four-spin interactions on a two-dimensional surface. A universal signature of this phase is topologicalentanglement entropy (TEE), but due to low signal to noise, it is extremely difficult to observe in these systems,and one usually resorts to measuring anyonic statistics of excitations or non-local string operators to revealthe order. We describe a continuous-variable analog to the surface code using quantum harmonic oscillatorson a two-dimensional lattice, which has the distinctive property of needing only two-body nearest-neighborinteractions for its creation. Though such a model is gapless, it satisfies an area law and the ground state canbe simply prepared by measurements on a finitely squeezed and gapped two-dimensional cluster state withouttopological order. Asymptotically, the continuous variable surface code TEE grows linearly with the squeezingparameter, and we show that its mixed-state generalization, the topological mutual information, is robust tosome forms of state preparation error and can be detected simply using single-mode quadrature measurements.Finally, we discuss scalable implementation of these methods using optical and circuit-QED technology.
Topological order describes a phase of matter whose corre-lations satisfy an area law while maintaining long-range en-tanglement and ground-state degeneracy impervious to all lo-cal perturbations. These properties make such systems at-tractive candidates for stable quantum memories or proces-sors [1]. However, the lack of a local order parameter makesmeasuring topological order an experimentally onerous task.Some possibilities include measuring non-local string oper-ators [2] or the statistics of anyonic excitations above theground state, as has been demonstrated experimentally withsmall photonic networks [3, 4]. However, due to finite cor-relation lengths of local operators [2, 5], these methods sufferfrom low visibility if the system is not prepared in a pure phasewith vanishing two-point correlations.An alternative is to study properties of the state itself thatare robust to small changes in the correlation length. For atopologically ordered phase, the entanglement entropy of asubsystem in state ρ A is the von-Neumann entropy S ( ρ A ) ≡− tr [ ρ A log ( ρ A )] = α | ∂ A | − γ + ε , where α ∈ R , | ∂ A | isthe boundary size of A , and ε → | ∂ A | → ∞ , [6–8].The parameter γ is termed the topological entanglement en-tropy (TEE) [1], which is an intrinsically non-local quantitythat characterizes topological phases in a variety of systems,including spin lattices such as the qubit surface code [6, 9],bosonic spin liquids [10], and fermionic Laughlin states [11].While useful for numerics, actually measuring TEE ina physical system is a daunting task since extracting thevon Neumann entropy requires knowledge of the completespectrum of the reduced state. A different option is to insteadmeasure the Renyi entropy S ( α ) ( ρ A ) ≡ − α log tr [ ρ α A ] since itwas shown [12] that γ is the same when replacing S ( ρ A ) with S ( α ) ( ρ A ) ∀ α . The value α = [ ρ ] is observable via a simple swap-test measurementon two copies of the state [13]. A pure topological phase,such as the qudit ( d -level spins) surface-code state [14], hastr [ ρ A ] = d −| ∂ A | , meaning γ = log ( d ) [15]. In contrast, thepurity of another area law state with no TEE, such as the qu- dit cluster state [16], is tr [( ρ (cid:48) A ) ] = d −| ∂ A | . Thus, even usingRenyi entropy one still requires a number of measurementsexponential in the size | ∂ A | to distinguish the two phases.In this work we study, for the first time, topological or-der in a continuous-variable (CV) Gaussian state [17] ana-logue of the discrete-variable surface-code state. We describehow to prepare it efficiently using an intermediate mapping tofirst an ideal (infinitely squeezed) and then a physical (finitelysqueezed) CV cluster state [18–25]. Remarkably, the TEEof the CV surface code can be easily computed simply fromquadrature measurements. We show that unlike their qubit(or qudit) counterparts, the CV surface-code state has a par-ent Hamiltonian that is gapless in the thermodynamic limit.Nonetheless, the state is topologically ordered with a TEE thatasymptotically grows linearly with the squeezing parameter.Other gapless models with topological order have been inves-tigated in different contexts [26]. We propose experimental re-alizations for this model that are accessible with today’s tech-nology, and we conclude by analyzing the stability of the CVtopological order against two forms of noise: thermalization(in the case of preparation by cooling) and noisy input states(in the case of active construction).The ideal CV cluster state [18, 19, 27] is the CV analog ofits qubit-based cousin [28, 29] and may be obtained by send-ing zero-momentum eigenstates through pairwise controlled- Z gates C ˆ Z j , k = e i ˆ q j ˆ q k in accord with an undirected, un-weighted graph with one qumode (quantum mode—i.e., har-monic oscillator) per vertex. A CV cluster state is describedcompletely by its nullifiers , which are comparable to stabi-lizers but with eigenvalue 0 instead of 1. Ideal CV clus-ter states have a complete set of nullifiers given by { ˆ η j = ˆ p j − ∑ k ∈ N ( j ) ˆ q j } , where j runs over all vertices, and N ( j ) is the neighborhood of j . From these nullifiers, we can con-struct a Hamiltonian ˆ H idealCS = ∑ j ˆ η j , which has the ideal CVcluster state as its (non-normalizable) ground state. Note thatˆ H idealCS has a continuous spectrum and is therefore gapless. See[19] and Appendix A for details. a r X i v : . [ qu a n t - ph ] O c t f v
12 3 456789 10 11 1213 141516 (a) (b) (c) ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s ⇤ s ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s ⇤ s ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s s ⇥ ⇥ ⇤ s ⇤ s ⇤ s ⇤ s ⇤ s ⇤ s FIG. 1: (a) Measurement scheme to project the CV cluster state (top)on a square lattice into the CV surface code (bottom) described by thegraph Λ . A ˆ q measurement removes the measured (grey) node andall the links departing from it, while a ˆ p measurement eliminates thecorresponding (black) node but creates new connections among thenearest neighboring nodes. (b) Structure of the resultant nullifiers forthe finitely squeezed surface code. For the vertex v indicated, ˆ a v = s (cid:48) √ (cid:104) ∑ j = (cid:0) ˆ q j + is (cid:48) ˆ p j (cid:1) + s s (cid:48) ∑ k = ˆ q k (cid:105) with s (cid:48) = √ s + s − , and forthe face f , ˆ b f = s √ (cid:2) ( ˆ p − ˆ p + ˆ p − ˆ p ) − is ( ˆ q − ˆ q + ˆ q − ˆ q ) (cid:3) .(c) The Gaussian pure-state graph Z [33] for a section of the CVsurface-code state. See main text and Appendix D for details. Inspired by the dynamical mapping of qubit cluster statesto qubit surface codes introduced in [30], there is a simplescheme [31] that transforms the CV cluster state into the cor-responding CV surface code. First, start with the CV clusterstate, and label vertices by row and column. Then, measurein ˆ p (in ˆ q ) those vertices on rows and columns that are bothodd (both even), as in Fig. 1a. This scheme is equivalent up totranslation and/or inversion of the ˆ p , ˆ q measurements. Afterthe measurements, we are left with a CV surface code statewith a new set of nullifiers.It is convenient to borrow notation from the qudit versionof surface codes [14] that describes the nature of the cou-pling involved in the nullifiers in terms of a surface-code graph Λ = { V , E , F } . This graph defines an oriented surface whereoscillators reside on the edges e ∈ E , and the oriented edgesmeet at vertices v ∈ V and surround oriented faces f ∈ F .Edge orientations are chosen such that for each vertex, all in-cident edges point toward it or all away from it (Fig. 1b). Thenew nullifiers are ˆ a v = ∑ e | v ∈ ∂ e ˆ q e and ˆ b f = ∑ e ∈ ∂ f o ( e , f ) ˆ p e ,where o ( e , f ) = ± e is oriented with ( + ) or against ( − ) f .The CV surface code is the non-normalizable ground state ofthe quadratic Hamiltonian ˆ H idealSC = ∑ v ˆ a † v ˆ a v + ∑ f ˆ b † f ˆ b f , whichhas a fully continuous spectrum and is therefore gapless.With experimental realization in mind, we need to con-sider the physical case of finite squeezing. Squeezing is aGaussian transformation, performed by the unitary operatorˆ S ( s ) = e − i ( log s )( ˆ q ˆ p + ˆ p ˆ q ) , with s >
