Large-scale GHZ states through topologically protected zero-energy mode in a superconducting qutrit-resonator chain
Jin-Xuan Han, Jin-Lei Wu, Yan Wang, Yan Xia, Yong-Yuan Jiang, Jie Song
aa r X i v : . [ qu a n t - ph ] F e b Large-scale GHZ states through topologically protected zero-energy mode in asuperconducting qutrit-resonator chain
Jin-Xuan Han , Jin-Lei Wu , ∗ Yan Wang , Yan Xia , Yong-Yuan Jiang , and Jie Song , , , † School of Physics, Harbin Institute of Technology, Harbin 150001, China Department of Physics, Fuzhou University, Fuzhou 350002, China Key Laboratory of Micro-Nano Optoelectronic Information System,Ministry of Industry and Information Technology, Harbin 150001, China Key Laboratory of Micro-Optics and Photonic Technology of Heilongjiang Province,Harbin Institute of Technology, Harbin 150001, China Collaborative Innovation Center of Extreme Optics, Shanxi University,Taiyuan, Shanxi 030006, People’s Republic of China
We propose a superconducting qutrit-resonator chain model, and analytically work out forms of itstopological edge states. The existence of the zero-energy mode enables to generate a state transferbetween two ends of the chain, accompanied with state flips of all intermediate qutrits, based onwhich N -body Greenberger-Horne-Zeilinger (GHZ) states can be generated with great robustnessagainst disorders of coupling strengths. Three schemes of generating large-scale GHZ states aredesigned, each of which possesses the robustness against loss of qutrits or of resonators, meetinga certain performance requirement of different experimental devices. With experimentally feasiblequtrit-resonator coupling strengths and available coherence times of qutrits and resonators, it has apotential to generate large-scale GHZ states among dozens of qutrits with a high fidelity. Further,we show the experimental consideration of generating GHZ states based on the circuit QED system,and discuss the prospect of realizing fast GHZ states. I. INTRODUCTION
Greenberger-Horne-Zeilinger (GHZ) states constitutean important class of entangled many-body states. Ageneral form of n -qubit GHZ states can be expressedas [1–5] 1 √ | α α α · · · α n i + e iφ | β β β · · · β n i ) . (1)where α j + β j = 1 ( α j , β j ∈ { , } ). These states playa key role in quantum science and technologies, includ-ing open-destination quantum teleportation [6], concate-nated error-correcting codes [7], quantum simulation [8],and high-precision spectroscopy [9]. In principle, thebenchmark for quantum information capability is thenumber of particles that can be reliably entangled in aquantum processor. In experiment, multi-body entangle-ment is achieved recently by capturing 20 trapped ionswith around the fidelity of 63 .
2% [10], 12 photons with59 .
8% [11], 18 photonic qubits exploiting three degrees offreedom of six photons with 72 .
4% [12], 12 superconduct-ing qubits with 55 .
6% [13], 18 superconducting qubitswith 53 .
0% [14] and 20 Rydberg atoms with 54 .
2% [15].Among various physical systems, the ubiquitous noiseand device imperfections are, however, unavoidable andlimit the range where multi-body entanglement can berealized with a high fidelity.Topological insulator [16, 17], a new kind of novel stateof matters, is characterized by the conducting edge states ∗ jinlei [email protected] † [email protected] and the insulating bulk states. The special conductingedge states are protected by the energy gap of the topo-logical system, leading the edge state to be insensitiveto local perturbations and disorders [18–20]. To thisend, the topological state of matter has been studied inmany systems, such as optical lattice systems [21, 22],spin Fermi systems [23, 24], supercooled atoms [25, 26]and synthetic materials [27]. These novel properties sup-port many potential applications of topological insula-tors in quantum information processing and computing.For instance, several proposals for quantum state transfer(QST) have been presented [28–35] with topologicallyprotected channels. A topologically protected channelfor QST between remote quantum nodes mediated bythe edge mode of a chiral spin liquid was proposed andanalyzed [28, 29]. Mei et al. presented an experimen-tally feasible mechanism for implementing robust QSTvia the topological edge states by connecting supercon-ducting