0, where log s is tradition-ally known as the squeezing parameter . In the Heisenbergpicture, ˆ S ( s ) † ˆ q ˆ S ( s ) = s ˆ q , and ˆ S ( s ) † ˆ p ˆ S ( s ) = ˆ p / s , such that thevariance of ˆ p (of ˆ q ) after squeezing is a factor of s − (of s )times its original value. To produce the CV cluster state inthe finite-squeezing case by the canonical method [18, 19],which is the easiest method to work with theoretically, onestarts from N vacuum modes | (cid:105) ⊗ N . Since ˆ a j | (cid:105) ⊗ N = ∀ j ,the initial nullifier set is { ˆ a j = √ ( ˆ q j + i ˆ p j ) } . These states are then all squeezed by ˆ S ( s ) , followed by the pairwise C ˆ Z couplings, yielding the transformed nullifiersˆ a j −→ s √ (cid:34) s − ˆ q j + i (cid:32) ˆ p j − ∑ k ∈ N ( j ) ˆ q k (cid:33)(cid:35) = ˆ η sj . (1)These operators satisfy the canonical commutation relationsfor normal-mode operators, [ ˆ η sj , ˆ η sk ] = [ ˆ η sj , ( ˆ η sk ) † ] = δ j , k ,and therefore the CV cluster-state Hamiltonian is [23]ˆ H CS ( s ) = N ∑ j = s (cid:18) ( ˆ η sj ) † ˆ η sj + (cid:19) . (2)For finite s , the system has a gap of 2 s − . The prefactor pro-vides for finite energy even in the limit of infinite squeezingwhere lim s → ∞ ˆ H CS ( s ) = ˆ H idealCS .Using the same measurement pattern as in the ideal case(Fig. 1), the finitely squeezed CV cluster state can bemapped to the finitely squeezed CV surface code. Takinglinear combinations of neighboring cluster-state nullifiers—specifically sums of neighboring nullifiers around the ˆ p -measured nodes—and alternating signed cyclic sums aroundˆ q -measured modes, one finds the general form of the surfacecode nullifiers [32]. Since the finitely squeezed cluster stateis Gaussian, and quadrature measurements are Gaussian oper-ations [33], so is the finitely squeezed surface code state. Inthe case of a square lattice with toroidal boundary conditions,the nullifiers areˆ a v = s (cid:48) √ (cid:34) ∑ e | v ∈ ∂ e (cid:18) ˆ q e + is (cid:48) ˆ p e (cid:19) + s s (cid:48) ∑ v (cid:48) | [ v (cid:48) , v ] ∈ E e | v (cid:48) ∈ ∂ e ∧ v (cid:54)∈ ∂ e ˆ q e (cid:35) , ˆ b f = s √ ∑ e ∈ ∂ f o ( e , f ) (cid:18) ˆ p e − is ˆ q e (cid:19) , (3)where s (cid:48) = √ s + s − . See Fig. 1b.We can construct a Hamiltonian using these nullifiers:ˆ H SC ( s ) = ∑ v s (cid:48) ˆ a † v ˆ a v + ∑ f s ˆ b † f ˆ b f . (4)The squeezing dependence of the prefactors is done to en-sure the Hamiltonian has finite energy for s → ∞ . Unlikethe discrete-variable case, this Hamiltonian is gapless in thethermodynamic limit. This arises because the nullifiers donot define normal modes. Rather, neighboring nullifiers havenontrivial commutation relations, which allow for low-energymode excitations. For a square n × m ( n or m odd) lattice with n ≤ m the gap is ∆ ( s ) ≈ π / s n , and generically the sys-tem is gapless, see Appendix B. Hence, in distinction to thecluster-state Hamiltonian ˆ H CS ( s ) , the surface-code Hamilto-nian ˆ H SC ( s ) is gapless in the thermodynamic limit, though forinfinite squeezing both models are gapless.A zero-mean N -mode Gaussian state is completely anduniquely described [34] via its symmetrized covariance ma-trix Γ j , k = Re tr [ ρ ˆ r j ˆ r k ] , where ˆ¯ r = ( ˆ q , ..., ˆ q N , ˆ p , ..., ˆ p N ) T is a2 N -dimensional column vector of quadrature operators [34] A B C (a) (b) A B C D
FIG. 2: Sections used for calculations of topological entanglemententropy by the methods of (a) Kitaev-Preskill [7] and (b) Levin-Wen [8]. The areas of the regions satisfy the equality depicted. . A Gaussian pure state’s entanglement entropy can be calcu-lated [35] in terms of the symplectic eigenvalues of Γ , whichare the positive elements of the N eigenvalue pairs {± σ j } ofthe matrix product i ΓΩ , with Ω j , k = − i [ ˆ r j , ˆ r k ] . The entropyfor an N A -mode Gaussian subsystem ρ A is S ( ρ A ) = ∑ { σ Ai } (cid:2) ( σ Ai + ) log ( σ Ai + ) − ( σ Ai − ) log ( σ Ai − ) (cid:3) , (5)calculated using the reduced symplectic spectrum { σ A , . . . , σ AN A } obtained deleting all of B ’s rows and columnsfrom the covariance matrix.In addition to the covariance-matrix representation, everyzero-mean N -mode Gaussian pure state can be uniquely rep-resented by an N -node, undirected, complex-weighted graphwhose adjacency matrix is called Z = V + i U [33, 36]. When Z is purely imaginary ( V = ), the state’s covariance matrixbecomes Γ ≡ ( Γ x ⊕ Γ p ) = ( U − ⊕ U ) . As shown in Fig. 1c,this is the case for the CV surface-code state, for which Z = i U SC = i ( s + s − ) I + is − A SC , where A SC is the cor-responding unweighted adjacency matrix without self-loops.From this, we see immediately that ˆ p - ˆ p correlations (deter-mined by U ) have range at most 1, and ˆ p - ˆ q correlations arezero. Further, as shown in Appendix E, the ˆ q - ˆ q correlationsare bounded above by (cid:104) ˆ q i ˆ q j (cid:105) ≤ Ce − ( d ( i , j )+ ) / ξ where d ( i , j ) is the shortest distance between modes i and j on the surfacecode graph. The constant is C = ( + √ s + ) / ( s + s − ) and the correlation length ξ is: ξ ≤ (cid:104) √ s + + √ s + − (cid:105) − . (6)Therefore, because for Gaussian states all higher order corre-lations are generated by the linear and quadratic ones, the CVsurface code state obeys an area law. Numerically we find forsqueezing log s = . ξ = .
42, seeFig. 4, and the scale factor of entropy with area is α = . S KPtopo ≡ − ( S A + S B + S C − S AB − S BC − S AC + S ABC ) = γ , (7) and another by Levin and Wen (LW) [8], S LWtopo ≡ − [( S A − S B ) − ( S C − S D )] = γ , (8)with regions shown in Figs. 2a and 2b, respectively. If the sys-tem is not topologically ordered, these combinations are ex-actly zero. Thus, we say that a model is topologically orderedonly when γ >
0. In our simulation we use the ground state ofa 36 ×
36 mode CV surface-code and the squeezing-dependentvalues of the TEE are calculated selecting from the covariancematrix the reductions corresponding to regions chosen as pre-scribed by Eq. (7) and Eq. (8). See Fig. 3 for numerical results.Further, we plot the
Topological Log-Negativity (TLN) for theKP regions based on recent results that show this quantity tobe a good witness of topological order for stabiliser states ofAbelian anyon models [39, 40]. The log-negativity of the re-duced state with support on a subsystem A is [41] N ( ρ A ) = − N ∑ i = log [ min ( , λ i ( Γ x µ A Γ p µ A ))] , (9)where µ A = P A ⊥ ⊕ ( − P A ) , with P X being the projector onto themodes in region X and λ i being the i -th eigenvalue of the ma-trix argument. Remarkably, we find that the TLN is a rathertight upper bound on the TEE with the same asymptotic slope.To derive this slope we consider the entanglement entropy forone mode of the smallest meaningful portion of the surfacecode, specifically a three-mode correlated state (see AppendixF for the details of the argument). The squeezing-dependentsymplectic eigenvalue of the reduced covariance matrix cor-responding to the mode used is given by σ = ( + s + s ) / ( + s ) − / . The mode entropy grows linearly withan asymptotic slope of lim s → ∞ d γ ( s ) d ( log s ) = / ln ( ) (cid:39) . s → ρ CS ( β ) = e − β ˆ H CS ( s ) / tr [ · ] as the pre-measurement initial state.This could be generated by engineering the Hamiltonianˆ H CS ( s ) , which is gapped for finite squeezing, and then wait-ing until the system reaches equilibrium with an environmentat temperature β − . Alternatively, using e.g. networks ofnon-interacting photons, one could start with separable modeseach in a thermal input state ρ ( β ) = ∏ j e − β s a † j a j / tr [ · ] andthen generate the thermal cluster state as before.For such mixed states we can detect topological order bymaking use of the Topological Mutual Information (TMI)[15]. The TMI is constructed replacing in Eq. (7) the von Neu-mann entropy S X with (half of) the mutual information I X = S X + S X c − S X ∪ X c between a region X and its complement X c : γ MI ≡ − ( I A + I B + I C − I AB − I BC − I AC + I ABC ) . (10)Assume that, under a given sequence of symplectic trans-formations and homodyne detections, the ground-state covari- (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) log (cid:72) s (cid:76) TEE (cid:144) T M I (cid:144) TL N (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) FIG. 3: Topological entanglement entropy (TEE) γ as a function ofthe squeezing parameter log s for the CV surface code on a squarelattice with 1296 modes. The asterisks ( ∗ ) indicate the TEE calcu-lated using Eq. (7); the crosses ( + ) label the TEE using Eq. (8); thesquares ( (cid:4) ) depicts the TLN; and the circles ( • ) indicate the lowerbound for the topological mutual information (TMI) γ l MI for noisystate preparation. The main graph shows the maximum single-modesqueezing of 12 . Inset : TEE/TMI for levels of multimode squeezing with 5 dBmarked as achievable with current optical technology [52]. The TLNis above the scale shown here. ance matrix Γ CS of the CV cluster-state Hamiltonian ˆ H CS ( s ) maps to a CV surface-code state Γ . Because all normal modesof ˆ H CS ( s ) have the same frequency, the thermal covariancematrix of ˆ H CS ( s ) at temperature β − is just κΓ CS and, underthe same evolution, maps to κΓ , where κ = coth ( βε / ) , and ε is the energy gap of ˆ H CS ( s ) . Note this is not a thermal statefor ˆ H SC ( s ) because the spectrum of ˆ H SC ( s ) is nonuniform.The TMI for this class of mixed states is lower bounded bythe value computed as κ → ∞ , as shown in Appendix F. Thisis γ l MI = − ∑ X ζ ( X ) ∑ (cid:48) i log ( e σ Xi ) , where X runs over all theregions in Fig. 2a and their complements, ζ ( X ) = ± σ Xi for which σ Xi > /
2. (see Fig. 3).The CV cluster states considered in this work are canonicalCV cluster states [33], so named because they are the statesthat would result if one were to use the canonical method ofconstructing them [18, 19, 27]. This method generates Gaus-sian states with graphs of the form Z = V + is − I , where theentries of V are either 0 or 1. While this method is straight-forward theoretically, the C ˆ Z gates are experimentally difficultand inefficient in an optical setting, where the most progresshas been made [42, 43].More efficient and scalable optical construction methodsexist [20–22, 24, 42, 45] but produce cluster states with uni-form non-unit edge weight. These optical methods can pro-duce very large states [50, 51], including a recently demon-strated 10,000-mode cluster state with linear topology [52].Very large square lattices with toroidal [21], cylindrical [21,24], and planar topology can also be made, as well as higher- dimensional lattices [46]. Similar states might also be createdby cooling a circuit-QED system to the ground state of ˆ H CS ( s ) [Eq. (2)] [23]. In Ref. [47] it was shown that using supercon-ducting co-planar waveguides coupled pairwise via dissipativeCooper pair boxes, one can engineer an effective ˆ q - ˆ q interac-tion between the microwave modes in neighbouring waveg-uides. By changing the location of the box in the waveguides,one can also generate ˆ p - ˆ p couplings. While the cluster-stateHamiltonian in Eq. (2) also has ˆ q - ˆ p couplings, there exist par-ent Hamiltonians for CV cluster states (up to phase shifts) thatconsist only of ˆ q - ˆ q and ˆ p - ˆ p couplings [25, 33].All of these methods can be used to efficiently producesquare-lattice CV cluster states with uniform edge weight g ,with 0 < g < . In the optical case, these states can be verylarge (thousands of modes) [22] and all produce a Gaussianstate with a graph of the form Z g = g V + i ε I . By squeezingeach mode in ˆ q by a factor of √ g , Z g (cid:55)→ g − Z g = V + i ( ε / g ) I .Despite being constructed by a completely different method,the resulting state is a canonical CV cluster state with effectiveinitial squeezing ˜ s = (cid:112) g / ε . Since entanglement measures arelocal-unitary invariant, all of our results apply to these states ifwe take s = ˜ s . Furthermore, we don’t need to actively performthe single-mode squeezing before measuring the TEE/TMI.We can simply rescale the outcomes of measurements on theoriginal ( g -weighted) state [48].We have presented a model of correlated quantum harmonicoscillators in a topologically ordered continuous-variablesurface-code state constructed using only Gaussian operationsand with the remarkable property that its topological entangle-ment entropy γ can be observed simply via quadrature mea-surements. In contrast to discrete-variable systems, now γ is a continuous function of a system parameter, specificallythe squeezing; however, this is not surprising since the parentHamiltonian for the system is gapless. The CV surface codestate can be prepared beginning in a gapped CV cluster statephase, and the topological entanglement entropy is robust topreparation errors modeled as thermal input. This provides apractical path to observe topological order in bosonic systemsusing current technology. Acknowledgments. —This research was supported in part bythe ARC Center of Excellence in Engineered Quantum Sys-tems (EQuS), Project No. CE110001013. G.K.B. and T.F.D.thank the KITP where part of this work was completed withsupport from the National Science Foundation under GrantNo. NSF PHY11-25915. N.C.M. was supported by the ARCunder grant No. DE120102204. G.K.B. thanks M. Hastings,Z. Wang, R. Ionicioiu, and S. Rebic for helpful discussions.T.F.D. wishes to thank A. Brodutch, M. Cirio, S. Flammia,M. Lewenstein, S. Singh and D. Terno for discussions, com-ments and support. NCM is grateful to O. Pfister, R. Alexan-der, P. van Loock, S. Flammia, A. Doherty, and S. Bartlett forhelpful discussions. T.L. thanks L. Lehman, D. Lombardo andV. Siddhu for illuminating conversations.
Appendix A: Infinitely squeezed CV cluster states andsurface-code states
In this appendix, we provide additional information aboutideal (i.e., infinitely squeezed) CV cluster states [19] and idealCV surface-code states [31].
Ideal CV cluster states
There are many ways to construct physical CV clusterstates [18–24, 42], all of which give slightly different states inthe finitely squeezed case [33], Each has important differencesthat manifest when using them for measurement-based quan-tum computation [48]. In the infinitely squeezed case, how-ever, these differences become largely irrelevant, and sinceinfinitely squeezed states are unphysical anyway, we are freeto choose for their analysis the method that is simplest from atheoretical perspective. For this reason, we choose the canoni-cal method [18], which is the most straightforward generaliza-tion of the qubit cluster-state preparation procedure, despite itsinefficiency when used in practice [42].A CV cluster state is described in terms of a continu-ous generalization of the Pauli group, namely, the Weyl-Heisenberg group [19]. This generalization is most easily ac-complished by thinking of the qubit ˆ σ x and ˆ σ z gates as imple-menting one-unit cyclic shifts in “position” ( | (cid:105) ↔ | (cid:105) ) and“momentum” ( | + (cid:105) ↔ |−(cid:105) ), respectively. The CV analogs ofthe qubit ˆ σ x and ˆ σ z gates are the translation (position-shift)operators ˆ X ( t ) and boost (momentum-shift) operators ˆ Z ( u ) ,respectively, with t , u ∈ R . While for qubits the shift is by anelement of Z , in the CV case, one may implement a shift inposition or momentum by any real-valued amount. These shiftoperators are generated by the canonical self-adjoint quadra-ture operators ˆ p and − ˆ q , respectively, which satisfy [ ˆ q , ˆ p ] = i .Specifically,ˆ σ x −→ ˆ X ( t ) = e − it ˆ p , ˆ σ z −→ ˆ Z ( u ) = e iu ˆ q , (A1)with the group commutatorˆ X ( − t ) ˆ Z ( − u ) ˆ X ( t ) ˆ Z ( u ) = e − itu . (A2)To construct a qubit cluster state [29], one starts with a col-lection of qubits in the | + (cid:105) state ( + σ x ) and applies controlled- ˆ σ z gates pairwise in accordwith a chosen graph (traditionally, a square lattice), with onequbit per node of the graph. To construct an ideal CV clusterstate by the canonical method [18], one begins with a collec-tion of N quantum harmonic oscillators, each prepared in a0-eigenstate of momentum, | (cid:105) ⊗ Np , where the subscript meansthat ˆ p j | (cid:105) p j = | CS (cid:105) isgenerated from these initial states through the pairwise ap-plication of controlled-Z gates C ˆ Z ( j , k ) = e i ˆ q j ˆ q k upon all thenearest-neighbor modes (cid:104) j , k (cid:105) of the initial state, N ∏ (cid:104) j , k (cid:105) C ˆ Z ( j , k ) | (cid:105) ⊗ Np = | CS (cid:105) , (A3) in accord with a given graph, with one mode per node of thegraph.It is worth pointing that there is some additional freedomin this construction procedure, even at the ideal level. Inparticular, ideal CV cluster-state graphs can have a nonzeroreal-valued weight g ∈ R ∗ associated with each edge. Thismodifies the strength of the C ˆ Z gate represented by that edge:C ˆ Z ( j , k ) [ g ] : = e ig ˆ q j ˆ q k . These weights were first introduced inRef. [20] as a way to enable new methods of constructingCV cluster states. They have shown themselves to be veryimportant when considering the computational properties ofthese states [48] and when considering efficient constructionof cluster states with very large graphs [22, 24, 25, 33]. For thepurposes of all derivations in this work, we set g =