Xmon qubits into a one-dimensional chain [30].Also, topologically protected entangled photonic statesand its transport via edge states have been reported [36–39]. In experiment, the topological protection of spatiallyentangled biphoton states was demonstrated [38]. Therobustness in crucial features of the topological bipho-ton correlation map in the presence of deliberately in-troduced disorder was found in the silicon nanophotonicstructure. Recently, a spatially entangled two-particleNOON state is proposed by topological Thouless pump-ing in one-dimensional disordered lattices [39].In this paper, we propose a superconducting-circuitmodel to generate large-scale GHZ states, where themodel is a chain consisting of N flux qutrits connectedby ( N −
1) resonators, and the entangled states are pro-tected by topological zero-energy mode. We analyticallyderive the wave function of the topological edge statewith zero energy of the qutrit-resonator chain, throughwhich a state transfer between two ends of the chain, ac-companied with the state flips of all intermediate qutrits,can be implemented by designing qutrit-resonator cou-pling strengths. The key advantage that topology offersin such processes is the inherent protection of boundaryedge states lying in the band gap of the dispersion rela-tion when the bulk is topologically nontrivial. Based onsuch a peculiar state transfer protected topologically, weshow three schemes of generating large-scale GHZ states,providing feasible and visible methods to generate the ro-bust large-scale GHZ states by meeting the performancerequirements of different experimental devices in the su-perconducting qutrit-resonator chain. Meanwhile, wetake into account the experimental consideration, suchas the implementation of the model, preparation of theinitial state, and realization of tunable couplings in thecircuit-QED chain. In addition, we discuss the potentialof realizing fast GHZ states by speeding up the adiabaticproscess, which provides more possibilities for obtaininghigh-fidelity multi-body entanglement.Our work may facilitate the potential applications oftopological matter in quantum information processing,due to the following advantages and interests. First, thetopological qutrit-resonator chain can be used to createa large-scale GHZ state theoretically with the size be-ing far more than N = 20 that is the particle numberof multi-body entanglement realized experimentally upto now [10–15]. Second, as the core foundation of im-plementing large-scale GHZ states, we derive theoreti-cally the wave function of an edge state with zero-energymode, whose form involves the state flips of all intermedi-ate qutrits, different from that in the frequently studiedstandard Su-Schrieffer-Heeger (SSH) model [30, 32–34].Finally, there are three schemes proposed for generatinglarge-scale GHZ states, which provide potential choices,depending on different device requirements, i.e., coher-ence times of qutrits and resonators. II. PHYSICAL MODEL AND WAVE FUNCTIONOF AN EDGE STATEA. Physical model
The setup of the superconducting qutrit-resonatorchain for generating large-scale GHZ states is illustratedin Fig. 1(a). The chain contains (2 N −
1) lattice sites, N qutrits and ( N −
1) resonators. Each unit cell in thechain contains one flux qutrit A n and one resonator B n ,whose circuit schematic is described by Fig. 1(b). Theenergy levels of qutrit and the coupling strengths are ad-justable via the magnetic flux provided by the flux-biasline (FBL) [40] and superconducting quantum interfer-ence device (SQUID). As described in Fig. 1(c), each fluxqutirt holds a three-level structure, involving two groundstates | L i and | R i , and one excited state | e i . The in- teraction in the chain can be described by the followinginteraction-picture Hamiltonian ( ~ = 1) H I = N − X n =1 (cid:0) J | e i n h j n | + J | e i n +1 h j n | (cid:1) b n + H . c ., (2)where j n = L ( R ) when n is odd (even), and b n theannihilation operator of the resonator B n . J and J can be tuned through adopting controlled voltage pulsesgenerated by an arbitrary waveform generator (AWG) totune the flux threading the loop [41]. Further, H I can berewritten as H = N − X n =1 J b n σ + n ( σ xn ) n + J b n σ + n +1 ( σ xn +1 ) n + H . c ., (3)where we define σ + n = | e