1, but inthe main text (“Experimental Implementation”), we showedhow our results apply to CV cluster states constructed withnon-unit—but still uniform—weight g .Just like in the qubit case, ideal CV cluster states admit a de-scription in terms of stabilizers. Given a stabilizer operator ˆ K j for a state | ψ (cid:105) (i.e., ˆ K j | ψ (cid:105) = | ψ (cid:105) ), under a unitary transforma-tion of the state, | ψ (cid:48) (cid:105) = ˆ U | ψ (cid:105) , ˆ K j transforms as ˆ K (cid:48) j = ˆ U ˆ K j ˆ U † in order to preserve its status as a stabilizer: ˆ K (cid:48) j | ψ (cid:48) (cid:105) = | ψ (cid:48) (cid:105) .Note that the transformation ˆ K j −→ ˆ K (cid:48) j under the action of ˆ U is opposite from the Heisenberg evolution of observables un-der the same unitary ˆ U . This is because we are not modelingthe evolution of an observable when we evolve stabilizers. In-stead, we are evolving the old stabilizer into a new stabilizerfor the new state. Thus, the unitary evolution applied to thestabilizer must counteract that applied to the state in order tomaintain the stabilizer’s role as such.To derive the Hamiltonian of the ideal CV cluster state, weconstruct the qubit case and then show the analogous steps inthe CV case. Consider, for simplicity, a square-lattice graphwith a qubit placed on each vertex. Each of the qubits is ini-tially in a | + (cid:105) state and hence is stabilized by a ˆ σ x opera-tor: ˆ σ xj | + (cid:105) j = | + (cid:105) j . Next, the cluster state is created by ap-plying a controlled- ˆ σ z gate on every pair of nearest-neighborqubits. Consequently, the stabilizers are transformed into newelements equal toˆ K j = ˆ σ xj ∏ k ∈ N ( j ) ˆ σ zk = ˆ σ xj ˆ σ zN ˆ σ zS ˆ σ zE ˆ σ zW , (A4)with N ( i ) identifying the nearest neighbors of qubit j , andwith N, S, E, and W indicating the qubit to the North, South,East, and West of qubit j , respectively. Finally, the qubit clus-ter state is the ground state of the Hamiltonian constructed byimposing an energy penalty for violating any of the stabilizerconditions: ˆ H qCS = − ∑ j ˆ K j . (A5)To similarly define the ideal CV cluster state, we substitutethe N qubits on the vertices of the square-lattice graph with N qumodes in the | (cid:105) p state as defined before. The global stateis therefore | (cid:105) ⊗ Np , and this is stabilized by the single-modeoperators ˆ X j ( s ) in the sense that ˆ X j ( s ) | (cid:105) ⊗ Np = | (cid:105) ⊗ Np . For CVstate, there exists an equivalent way to express the stabilizerrelations by using nullifiers [33]. An operator ˆ η is called anullifier for a state | ψ (cid:105) when ˆ η | ψ (cid:105) = X j ( s ) | (cid:105) p j = e − is ˆ p j | (cid:105) p j = | (cid:105) p j ←→ ˆ p j | (cid:105) p j = , (A6)and the state | (cid:105) ⊗ Np is nullified by the set { ˆ p j } of generatorsof the stabilizer group. As in the qubit case, the ideal CVcluster state | CS (cid:105) is the result of the pairwise application ofnearest-neighbor controlled- Z gates C ˆ Z ( j , k ) on | (cid:105) ⊗ Np . Underthe C ˆ Z ( j , k ) evolution, the quadrature operators transform asC ˆ Z ( j , k ) ˆ q j C ˆ Z † ( j , k ) = ˆ q j , C ˆ Z ( j , k ) ˆ p j C ˆ Z † ( j , k ) = ˆ p j − ˆ q k , (A7)and thus the initial state stabilizers { ˆ X j ( s ) } are changed intothe cluster state stabilizers { ˆ X j ( s ) ∏ k ∈ N ( j ) ˆ Z k ( s ) } , which havethe same form of the operators in Eq. (A4). We can also ex-press the stabilizers by [33]ˆ X j ( s ) ∏ k ∈ N ( j ) ˆ Z k ( s ) | CS (cid:105) = e − is ( ˆ p j − ∑ k ∈ N ( j ) ˆ q k ) | CS (cid:105) , (A8)equivalent to the elements of the CV cluster-state nullifier set { ˆ η j } being defined asˆ η j = ˆ p j − ∑ k ∈ N ( j ) ˆ q k . (A9)All the ˆ η j commute and are the elements of the algebra thatgenerates the stabilizer group of | CS (cid:105) . Again, we can con-struct a Hamiltonian ˆ H CS whose ground state is | CS (cid:105) by im-posing an energy penalty for violating any of the nullifier con-ditions: ˆ H CS = N ∑ j = ˆ η j . (A10)Since all nullifiers { η j } commute and have a continuous spec-trum of eigenvalues ( R ), this Hamiltonian also has a continu-ous spectrum, [ , ∞ ) , and is therefore gapless. Ideal CV surface-code states
Now we present a general description of the ideal CVsurface code using a representation reminiscent of case forqudits [14]. As stated in the main text, consider a graph Λ = { V , E , F } where V is a vertex set, E is an edge set,and F is a face set. We assume the graph is oriented and thatthe faces inherit this orientation. Each quantum mode is lo-cated on an edge e j ∈ E , with the orientation of any edge e determined by e = [ v , v (cid:48) ] for the base starting at vertex v andthe head a vertex v (cid:48) .The construction for the ideal CV surface-code state in themain text focussed on creating the state by starting with an ideal CV cluster state and then performing quadrature mea-surements on selected modes of the system in accord with asimple pattern [31] in analogy to the qubit case [30]. This wasinvolved specializing to the simple case of Λ being a planarsquare lattice with edge orientations chosen such that at anyvertex v , all incident edges point toward v , or all point awayfrom v , and nullifiers were derived under these assumptions.This Appendix will diverge from the main text and make nosuch assumptions about the edge orientations.In this more general case, the stabilizers for the surface codeareˆ A v ( t ) = ∏ e | v ∈ ∂ e ˆ Z e [ o ( e , v ) t ] , ˆ B f ( s ) = ∏ e ∈ ∂ f ˆ X j [ − o ( e , f ) s ] , (A11)where o ( e , v ) = (cid:40) + e ∈ [ v , · ] , − e ∈ [ · , v ] , o ( e , f ) = (cid:40) + e is oriented the same as f , − , (A12)where the dot ( · ) stands for any vertex. By construction, thestabilizers commute: [ ˆ A v , ˆ A v (cid:48) ] = [ ˆ B f , ˆ B f (cid:48) ] = [ ˆ A v , ˆ B f ] =
0. The + A v and ˆ B f is the surface-code subspace.Oriented edges are useful when making the connection toqudits, but they are unnecessary in the special (and original)case of the qubit surface code [9]. In this case, the CV stabi-lizers in Eq. (A11) function once again as the CV analogs ofthe surface-code stabilizers for qubits:ˆ A v ( t ) ←→ ˆ A qv = ∏ j ∈ v ˆ σ zj , ˆ B f ( s ) ←→ ˆ B qf = ∏ j ∈ ∂ f ˆ σ xj . (A13)In the special case considered in the main text—i.e., when Λ is a planar square lattice with edge orientations chosen suchthat at any vertex v , all incident edges point toward v , or allpoint away from v , o ( e , v ) is a constant ( ±
1) for any vertex v .Thus, it amounts to only a sign flip on t in Eq. (A11), whichhas no effect on a stabilizer’s role as such, and in the maintext we ignore it for simplicity. Under these conditions, thestabilizers in Eq. (A11) becomeˆ A v ( t ) −→ e i ˆ a v t , ˆ B f ( s ) = e i ˆ b f s , (A14)where, as noted in the main text, the nullifiers ˆ a v , ˆ b f areˆ a v = ∑ e | v ∈ ∂ e ˆ q e , (A15)ˆ b f = ∑ e ∈ ∂ f o ( e , f ) ˆ p e . (A16)If, on the boundary, one of the edges is missing, then thatmode is not included in the nullifiers. The Hamiltonian whoseground state is the ideal CV surface code is given byˆ H = ∑ v ˆ a † v ˆ a v + ∑ f ˆ b † f ˆ b f . (A17)While we have written this Hamiltonian as if it were that ofa system of coupled oscillators, this connection is spuriousin the ideal case [although it will be correct in the finitelysqueezed case; see Eq. (B3)]. Since the “mode operators” ˆ a v and ˆ b f are actually (Hermitian) quadrature operators that allcommute, this Hamiltonian has a continuous spectrum [ , ∞ ) and is therefore gapless for any number of systems. Appendix B: Hamiltonian of the finitely squeezed CVsurface code
In this section we provide details on the form of the finitelysqueezed CV surface-code Hamiltonian and its spectrum. Theexplicit form of the surface-code nullifiers for a square lattice(while allowing for smooth or rough open boundaries) are ob-tained by taking linear combination of neighboring nullifiersof the finitely squeezed CV cluster state. Given a set of exactnullifiers for a Gaussian state [33], one can obtain new nulli-fiers by a three-step process:1. Given a quadrature measurement ˆ x j to be made onmode j , where ˆ x ∈ { ˆ q , ˆ p } , using linear combinationsof the original nullifiers, write a new set of nullifierssuch that the canonically conjugate local quadrature ˆ y j (where [ ˆ x j , ˆ y j ] = ± i ) appears in only one nullifier in thenew set.2. In each new nullifier, replace ˆ x j with the real-valuedmeasurement outcome.3. Eliminate the nullifier that contains ˆ y j .We always assume that the outcome of the measurement is 0because any other outcome would merely result in the samestate up to displacements in phase space. These displacementscan always be undone by local unitaries and therefore do notchange any entanglement measure we might want to calculate.To obtain the face nullifiers ˆ b f , one sums the cluster-statenullifiers immediately adjacent to the node in question (alongthe cardinal directions) with orientation-dependent signs (e.g.,ˆ η sN + ˆ η sS − ˆ η sE − ˆ η sW ). The vertex nullifiers ˆ a v are more com-plex and require next-nearest-neighbor nullifiers to be addedto the sum in order to achieve step 1 in the procedure above.The result for a lattice with possibly incomplete vertices andfaces isˆ a v = s v (cid:112) V ( v )( + ( s / s v ) ) × (cid:34) ∑ e | v ∈ ∂ e (cid:18) ˆ q e + is v ˆ p e (cid:19) + s s v ∑ v (cid:48) | [ v (cid:48) , v ] ∈ E ∑ e | v (cid:48) ∈ ∂ e ˆ q e (cid:35) ˆ b f = s (cid:112) | ∂ f | ∑ e ∈ ∂ f o ( e , f ) (cid:18) ˆ p e − is ˆ q e (cid:19) , (B1)with s v = (cid:112) V ( v ) s + s − , V ( v ) valence of vertex v and | ∂ f | boundary size of the lattice face.Note that now there is a dependence on the lattice positionof the vertex or plaquette considered. The nullifiers satisfy the following commutation relations: [ ˆ a v , ˆ a † v (cid:48) ] = d ( v , v (cid:48) ) = ( s v + s v (cid:48) + s ( V ( v )+ V ( v (cid:48) ))) / s v s v (cid:48) [ V ( v ) V ( v (cid:48) )( +( s / s v ) )( +( s / s v (cid:48) ) )] / if d ( v , v (cid:48) ) = s / s v s v (cid:48) [ V ( v ) V ( v (cid:48) )( +( s / s v ) )( +( s / s v (cid:48) ) )] / if d ( v , v (cid:48) ) = √ s / s v s v (cid:48) [ V ( v ) V ( v (cid:48) )( +( s / s v ) )( +( s / s v (cid:48) ) )] / if d ( v , v (cid:48) ) = d ( v , v (cid:48) ) > [ ˆ b f , ˆ b † f (cid:48) ] = f = f (cid:48) , √ | ∂ f || ∂ f (cid:48) | if [ f , f (cid:48) ] ∈ E ,0 otherwise, [ ˆ b f , ˆ b f (cid:48) ] = [ ˆ a v , ˆ a v (cid:48) ] = [ ˆ a v , ˆ b f ] = [ ˆ a v , ˆ b † f ] = . (B2)Here d ( v , v (cid:48) ) is the Euclidean distance between the two ver-tices where the edge lengths of the graph are unit length. TheHamiltonian is given byˆ H SC ( s ) = ∑ v V ( v )( + s / s v ) s v ˆ a † v ˆ a v + ∑ f | ∂ f | s ˆ b † f ˆ b f , (B3)and the squeezing dependence of the prefactors for ˆ h V andˆ h F ensures the Hamiltonian has finite energy in the infinitelysqueezed limit:lim s → ∞ ˆ H SC ( s ) = ∑ v (cid:32) ∑ e | v ∈ ∂ e ˆ q e (cid:33) + ∑ f (cid:32) ∑ e ∈ ∂ f o ( e , f ) ˆ p e (cid:33) . (B4)Here we have used the fact that in the infinite-squeezing limit,each vertex nullifier involves a sum of ˆ q ’s around that ver-tex and its four neighboring vertices, and since they all com-mute, the parent Hamiltonian is simply the squared sum of ˆ q ’saround each vertex.Now we compute the gap of the surface code Hamiltonian,Eq. (B3). We first consider a square n × m lattice with toroidalboundary conditions: [ ˆ a v , ˆ a † v (cid:48) ] = w [ d ( v , v (cid:48) )] , [ ˆ b f , ˆ b † f (cid:48) ] = x [ d ( f , f (cid:48) )] , [ ˆ a v , ˆ a v (cid:48) ] = [ ˆ b f , ˆ b f (cid:48) ] = [ ˆ a v , ˆ b f ] = [ ˆ a v , ˆ b † f ] = , (B5)where d ( v , v (cid:48) ) (respectively, d ( f , f (cid:48) ) ) is the Euclidean distancebetween vertices (faces) on the unit-edge-length lattice (duallattice): w ( ) = , w ( ) = ( + s ) ( + s ) , w ( √ ) = s ( + s ) , w ( ) = s ( + s ) , w ( d > ) = , (B6)and x ( ) = , x ( ) = , x ( d > ) = . (B7)On a torus | E | = nm , | F | = nm , and | V | = nm . We first focuson the case where n × m is odd such that there are | E | inde-pendent nullifiers which therefore span the space of all thephysical-mode annihilation operators. Introducing normal-mode operatorsˆ c j = n − ∑ r = m − ∑ s = α ( j ) r , s ˆ a v r , s , ˆ d j = n − ∑ r = m − ∑ s = β ( j ) r , s ˆ b f r , s , (B8)where the vertices at the lattice sites have vertex coordinates { v r , s } and the sites of the dual lattice have face coordinates { f k , l } , the Hamiltonian isˆ H SC ( s ) = ∑ j ω j s (cid:48) ˆ c † j ˆ c j + ∑ j δ j s ˆ d † j ˆ d j . (B9)To find the normal-mode frequencies, we need to solve theequations (cid:20) ˆ c j , ∑ v ˆ a † v ˆ a v (cid:21) = ω j ˆ c j , (cid:34) ˆ d j , ∑ f ˆ b † f ˆ b f (cid:35) = δ j ˆ d j . (B10)These two linear equations can be vectorized and rewritten as M v | α ( j ) (cid:105) = ω j | α ( j ) (cid:105) , M f | β ( j ) (cid:105) = δ j | β ( j ) (cid:105) , (B11)where | α ( j ) (cid:105) is the vectorized form of the operator ˆ c j , and | α ( j ) (cid:105) is that for ˆ d j . Defining the shift operator ˆ X r = ∑ r − k = | k ⊕ r (cid:105)(cid:104) k | , we have M v = ˆ I nm + w ( ) (cid:2) ˆ I n ⊗ ( ˆ X m + ˆ X † m ) + ( ˆ X n + ˆ X † n ) ⊗ ˆ I m (cid:3) + w ( √ ) (cid:2) ˆ X n ⊗ ˆ X m + ˆ X † n ⊗ ˆ X † m + ˆ X n ⊗ ˆ X † m + ˆ X † n ⊗ ˆ X m (cid:3) + w ( ) (cid:2) ˆ I n ⊗ ( ˆ X m + ˆ X m ) + ( ˆ X n + ˆ X n ) ⊗ ˆ I m (cid:3) , M f = ˆ I nm + x ( ) (cid:2) ˆ I n ⊗ ( ˆ X m + ˆ X † m ) + ( ˆ X n + ˆ X † n ) ⊗ ˆ I m (cid:3) . (B12)The linear equations Eq. (B11) can be solved in the Fourierbasis via ˆ F n ⊗ ˆ F m where ˆ F r = √ r ∑ r − j , k = e i π jk / r | j (cid:105)(cid:104) k | . { ω j } = (cid:26) + w ( ) (cid:20) cos (cid:18) π j x n (cid:19) + cos (cid:18) π j y m (cid:19)(cid:21) + w ( √ ) (cid:20) cos (cid:18) π j x n + π j y m (cid:19) + cos (cid:18) π j x n − π j y m (cid:19)(cid:21) + w ( ) (cid:20) cos (cid:18) π j x n (cid:19) + cos (cid:18) π j y m (cid:19)(cid:21)(cid:27) n − , m − j x = , j y = , { δ j } = (cid:26) + x ( ) (cid:20) cos (cid:18) π j x n (cid:19) + cos (cid:18) π j y m (cid:19)(cid:21)(cid:27) n − , m − j x = , j y = , | β ( j ) (cid:105) = | α ( j ) (cid:105) = ˆ F n | j x (cid:105) ⊗ ˆ F m | j y (cid:105) , (B13)treating j = ( j x , j y ) ∈ Z n × Z m as a collective index. The gapenergy is the lowest-frequency mode energy: ∆ ( s ) = min j x , j y (cid:26) s ω j + s , δ j s (cid:27) . (B14) For large systems sizes, n , m (cid:29)
1, and choosing without lossof generality n ≤ m , the gap is ∆ ( s ) ≈ π s n . (B15)If n , m even on the torus, then not all face nullifiers are inde-pendent (simply bicolor the faces, and assign plus signs to faceoperators on one color and minus signs on the other, and thenadd to get zero). Thus, the Hamiltonian ˆ H SC ( s ) is undercon-strained, and there is an exact gapless zero mode. For a squarelattice with planar boundaries, there are boundary effects, butthis makes only a small modification to the gap, which stillscales like the inverse of the system size for large lattices. Appendix C: Topological S-matrix for symmetric CV surfacecode
One of the defining characteristics of topologically or-dered matter is the existence of non-local interactions betweenparticle-like excitations, which are described in terms of ascattering matrix. The scattering matrix can be evaluated interms of expectation values of products of ribbon operatorsaround a torus [1]. We show how these arise in the context ofcontinuous-variable topologically ordered states.For simplicity, consider the symmetric Hamiltonian men-tioned in the text:ˆ H (cid:48) SC ( s ) = s ∑ v ˆ a (cid:48) † v ˆ a (cid:48) v + s ∑ f ˆ b † f ˆ b f , (C1)where ˆ b f = s √ ∑ e ∈ ∂ f o ( e , f ) (cid:18) ˆ p e − is ˆ q e (cid:19) , ˆ a (cid:48) v = s √ ∑ e | v ∈ ∂ e (cid:18) ˆ q e + is ˆ p e (cid:19) . (C2)For the case n , m even, because of the simple structure of thevertex and face nullifiers, there are only n − m − + − Z P ( t ) = exp (cid:34) it ∑ e ∈ P o ( e ) √ − s − (cid:18) ˆ p e − is ˆ q e (cid:19)(cid:35) , ˆ X ˜ P ( r ) = exp (cid:34) − ir ∑ e ∈ ˜ P f ( e ) √ − s − (cid:18) ˆ q e + is ˆ p e (cid:19)(cid:35) , (C3)where P and ˜ P are oriented paths on the lattice and dual lat-tice, respectively. Here o ( e ) = ± e is oriented inthe same (opposite) direction as P , and f ( e ) = ± e has the same (opposite) framing to the path ˜ P . The framingof the path ˜ P is to the right, normal to its direction. Sinceeach string touches an even number of modes of each nullifierwith opposite signs due to edge orientations, then we have byconstruction: [ ˆ Z P ( t ) , ˆ a (cid:48) v ] = [ ˆ Z P ( t ) , ˆ a (cid:48) † v ] = [ ˆ Z P ( t ) , ˆ b f ] = , [ ˆ X ˜ P ( r ) , ˆ b f ] = [ ˆ X ˜ P ( r ) , ˆ b † f ] = [ ˆ X ˜ P ( r ) , ˆ a (cid:48) v ] = . (C4)However, [ ˆ Z P ( t ) , ˆ b † f ] (cid:54) =
0, and [ ˆ X ˜ P ( t ) , ˆ a (cid:48) † v ] (cid:54) =