i n h R | , σ xn = | L i n h R | + | R i n h L | .The existence of the ( σ xn ) n renders two different tran-sitions | L i ↔ | e i and | R i ↔ | e i for qutrit A n de-pending on the odevity of n . For instance, σ + n ( σ xn ) n = | e i n h R | ( | e i n h L | ) with n being even (odd).Such a adjustable chain can be analogous to anSSH model [42–44] which describes quanta (e.g., elec-trons, photons, or phonons) hopping on a chain (one-dimensional lattice), with staggered hopping amplitudes.The chain of SSH model consists of N unit cells, each unitcell hosting two sites, one on sublattice A , and the otheron sublattice B . The Hamiltonian of the standard SSHmodel is of the form H SSH = ( ν P Nm =1 | m, B ih m, A | + µ P N − m =1 | m +1 , A ih m, B | )+H . c . . Here | m, A i and | m, B i ,with m ∈ , , · · · N , denote the states of the chain wherethe hopping quantum is on sites A and B , respectively,in the unit cell m [42, 43]. The Hamiltonian (3) holds aform of the SSH model whose topological phase is charac-terized by winding number [45], except for an additionaloperator ( σ xn ) n that works for flipping qutrit states andis the key to realizing GHZ states. Based on the bulk-edge correspondence [20, 46, 47], the SSH model pos-sesses zero-energy edge modes at open boundaries in thetopologically nontrivial phase, which are protected by thetopological properties of the system. B. Wave function of an edge state
The appearance of topologically protected gapless edgestates within the bulk gap is a manifestation of the topo-logical insulator. The number of such gapless edge modesis specified by topological invariants. As for an SSHmodel, the zero energy mode, one of the characteristics ofthe topological nontrivial SSH phase, is regarded as thetopological invariant in the edge state and will protectthe edge state topologically from local disorders. Thus,it is critical to obtain the wave function of the edge statewith zero energy mode in our model.The edge states of the chain with a single excitationare exponentially localized at the boundaries. The wave (cid:2159) (cid:2191)(cid:2196) (cid:2159) (cid:2197)(cid:2203)(cid:2202) (cid:1780)(cid:1776)(cid:1786) (cid:1793)(cid:1791)(cid:1795)(cid:1783)(cid:1778) (cid:1803)(cid:1815)(cid:1821)(cid:1816)(cid:1812)(cid:1805)(cid:1818) (cid:513) (cid:1767)(cid:1844)(cid:513) (cid:1767)(cid:1838)(cid:513) (cid:1767)(cid:1857) (cid:2166) (cid:2778) (cid:2157) (cid:2778) (cid:2166) (cid:2778) (cid:2158) (cid:2778) (cid:2166) (cid:2779) (cid:2157) (cid:2779) (cid:2157) (cid:2170) (cid:2166) (cid:2779) (cid:2158) (cid:2170)(cid:2879)(cid:2778) (cid:2157) (cid:2170)(cid:2879)(cid:2778) (cid:2158) (cid:2196) (cid:2157) (cid:2196) (cid:2166) (cid:2778) (cid:2157) (cid:2196)(cid:2878)(cid:2778) (cid:2166) (cid:2779) (a) (b)(c) (cid:513) (cid:1767)(cid:1844)(cid:513) (cid:1767)(cid:1838)(cid:513) (cid:1767)(cid:1857) (cid:513) (cid:1767)(cid:1844)(cid:513) (cid:1767)(cid:1838)(cid:513) (cid:1767)(cid:1857)(cid:2157) (cid:2778) (cid:2157) (cid:2196) (cid:2157) (cid:2170) (cid:2166) (cid:2779)(cid:4666)(cid:2187)(cid:2204)(cid:2187)(cid:2196)(cid:4667) (cid:2166) (cid:2779)(cid:4666)(cid:2197)(cid:2186)(cid:2186)(cid:4667) (cid:2166) (cid:2778) (cid:2166) (cid:2168) (cid:2166) (cid:2174) (cid:2157) (cid:2196) (cid:2158) (cid:2196)
FIG. 1. (a) A superconducting qutrit-resonator chain. The n -th unit cell contains one flux qutrit and one single-mode resonator,labeled as A n and B n , respectively, and holds an intra-cell qutrit-resonator coupling strength J . Between two adjacent cells,a qutrit A n +1 is coupled to the resonator B n with an inter-cell coupling strength J . The resonator B n drives the transition | L i ↔ | e i ( | R i ↔ | e i ) of the two nearest-neighbor qutrits A n and A n +1 when n is odd (even). (b) Circuit schematic of one unitcell in the superconducting qutrit-resonator chain. The coupling strength can be dynamically tuned by a coupler of SQUID. Theenergy level space of the qutrit can be tuned by FBL. (c) Schematics of energy level transitions for qutrits A , A n (1 < n < N ),and A N . The energy level structure of a flux qutrit holds two ground states ( | L i and | R i ) and one excited state ( | e i ). Wedenote that coupling strengths J L = J and J R = J ( J L = J and J R = J ) when n is even (odd). function of an edge state can be described by the follow-ing ansatz, analogous to the standard SSH model [42–44] | ϕ e i = N X n =1 λ n h γσ + n ( σ xn ) n − n − O l =1 σ xl + ηb † n n O l =1 σ xl i | G i , (4)where | G i = | RLR · · · | {z } N i A N ⊗ | · · · | {z } N − i B N − denotes a decoupled state of the qutrit-resonator chainwith all resonators in | i (i.e., zero-photon Fock state),while qutrits A , A , A , · · · in states | R i , | L i , | R i , · · · ,repsectively. λ is the localized index, γ and η being theprobability amplitudes of the gap states. When λ< λ> n decays(increases) exponentially with distance n which meansthe left (right) edge state with the wave function nor-malized. When γ = 0 ( η = 0), according to Eq. (4)the resonators (qutrits) are occupied by the edge statewhose eigenenergy is E = 0. In particular, in orderto generate large-scale GHZ states with binary quan-tum information carried by two states in qutrits, wechoose γ = 1 and η = 0 to render the qutrit in eachunit cell to occupy the left ( λ <
1) and right ( λ >
1) edge states with E = 0. Thus, the wave function ofan edge state occupied by qutrits can be written as | ϕ ′ e i = P Nn =1 λ n σ + n ( σ xn ) n − N n − l =1 σ xl | G i . Substituting | ϕ ′ e i into the eigenvalue equation E | ϕ ′ e i = H | ϕ ′ e i , onecan obtain E N X n =1 λ n σ + n ( σ xn ) n − n − O l =1 σ xl | G i = (cid:16) J λ n b † n n O l =1 σ xl + J λ n +1 b † n n O l =1 σ xl (cid:17) | G i . (5)Through these analyses, the edge state can be workedout as | ϕ E i = N X n =1 λ n σ + n ( σ xn ) n − n − O l =1 σ xl | G i , (6)with λ = − J /J .To illustrate the topological property of the qutrit-resonator chain more clearly, we take the size L ≡ N − J /J . We find that the energy spectrumpossesses a unique zero energy mode keeping unchanged FIG. 2. (a) Energy spectra of the chain versus J /J . (b) Distribution of the zero energy mode versus J /J . (c) Energyspectra of the chain versus g t . (d) Distribution of the zero energy mode versus g t . The size of the chain is L = 2 N − with varying J /J , which denotes a non-evolutive state.The closest distance between the zero energy state andthe bulk appears at the point J /J = 1. In Fig. 2(b),the zero energy state is localized near the left extrem-ity when J /J <
1, while for J /J > J /J , which is consistent with conclusionobtained from Eq. (6). Therefore, the Hamiltonian H isa modulated model if varying the coupling strength J and J . Specifically, the shapes of the coupling strengthsare engineered as Gauss functions J = g exp [ − ( t − τ ) /τ ] ,J = g exp [ − ( t − τ ) /τ ] , (7)where τ = T / J and J , with T being the evolutiontime. The forms of J and J in Eq. (7) satisfy the statetransfer conditions J /J | t → = 0 and J /J | t → T = + ∞ well. The idea of engineering the coupling strengths asGauss functions is to achieve a temporal soft quantumcontrol starting from and ending at a zero amplitude,which enables on-resonant couplings among a desired setof target levels, while efficiently avoiding unwanted off-resonant contributions coming from others [48, 49].Based on time-dependent Gaussian coupling strengths, as shown in Fig. 2(c), there exists a zero energy modeamong all eigenstates during the whole evolutionary pro-cess. Simultaneously, the topological edge state at zeroenergy is well separated from bulk states. In Fig. 2(d),we plot the state distribution of the zero mode. Thequtrit-resonator chain has not only a bulk part but alsoboundaries (which we refer to ends or edges). The qutrits A and A N are regarded as two edges, while the otherqutrits and resonators are the bulk part. The distribu-tion of the zero mode state decays exponentially on thequtrits under the condition of g t < A (left edge) and the distribution of left edgestate with zero mode is equal to unity when g t < g t increases continuously, the distribution of zeromode increases exponentially on the qutrits. Similarly,the eigenstate is localized near the last lattice site A (right edge) and the distribution of right edge state withzero mode is equal to unity when g t > III. LARGE-SCALE GHZ STATESA. Scheme A for generating large-scale GHZ states