0, so these stringoperators are not symmetries of the Hamiltonian ˆ H (cid:48) SC ( s ) .However they are symmetries of the ground subspace H (cid:48) GS .To see this notice that the normal mode operators for this sym-metric model are defined as in Eq. (B8), call them ˆ c (cid:48) j and ˆ d j .Since ˆ c (cid:48) j and ˆ d j are linear combinations of the nullifiers ˆ a (cid:48) v andˆ b f operators respectively, we haveˆ c (cid:48) j [ ˆ Z P ( t ) | GS (cid:105) ] = ˆ Z P ( t ) ˆ c (cid:48) j | GS (cid:105) = , ˆ d j [ ˆ Z P ( t ) | GS (cid:105) ] = ˆ Z P ( t ) ˆ d j | GS (cid:105) = , ˆ c (cid:48) j [ ˆ X ˜ P ( r ) | GS (cid:105) ] = ˆ X ˜ P ( r ) ˆ c (cid:48) j | GS (cid:105) = , ˆ d j [ ˆ X ˜ P ( r ) | GS (cid:105) ] = ˆ X ˜ P ( r ) ˆ d j | GS (cid:105) = . (C5)For contractable paths P and ˜ P , the generators of ˆ Z P ( t ) andˆ X ˜ P ( r ) are linear combinations of ˆ b f or ˆ a (cid:48) v operators inside theloops.We can compute the topological S-matrix for this modelby computing the monondromy [56]. We loosely refer tothe string operators as worldlines for particles that are cre-ated out of the vacuum, make an excursion around one non-contractible loop on the torus, and then annihilate. There aretwo types of particles: electric charges with charge t associ-ated with loops ˆ Z P ( t ) and magnetic fluxes with flux r asso-ciated with loops ˆ X ˜ P ( r ) . The scattering-matrix element asso-ciated with a t electric charge braiding around an r magneticflux is S r , t = (cid:104) GS | ˆ Z − P ( t ) ˆ X − P ( r ) ˆ Z P ( t ) ˆ X ˜ P ( r ) | GS (cid:105) , (C6)where P , are two loops along the vertical direction, and ˜ P , are two loops along the horizontal direction. This can be eval-uated by using the symmetry of the loop operators: S r , t = (cid:104) GS | ˆ Z P ( t ) ˆ Z P ( − t ) ˆ X ˜ P ( r ) ˆ X ˜ P ( − r ) ˆ Z P ( − t ) ˆ X ˜ P ( − r ) × ˆ Z P ( t ) ˆ X ˜ P ( r ) | GS (cid:105) = e irt , (C7)which is the expected statistics based on the group commuta-tor for the Weyl representation of the Heisenberg group: e − itq e irp e itq e − irp = e irt . (C8)Here we have used the facts that ˆ Z − P ( t ) = ˆ Z P ( − t ) andˆ X − P ( r ) = ˆ X ˜ P ( − r ) and that the only nontrivial action is at fourintersection mode locations { e a , e b , e c , e d } . Note there is a fac-tor ( − ) f ( e a ) f ( e c )+ f ( e d )+ o ( e a )+ o ( e c )+ o ( e d ) also in the exponent which we have assumed is equal to one. If for our chosen edgeorientations this is −
1, we simply redefine either r → − r or t → − t . Note that in the limit of infinite squeezing, s → ∞ , thestring operators become unitary. Appendix D: General properties of Gaussian states, Z matrixand entanglement entropy
Both qubit cluster states and the qubit surface code canbe represented by graphs. Strictly speaking, however, theirdefinitions in terms of graphs are incompatible. In particu-lar, qubit cluster states (or qubit “graph states,” as they areoften confusingly called in the literature) are represented bygraphs that act as a well-defined recipe for creating the state:nodes represent qubits prepared in | + (cid:105) , and edges representCPHASE gates between them. While one does not have tomake cluster states this way, any cluster state can be so con-structed. If we try to apply this recipe to the graphs for qubitsurface-code states, however, we fail because for the surfacecode, qubits live on edges, while plaquettes and vertices de-fine stabilizers in terms of all ˆ σ z ’s or all ˆ σ x ’s. There is noanalog for this in the graph recipe used to define qubit clusterstates. While one can define by fiat a connection between thetwo types of graphs, it is at best a patch-up job.In contrast, the graphical calculus for Gaussian purestates [33] provides a unified graphical representation of anyGaussian pure state in terms of a recipe for its creation. Again,one does not have to follow the recipe to make a Gaussianstate, but any Gaussian state can be made by the recipe defined(uniquely) by its graph. As such, for CV cluster states and theCV surface code, only one type of graph is needed. Further-more, the measurement patterns that connect the two types ofstates have definite graph transformation rules, so the connec-tion between the two graphs can be derived using the graphi-cal calculus instead of invented ad hoc as a patch between twoinequivalent types of graphs, as in the qubit case. This elegantconnection is described in detail in this Appendix, where wealso review the main features of Gaussian states.Entanglement entropy of a generic state can be rather hardto compute since it generally requires knowledge of the spec-trum of the density matrix. On the other hand, a zero-mean, N -mode Gaussian state can be described conveniently and com-pletely by an easy algebraic formalism that follows from theform of its characteristic function [57], which is solely a func-tion of the vector of the first statistical moments and the ma-trix Γ that carries the information about the second moments.Recall that Γ is the covariance matrix of the Gaussian statedefined as Γ j , k = Re tr [ ρ ˆ¯ r i ˆ¯ r j ] , (D1)where ˆ¯ r = ( ˆ q , ..., ˆ q n , ˆ p , ..., ˆ p n ) T is the 2 N -dimensional col-umn vector of the hermitian quadrature operators of the N modes. The information contained in the covariance matrixcompletely determines the entanglement properties of a Gaus-sian state [35]. Explicit calculations of Gaussian states entan-glement entropy are performed making use of the symplectic spectrum of Γ . Let us introduce the symplectic form Ω , Ω j , k = − i [ ˆ¯ r i , ˆ¯ r j ] , (D2)which is a skew-symmetric matrix that incapsulates the canon-ical commutation relations of the quadrature operators. For aGaussian state ρ with covariance matrix Γ , the positive ele-ments of the N pairs of eigenvalues {± σ i } of the matrix prod-uct i ΓΩ are called symplectic eigenvalues . The connection be-tween entanglement entropy S ( ρ ) and symplectic eigenvaluesof ρ is given by the formula S ( ρ ) = ∑ { σ i } (cid:20)(cid:18) σ i + (cid:19) log (cid:18) σ i + (cid:19) − (cid:18) σ i − (cid:19) log (cid:18) σ i − (cid:19)(cid:21) . (D3)The Gaussianity of the state is preserved under Gaussian uni-tary transformations ˆ U G . Each of these operations has a corre-spondent matrix representation Y that belongs to the Symplec-tic group Sp ( n , R ) , while quadratures measurements have awell-defined action on the covariance matrix Γ of the state,which is described below.A symplectic transformation Y preserves the canonical com-mutation relations as follows Y Ω Y T = Ω , ∀ Y ∈ Sp ( n , R ) , (D4)while the action of a Gaussian transformation ˆ U G on thequadratures can be expressed byˆ¯ r (cid:48) = ˆ U † G ˆ¯ r ˆ U G −→ ˆ¯ r (cid:48) = Y ˆ¯ r , (D5)where the right-hand side corresponds to a matrix multiplica-tion on the quadratures vector. At the level of the covariancematrix, this is reflected in the transformation rule Γ (cid:48) = Re (cid:104) ˆ¯ r (cid:48) ˆ¯ r (cid:48) T (cid:105) −→ Γ (cid:48) = Re (cid:104) Y ˆ¯ r ( Y ˆ¯ r ) T (cid:105) = Y Re (cid:104) ˆ¯ r ˆ¯ r T (cid:105) Y T = Y Γ Y T . (D6)A state of N independent vacua is described by the covariancematrix Γ = I N . (D7)After a Gaussian unitary transformation ˆ U Y represented bythe matrix Y is applied to Γ , the resulting Gaussian state is Γ Y = YY T . (D8)Exploiting the decomposition properties of symplectic ma-trices [33, 36], the product YY T is uniquely specified by Y ( UV ) Y T ( UV ) , with Y ( UV ) = (cid:18) U − / VU − / U / (cid:19) , (D9) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) log (cid:72) s (cid:76) (cid:61) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) log (cid:72) s (cid:76) (cid:61) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) log (cid:72) s (cid:76) (cid:61) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) • log (cid:72) s (cid:76) (cid:61) (cid:200) j (cid:45) k (cid:200) (cid:60) q j q k (cid:62) FIG. 4: ˆ q correlations for the qumodes along the main axes of theopen boundary CV surface code for different values r of the initialsqueezing. They decay as ae − | j − k | ξ a + be − | j − k | ξ b and for log ( s ) > ξ a = . , ξ b = . p correlations immediately drop to zero beyond one unitseparation. where, for an N -mode state, both U and V are N × N symmet-ric matrices, and U >
0. Hence, the complex linear combina-tion Z : = V + i U , (D10)offers an alternative description for a pure Gaussian state. Thegraph Z shows up directly in the position-space wavefunc-tion ψ Z ( q ) for an N -mode Gaussian state | ψ Z (cid:105) : ψ Z ( q ) = π − N / ( det U ) / exp (cid:18) i q T Zq (cid:19) , (D11)where q = ( q , . . . , q ) T is a column vector of c-number posi-tion variables. For this state, Γ Y from Eq. (D9) can be rewrit-ten as Γ Y = (cid:18) U − U − VVU − U + VU − V (cid:19) . (D12)The matrix Z has a simple transformation rule under a sym-plectic transformation Y . If Y is decomposed into block form Y = (cid:18) A BC D (cid:19) , (D13)then the Z (cid:48) matrix associated to the transformed state is givenby Z (cid:48) = ( C + DZ )( A + BZ ) − . (D14)The usefulness of this approach lies in the simple transforma-tion rules of Z for the most common laboratory procedurescorresponding to Gaussian unitary transformations, as listedin [33]. This permits the study of Gaussian state evolutionsimply in terms of appropriate transformations on Z . Further-more, it is possible to define the Z CS matrix for the CV clusterstate directly, solely making use of the adjacency matrix A d that describes the square-lattice pattern of connections amongthe modes. Explicitly: Z CS ( s ) : = A d + is − I N , (D15)1 s s s s s s s s s s s s s s s s s s s s