Now we focus on the generation of large-scale GHZstates among the N qutrits in the superconductingqutrit-resonator chain. In Fig. 2(d), the edge stateconcentrates towards the left (right) end when g t < g t > g t → g t → + ∞ , the edge states become | l i = | eLR · · · m i A N ⊗ | · · · i B N − , | r i = | LRL · · · e i A N ⊗ | · · · i B N − , (8)where m = R ( L ) when N is odd (even). For generatingGHZ states, the superconducting qutrit-resonator chainis assumed initially in the state | φ i = 1 √ | G i + | l i ) . (9)Going through the evolution, the first decoupled statecomponent in the Eq. (9) does not evolve because nophoton in the resonators can be absorbed to excite theground-state qutrits. From the Fig. 2(d), we learn thatthe second term of Eq. (9) is essentially the left edge statewith zero energy, which can evolve into the right edgestate along the topologically protected process. Thus,the following evolution occurs | φ i 7→ √ (cid:2) | G i − ( − N | r i (cid:3) . Here we set that | R i is used to carry the logical state 1and | L i is used to carry the logical state 0 for the qutrit A n (1 ≤ n < N ), while | e i of the last qutrit A N is en-coded as the logic state 1(0) when N is even (odd). Ex-cept for the last qutrit A N , the excitation state | e i is anauxiliary state without carrying any quantum informa-tion. Accordingly, the final state of N qutrits after theevolution with omitting the zero-photon product state ofresonators is1 √ | · · · i A N − ( − N | · · · i A N ) , (10)which is exactly an N -body GHZ state according toEq. (1).As an example, with the full Hamiltonian Eq. (3) wetake N = 25 to numerically plot in Fig. 3 the populationevolution of the initial state, the ideal state | φ ideal i = √ ( | RLR · · · R i A + | LRL · · · e i A ) ⊗ | · · · i B , andtwo edge states | l A i = | eLR · · · R i A ⊗ | · · · i B and | r A i = | LRL · · · e i A ⊗| · · · i B . Evidently, the pop-ulation of | φ ideal i evolves from 0 .
25 to 1, which indicatesthe successful creation of a 25-body GHZ state. When g t ≈ | φ ideal i and | r A i suddenlyincrease, which shows the same result as the distributionof the zero mode in Fig. 2(d). B. Shortest time for generating high-fidelity N -body GHZ states and robustness against disorderof coupling strengths In the protocol above, the N -body GHZ state isachieved through an excitation displacement of the zero-energy mode along the chain. The evolution time T FIG. 3. Time evolution of populations for | φ i , | φ ideal i , | l A i and | r A i with the size of the superconducting qutrit-resonatorchain L = 49. We choose g T = 3600 as a total evolutiontime. must be chosen as long as possible in order to obtainan adiabatic excitation displacement from one extrem-ity to the other of the qutrit-resonator chain. The adi-abaticity condition can be written as ˙ θ ≪ ∆ E , where θ = arctan( J /J ) and ∆ E is the energy gap betweenthe zero-energy mode and the bulk modes [30, 50]. Inthe above numerical simulation, the evolution time is ap-propriate but not the shortest time of generating a high-fidelity (0 . N -body GHZ states, we numerically cal-culate the fidelity of ideal N -body GHZ states, and plotin Fig. 4(a) the time evolution of log (1 − F ) with N from 10 to 60 at intervals of 5. The numerical resultsexhibit that a higher fidelity needs a longer evolutiontime with increasing N . For example, the evolution timefor 10-body GHZ state is g t = 661 with 99 .
9% fidelity.However, 60-body GHZ state with 99 .
9% fidelity requires g t = 2 . × . In practice, the value of g can bechosen about 2 π ×
50 MHz [51]. Thus, the evolutiontime of realizing 60-body GHZ state is T ≈ µ s. Upto now, the superconducting resonator lifetimes can beachieved between 1 and 10 ms [52–54]. And the decoher-ence time of the superconducting magnetic flux qubtritachieved on 1 ms by designing a π -phase difference acrossthe Josephson junction in circuit has been reported [55],which is far more than the evolution time T . As shown inFig. 4(b), by selecting different N and corresponding evo-lution times of generating GHZ states with the fidelity of99 .
9% as numerical samples, g t versus N can be fitted bya quadratic function g t = 6 . N + 2 . N − .
10 20 30 40 50 600123 (b) (d) (a) (c)
FIG. 4. (a) Time evolution of log (1 − F ) for generating N -body GHZ states with N from 10 to 60 at intervals of 5. (b) Thefitting function and numerical scatters between the number of qutrits and evolution time with 99 .
9% fidelity. (c) The fidelityof GHZ state against the coupling strength with disorder δ corresponding to 11, 18 and 25 qutrits. (d) The fidelity of GHZstate versus the varying δ/g and g t/ for 11-body GHZ state. study the robustness of the scheme, we add a random dis-order into coupling strength J ′ = J (1+rand [ − δ, δ ])for each qutrit-resonator coupling, where rand [ − δ, δ ] de-notes a random number within the range of [ − δ, δ ]. Fig-ure 4(c) shows the relationship between the fidelity ofGHZ state and the disorder δ ∈ [0 , .
5] for 11, 18 and 25qutrits. Note that the disorder is randomly sampled 101times, and then the fidelity is taken as the mean valueof the 101 results. Corresponding to a shorter size ofthe superconducting qutrit-resonator chain, the fidelitydecreases more slowly and the GHZ state is more robustto the disorder of coupling strength. On account of theprocess of generating the GHZ state via the topologicalzero mode protected by the energy gap, the width of thegap in a superconducting qutrit-resonator chain usuallyexhibits an exponential decay behavior with the size ofchain increasing [17]. Furthermore, the disorder has littleeffect on the GHZ state with respect to δ/g < [0 , . δ/g ∈ [0 . , . . δ for 11 qutrits. We can learn that the dam-age to fidelity caused by coupling defects δ/g ∈ [0 . , . { g t = 830 , δ = 0 . } , the fidelity of 11-body GHZ state can be achieved with fidelity over 0 . N -body GHZ state to the disorder andperturbation provides much more convenience for the ex-perimental realization and the practical application ofmulti-particle entanglement state. C. Influence of the losses in the superconductingqutrit-resonator chain
We now give a discussion for the effect of qutrit-resonator losses on the fidelity of generating GHZ states.Two dominant channels are considered in the loss mech-anism: (i) The loss of qutrits with decay rates γ n ; (ii)The loss of resonators with decay rates κ n .The effect of losses during the evolution time can beevaluated by using a conditional Hamiltonian [56, 57] H cond = H − iκ n N − X n =1 b † n b n − iγ n N X n =1 | e i n h e | , (11)where H is the lossless Hamiltonian for the system inEq. (3). The second and third terms represent the lossesof resonators and qutrits, respectively. For convenience,we assume that κ n = κ and γ n = γ . We take N =11, 18, and 25 as examples. The fidelity is formulatedas F = h Ψ ideal | ρ | Ψ ideal i , where | Ψ ideal i = 1 √ | G i − ( − N | r i ) is the output state of an ideal system, and ρ is the density operator of the system dominated by theconditional Hamiltonian Eq. (11).We now numerically simulate the fidelity of ideal GHZstates by solving the non-Hermitian Liouville equation˙ ρ = − i ( H cond ρ − ρH † cond ). Figure 5 shows the relationshipbetween the fidelity F and decay of qutrits or resonators.As the decay rate of qutrits or resonators increases, thefidelity of the ideal state shows a trend of decline. Com-paring those lines, the size of chain increases, the fidelitydecreases slightly. And the higher fidelities indicate thatthe Scheme A for generating N -body GHZ states is in-sensitive to the value of κ . The fidelity of the 11-, 18-or 25-body GHZ state plummets to 25% roughly with γ/g = 0 .
01. Evidently, the value of γ has a greater in-fluence on the fidelity. Owing to the distribution of edgestates | l A i and | r A i closed to 0.5 at end of the evolu-tion time in Fig. 2(d), the loss of qutrits in the excitedstate | e i in edge states during the whole evolution timecauses a significant destructive effect on the fidelity ofGHZ states. As shown in Fig. 5, we plot the fidelityof GHZ states without considering the losses in left andright edge states, i.e, setting γ = γ n = 0. Under thecondition of γ n /g = 0 .
01 for 1 < n < N , the fidelities of11-, 18- and 25-body GHZ states are 55 . .
72% and33 . | e i of qutrits, especiallyof A and A N , is the key to enhancing the fidelity of GHZstates. In the next section, we propose two alternativeschemes that are of improved robustness against losses ofqutrits. IV. ALTERNATIVE SCHEMES
In the process of generating GHZ states, the distribu-tion of excitations in two edge states is 0.5 at the startand end of the evolution time (see Fig. 3). The losses ofqutrits in the excited states have a great destructive effecton the final fidelities of GHZ states. In this section, wepropose two improvement protocols, labeled Scheme Band Scheme C, to focus on suppressing the loss of qutritsin the excited state.