11 11 11 11 1 s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s (a) CV cluster state (b) CV surface-code state FIG. 5: Gaussian pure-state graph Z [33] for a section of a canonicalCV cluster state (a) and for a section of the CV surface-code state (b).Applying the measurement pattern prescribed in Fig. 1a in the maintext to the state represented by the graph in (a) produces a state withthe graph in (b). In both graphs, log s is called the squeezing param-eter as used throughout this document. The darkness of each lineis proportional to its magnitude, with color indicating phase (red =positive real, cyan = positive imaginary). with squeezing parameter log s and I N the N × N unit matrix.This is illustrated in Fig. 5a. For the finitely squeezed clusterstate with open boundary conditions, we start with Z CS ( s ) andperform the measurement scheme displayed in Fig. 1 in themain text. Measurements have a straightforward translation inthe Z transformation rules language. A ˆ q measurement on the k th mode is equivalent to deleting the k th row and column ofthe Z matrix, while a ˆ p measurement is equivalent to applyinga π / k th mode and then measuring the ˆ q quadrature. At the level of the Z matrix, any ˆ p measurementdeletes the measured mode while generating new connectionsamong its nearest neighbors.The Z CS ( s ) transforms into the CV finitely squeezedsurface-code state Z SC ( s ) , shown in Fig. 5b. In this case, itis a matrix having only imaginary entries given by Z SC ( s ) = i U SC ( s ) , (D16)with U SC ( s ) = s A SC + (cid:0) s − + s (cid:1) I N , (D17)and A SC adjacency matrix of the surface code. This immedi-ately provides a simple expression for the covariance matrixof the newly evolved state: Γ SC ( s ) = (cid:18) U − ( s ) U SC ( s ) (cid:19) . (D18)Complete knowledge of the covariance matrix makes straight-forward calculations of entanglement entropy for different re-gions of the system. It is enough to derive the proper reducedcovariance matrices, calculate the symplectic eigenvalues, andapply the formula from Eq. (D3).The covariance matrix Γ SC ( s ) allows us to study the ˆ q and ˆ p -correlations for the modes of the surface code. FromEq. (D17) one can immediately notice that the ˆ p -correlationsare non-zero exclusively in the nearest neighborhood of each mode, due to the particular form of the matrix U SC ( s ) . Thegraph in Fig. 4 shows the behavior of the ˆ q -correlations alongthe main axes of the system described by Eq. (D18), which arenot simple because U − ( s ) is complicated. These correlationsdecay exponentially with the distance. Appendix E: Correlations on the lattice
A necessary assumption to use the topological entropy for-mulas is the exponential decay of the quadrature correla-tions. The covariance matrix Γ SC ( s ) allows us to study theˆ q and ˆ p -correlations for the modes of the surface code. FromEq. (D17) one can immediately notice that the ˆ p -correlationsare non-zero exclusively in the nearest neighborhood of eachmode, due to the particular form of the matrix U SC ( s ) . Thiscorresponds to a correlation length ξ p =
1. On the other hand,the ˆ q -correlations require a more elaborate analysis becausethe matrix U − ( s ) is more complicated.It is however possible to prove analytically that also theˆ q -correlations decay quickly enough. First we show thatthe spectral range of U SC ( s ) , denoted σ ( U SC ( s )) , satisfies σ ( U SC ( s )) ⊂ [ a , b ] where a = s − and b = s ( + s − ) . For theminimum eigenvalue, note that for finite squeezing the matrix U SC ( s ) / s is positive definite but for s → ∞ , some ˆ q − ˆ q corre-lations become infinite. This indicates that the matrix is singu-lar in that limit and the smallest eigenvalue is therefore zero.Adding finite squeezing shifts the spectrum of U SC ( s ) / s by s − so the minimum eigenvalue of U SC ( s ) is a = s − .To derive the largest eigenvalue of U SC ( s ) , observe that forthe surface code on a lattice with periodic boundaries, the Z graph associated with the adjacency matrix A SC is regu-lar with degree 6, i.e. each node connects to six others. Thelargest eigenvalue of the adjacency matrix of a regular graph isequal to the degree with an associated eigenvector ν = , ..., A SC is 6. Fromthese considerations it follows that the largest eigenvalue of U SC ( s ) is b = s ( + s − ) . For lattices with open boundaryconditions, the eigenvalues of A SC are upper bounded by themaximal degree with corrections that fall off with the systemsize, so the spectrum of U SC ( s ) lies in the interval [ a , b ] .On a n × m lattice, the nm × nm matrix U SC ( s ) is blocktridiagonal with n identical m × n matrices A on the diag-onal and identical m × m matrices B on the immediate up-per and lower blocks. Here the matrix coordinates ( i , j ) correspond to Euclidean coordinates (( i x , i y ) , ( j x , j y )) on thelattice where i = mi x + i y for i x ∈ { , ..., n − } and i y ∈{ , ..., m − } , etc. It is convenient to define a graph dis-tance d ( i , j ) = max {| i x − j x | , | i y − j x |} between coordinates ( i x , i y ) and ( j x , j y ) . Since away from the edges the Z graphis a union of square graph with a graph having two diagonaledges passing though every other plaquette, the graph distanceis the number of edges on the shortest path between ( i x , i y ) and ( j x , j y ) and it satisfies ed ( i , j ) / √ ≤ d ( i , j ) ≤ ed ( i , j ) where ed ( i , j ) = (cid:112) ( i x − j x ) + ( i y − j y ) is the Euclidean distance.The matrix A is itself tridiagonal with elements α = s + s − on the main diagonal and β = s on the immediate upper and2lower diagonal. The matrix B is also tridiagonal with diag-onal elements γ = s and immediate upper and lower diago-nal elements either equal to zero or γ . A theorem of Demko,Moss, and Smith [37, 38] shows that banded matrices of acertain class have inverses with matrix elements that decay ex-ponentially with the distance from the diagonal. Specificallythey show for matrices M of size N × N and spectral range σ ( M ) ⊂ [ a , b ] with a > {| M − i , j | : ( i , j ) ∈ D n ( M ) } ≤ C q n + (E1)where the decay sets are D n ( M ) = ( { , . . . N } × { , . . . N } ) \ S p ( M ) and the support sets are S p ( M ) = p (cid:91) k = { ( i , j ) : M ki , j (cid:54) = } . Here C = ( + √ b / a ) b and q = √ b / a − √ b / a + < U k SC ( s ) is a banded block symmetric matrix withblocks of size m × m and block band width 2 k +
1. Further-more, each such block is banded with band width 2 k + S p ( U SC ( s )) is the set of those matrixcoordinates ( i , j ) such that the graph distance d ( i , j ) , is nomore that 2 p +
1. Similarly, the decay set is all matrix co-ordinates outside the support set. The statement in Eq. E1is that for nodes separated in graph distance d ( i , j ) > p + ( i , j ) the inverse matrix el-ement U − ( s ) i , j falls off exponentially with graph distanceas q ( d ( i , j )+ ) / . This implies that the ˆ q -correlations betweennodes i and j separated in graph distance by d ( i , j ) satisfy (cid:104) ˆ q i ˆ q j (cid:105) ≤ Ce − ( d ( i , j )+ ) / ξ (E2)where the constant is C = ( + √ s + ) ( s + s − ) and the correlation length is ξ = (cid:104) √ s + + √ s + − (cid:105) . (E3) Appendix F: Topological entanglement entropy
Let us give a physical interpretation of the different for-mulas used to identify the TEE. Topological order is a phaseof matter resulting from non-local long-range correlations oftopological origin that extend along all the modes of such sys-tems. For this reason, local parameters cannot detect it, andtherefore more effort is required in order to identify it. Thefirst hint regarding the nature of the TEE was given in [6], where considerations about the toric code led the authors tonote that the entanglement entropy of a region of the systemis proportional to the perimeter of that region minus a con-stant. This idea was then broadened, and interesting relation-ships among entanglement entropy, total quantum dimensionof the model, and TEE were introduced. The topology of thesurface where a topologically ordered model is defined deter-mines the species of anyons that are supported by the model.If we assign to each anyonic species a quantum dimension d ,and we sum over all the quasiparticles species, we call the total quantum dimension the quantity D = (cid:114) ∑ i d i . (F1)Both Kitaev and Preskill (KP) [7] and Levin andWen (LW) [8] had the intuition that for a topologically or-dered phase the entanglement entropy of a region scales as S A = α | ∂ A | − γ + ε , (F2)where α ∈ R , | ∂ A | is the size of the boundary of A , ε is acontribution that goes to zero in the limit of | ∂ A | → ∞ , and γ = log D (F3)is the TEE. KP focus on constructing a linear combination ofentanglement entropies such that all the terms proportional tothe length of the boundaries of the regions cancel out, and onlythe topological term remains. Specifically, this correspondsto: γ ≡ − ( S A + S B + S C − S AB − S BC − S AC + S ABC ) . (F4)Once the regions are chosen to be reasonably bigger than thecorrelation length of the system, it is also possible to demon-strate that deformations of the boundaries of the regions andsmooth deformations of the Hamiltonian do not change thevalue of γ . This means that γ is a quantity that is both in-variant (i.e., only determined by the underlying topology) anduniversal (i.e., local modifications of the Hamiltonian do notaffect the global topological properties).LW moved along a different line of thought, consideringpartitions of the system with different topologies rather thanmore general regions as KP, see Fig. 2. The partitions areconstructed such that their pairwise differences are equal, andchosen to be large enough, so that short-range correlations de-cay inside the boundaries. In analogy to Eq. (F4), an explicitexpression for the TEE can be derived: γ ≡ − [( S A − S B ) − ( S C − S D )] . (F5)Again, if the system is not topologically ordered, and if theentanglement entropies only depend on terms proportional tothe boundaries, then the difference in Eq. (F5) is zero. On theother hand, if topological long-range correlations are presenton the lattice, LW argue that non-local closed string operatorswith non-vanishing expectation value exist on the lattice. Op-erators that wind around the region A must then contribute to3a nonzero value of Eq. (F5). They indeed call this contribu-tion TEE, which is a witness for topological order in the fol-lowing sense: the value of γ characterizes the global anyonicand entanglement properties of the topological state. When γ =
0, then it follows from Eq. (F3) that D =
1, which phys-ically means that no anyons are supported by the system, andno long-range topological correlations contribute to the entan-glement. The idea behind both the KP and LW derivations ofTEE formulas is to suppress the non-topological correlations,extracting only the topological information.For mixed states, the appropriate signature of topologicalorder is given by a quantity called topological mutual infor-mation (TMI), introduced in [15]: γ MI = − ( I A + I B + I C − I AB − I BC − I AC + I ABC ) , (F6)using the same regions used by KP for the calculations of theTEE. Now the correlations are measured using (one half of)the mutual information I A between a region A of a system andits complement A c , i.e., I A = S A + S A c − S A ∪ A c , which replacesthe von Neumann entropy S A (up to a factor of 2).In [44] a modified version of the TMI was proposed, toavoid treacherous ambiguities in the recognition of topolog-ical order. In fact, certain mixed states (in particular mix-ture of degenerate ground states of gapped local Hamiltoni-ans) can exhibit long-range non topological correlations andtherefore affect the value of the TMI as it was defined in[7, 8]. Together with a new formulation for the TMI, theyalso introduced lower and upper bounds that precisely deter-mine the TMI whenever they coincide. Consider now the LWregions as defined in Fig. 2 and label E = A \ B , set differ-ence of A and B , F = A \ C , and F = D (from which follows F = F ∩ F = C ) . Then the value of the TMI lies betweenthese two bounds:min ( I E , F − I E , F − I E , F , ) ≤ γ ≤ max ( I E , F , I E , F ) . (F7)The circles ( • ) in Fig. 3 in the main text, are a plot of the lowerbound in Eq. (F7) in the worst-case scenario of very high ini-tial temperature ( κ → ∞ ); see Appendix G. This value for thebound is the lowest possible bound when creating the surface-code state from a thermal cluster state. It therefore illustratesthe maximum extent to which the TMI can sink below theTEE for any given value of the squeezing parameter for theparticular construction procedure described in the main text. Upper bound on TEE
Here we compute an upper bound on the TEE for the CVsurface-code state by calculating the subsystem entropy of asimpler network of entangled modes. To do this, we invokea the calculation of subsystem entropy appropriate to stabi-lizer states, which have vanishing two-point correlation func-tions. Consider stabilizer states that are quantum doubles ofa finite group G (such as the toric code with group G = Z ).As shown in Ref. [6], the TEE is calculated by dividing thesystem into two subsystems A , B and identifying the redun-dant gauge transformations defined on the boundary between A B A B v v v v v v v v f f f f (a) (b) (c) (d) A B
FIG. 6: Simplified network of modes with which to derive an upperbound on the TEE for the CV surface code: (a) This shows a surface-code state for a quantum double model with discrete variables. Thereare six vertices, two faces, and seven edges where physical modesreside. Not all the vertex stabilizers are independent. Rather, theirproduct is the identity, so the number of independent stabilizers is5 +
2, equal to the number of physical modes. We can deform the lat-tice while preserving the topological order by replacing three modesof each face, excluding the mode on the shared boundary, with onemode as shown in (b). This network has one independent vertex sta-bilizer, and two independent face stabilizers which equals the num-ber of physical modes. For the toric code, the network represents aground state that is a GHZ state. (c) A finitely squeezed cluster-stategraph with seven modes maps to the three-mode network by measur-ing the grey modes in the ˆ q basis and the black modes in the ˆ p basis.In fact, the nullifiers on the top and bottom black modes of the graphact equivalently on the white modes meaning one of them is redun-dant. Therefore, an even simpler CV cluster-state graph suffices asdepicted in (d). Upon measurements, the reduced network has threemodes with a correlation matrix that can be computed exactly to yieldan upper bound to the CV surface code TEE. the two regions. The entanglement entropy of subsystem A is the logarithm of the number of the (all equivalent) Schmidtcoefficients of the state. Exploiting the group properties of G allows one to write the entropy as S ( ρ A ) = ( | ∂ A | − ) log | G | ,implying γ = log | G | = log ( D ) .For the CV surface codes, it is complicated to extract ananalogous exact expression for the entropy of a subsystem be-cause the Schmidt coefficients are not equal as in the discretecase. Furthermore, the TEE is infinite for infinitely squeezedCV surface code states, and the definition of quantum dimen-sion is not so clear for finitely squeezed CV surface code statessince we do not yet have a description of this model in termsof a quantum double of a group. Nevertheless, we can goahead and compute the subsystem entropy in the same waythat would be done for the discrete case and treat this as abound for the TEE of the CV surface code state. It is simplyan upper bound because we are ignoring longer-range corre-lations that degrade the topological order, but since the corre-lation length is bounded for any finite amount of squeezing,this should be a reasonably tight bound.A simple configuration to start with is a quantum dou-ble model with a discrete group on a lattice with two faces.This can be realised using a graph with just three edges(physical modes) and two vertices, as shown in Figs. 6a,b.For the toric code the ground state would be the GHZ state ( | (cid:105) + | (cid:105) ) / √ σ x ˆ σ x ˆ σ x , and one face enforces ˆ σ z ˆ σ z , while the other faceenforces ˆ σ z ˆ σ z . Identifying two qubits on one of the faceswith subsystem B , the subsystem entropy is S ( ρ A ) = S ( ρ B ) = − γ =
1, where 2 comes form the size of the boundary of re-4gion B and therefore γ =
1. This simplified surface-code net-work of three modes can be obtained from a finitely squeezedCV cluster state with six modes after measuring out three ofthe modes (Fig. 6d). The resultant CV network has a corre-lation matrix that can be computed exactly. The symplecticspectrum for one subsystem consisting of one mode has twoeigenvalues {± σ } with σ = (cid:115) + s + s + s . (F8)Hence the TEE upper bound can be expressed as γ ub ( s ) = (cid:104)(cid:16) σ + (cid:17) log (cid:16) σ + (cid:17) − (cid:16) σ − (cid:17) log (cid:16) σ − (cid:17)(cid:105) . (F9)Asymptotically, the upper bound on TEE grows linearly witha slope of lim s → ∞ d γ ub ( s ) d ( log s ) = / ln ( ) . Appendix G: Surface-code TMI from high-temperature CVcluster states