A. Scheme B for suppressing excitation of qutrits A and A N The energy level space of a qutrit can be readily ad-justed via changing the external flux applied to theSQUID loop [58–60] or/and provided by an FBL [40]. Asshown in Figs. 6(a) and (b), an extra long-lived state | P i is introduced into the right potential well of the qutrit A ,upper than and separated enough from the ground state | R i . The transition between | e i ↔ | L i in qutrit A is without edge states FIG. 5. Fidelities for GHZ states versus decay rates ofqutrits or resonators for different N , where γ ( κ ) in the leg-end represents decoherence involving only the qutrit (res-onator) decay. The evolution time T is chosen as g T =6 . N + 2 . N − . .
9% fidelity forgenerating a lossless N -body GHZ state. coupled off-resonantly to the resonator B with couplingstrength J ′ and detuning ∆ . For the qutrit A N with N being odd (even), the transition between | e i ↔ | R i ( | L i )is coupled off-resonantly to resonator B N − with the cou-pling strength J ′ and detuning ∆ . In addition, an aux-iliary classical field is introduced to drive the transitionoff-resonantly between | e i ↔ | P i for qutrit A with Rabifrequency Ω and detuning ∆ . The other classical fieldwith Rabi frequency Ω and detuning ∆ drives the tran-sition of A N off-resonantly, | e i ↔ | L i ( | R i ), with N beingodd (even). The interaction Hamiltonians for the firstand last qutrits coupled to their adjacent resonators canbe written as H = J ′ b | e i h L | e i ∆ t + Ω | e i h P | e i ∆ t + H . c ., (12a) H N = J ′ b N − | e i N h m | e i ∆ t + Ω | e i N h n | e i ∆ t + H . c ., (12b)with m = R ( L ) and n = L ( R ) when N is odd (even),while the interaction Hamiltonian for other cells is iden-tical with Eq. (3).For suppressing losses of qutrits in the excited state | e i , the value of ∆ is supposed to be as large as pos-sible to satisfy the condition ∆ ≫ { J ′ , Ω } . Ac-cording to the theory of second-order perturbation [61],one can eliminate adiabatically the excited states ofqutrits A and A N . Besides, off-resonant-interaction-induced ground-state Stark shifts can be offset by in-troducing auxiliary off-resonant fields [62], phase com-pensations [63], or detuning compensations [64]. Then,the Hamiltonians in Eqs. (12a) and (12b) can be reduced (cid:1827) (cid:2869) (cid:1836) (cid:2869)(cid:4593) (cid:959) (cid:2869) (cid:563) (cid:2869) (cid:513) (cid:1767)(cid:1842)(cid:513) (cid:1767)(cid:1844)(cid:513) (cid:1767)(cid:1857) (cid:513) (cid:1767)(cid:1838) (cid:513) (cid:1767)(cid:1842) (cid:1836) (cid:2869) (cid:513) (cid:1767)(cid:1844) (cid:513) (cid:1767) (cid:1857) (cid:513) (cid:1767)(cid:1838) (cid:959) (cid:2870) (cid:513) (cid:1767)(cid:1857) (cid:513) (cid:1767)(cid:1838) (cid:513) (cid:1767)(cid:1844) (cid:513) (cid:1767)(cid:1844) (cid:513) (cid:1767)(cid:1838) (cid:1827) (cid:3015) (cid:513) (cid:1767)(cid:1857) (cid:1836) (cid:2870)(cid:4593) (cid:3032)(cid:3049)(cid:3032)(cid:3041) (cid:563) (cid:2870)(cid:4666)(cid:3042)(cid:3031)(cid:3031)(cid:4667) (cid:563) (cid:2870)(cid:4666)(cid:3032)(cid:3049)(cid:3032)(cid:3041)(cid:4667) (cid:1836) (cid:2870)(cid:4593) (cid:3042)(cid:3031)(cid:3031) (cid:1827) (cid:2869) (cid:1827) (cid:3015) (a) (d) (c) (b) (cid:3561)(cid:563) (cid:2870) (cid:4666)(cid:3042)(cid:3031)(cid:3031)(cid:4667) (cid:3561)(cid:563) (cid:2870)(cid:4666)(cid:3032)(cid:3049)(cid:3032)(cid:3041)(cid:4667) (cid:3561) (cid:563) (cid:2869) (cid:1836) (cid:2869)(cid:4666)(cid:3042)(cid:3031)(cid:3031)(cid:4667) (cid:1836) (cid:2870) (cid:4666)(cid:3032)(cid:3049)(cid:3032)(cid:3041)(cid:4667) FIG. 6. (a) and (b): Schematics of energy level transitions for qutirts A and A N in Scheme B. (c) and (d): Schematics ofenergy level transitions for qutirts A and A N in Scheme C. The interaction diagram of other cells is the same as Fig. 2 in bothScheme A and B. The choice of coupling strengths and driving fields of last qutrits A N depends on the odevity of N . into, respectively H = J ( | P i h L | + | L i h P | ) , (13a) H N eff = J ( | R i N h L | + | L i N h R | ) , (13b)with J = J ′ Ω / ∆ , which involve solelylong-lived states. When the chain with the size L =2 N − | l B i = | ψ i = | P LR · · · LR i A ⊗ | · · · i B under the inter-action Hamiltonians (12a) and (12b), the system evolvesin the finite space {| ψ n i}| ψ i = | P LR · · · LR i A ⊗ | · · · i B , | ψ i = | eLR · · · LR i A ⊗ | · · · i B , | ψ i = | LLR · · · LR i A ⊗ | · · · i B , | ψ i = | LeR · · · LR i A ⊗ | · · · i B , ... | ψ i = | LRL · · · RR i A ⊗ | · · · i B , | ψ i = | LRL · · · Re i A ⊗ | · · · i B , | ψ i = | LRL · · · RL i A ⊗ | · · · i B . (14)To further evaluate the topological properties of zero en-ergy states in the Scheme B, we plot the distribution ofthe zero mode on component states | ψ n i , as shown inFig. 7(a). The components | ψ i and | ψ i , which playroles of left and right edge states, | l B i and | r B i , respec-tively, are populated with the maximal distributions inthe regions of g t ∈ [0 , g t ∈ (2200 , N − N + 1 due to the introductions of classicaldrives on the two extremity qutrits of the chain. In theScheme B, an extra long-lived state | P i is introduced inthe qutrit A , and the large detuning condition is satis-fied so as to eliminate adiabatically the excited state | e i . In other words, the state transformations of Scheme Aand Scheme B are, respectively | l A i = | eLR · · · R i A ⊗ | · · · i B ⇓| r A i = | LRL · · · e i A ⊗ | · · · i B , and | l B i = | P LR · · · LR i A ⊗ | · · · i B ⇓| r B i = | LRL · · · RL i A ⊗ | · · · i B . Thus, the distribution and spacing of bright and darkfringes in Fig. 7(a) are different from that in Fig. 2(d).The dark fringes in Fig. 7(a) appearing on | ψ i and | ψ i indicate that the population elimination of the excitedstate | e i in qutrits A and A .For sake of generating large-scale GHZ states, the su-perconducting qutrit-resonator chain is assumed initiallyin the state | Φ ′ i = 1 √ (cid:16) | G i + | l ′ B i (cid:17) . (15)where | l ′ B i = | P LR · · · m i A N ⊗ | · · · i B N − and m = R ( L ) when N is odd (even). After the evolution alongthe topologically protected zero-energy mode, similar tothe Scheme A, one can obtain the final state | Φ ′ ideal i = 1 √ (cid:16) | G i − ( − N | r ′ B i (cid:17) . (16)in which | r ′ B i = | LRL · · · n i A N ⊗| · · · i B N − with n = L ( R ) when N is odd (even). Here | R i is used to carry thelogical state 1 while | L i carries the logical state 0 for allqutrits. Accordingly, the initial state without consideringthe zero-photon resonators after the evolution becomes1 √ (cid:0) | · · · i A N − ( − N | · · · i A N (cid:1) . (17) (cid:2032) (cid:3041) (cid:2038) (cid:3041) FIG. 7. Distributions of the zero energy mode for (a) Scheme B and (b) Scheme C, respectively, on component states | ψ n i and | φ n i . The size of the superconducting qutrit-resonator chain is L = 2 N − which is exactly an N -body GHZ state.In Fig. 8(a), with the same parameters as in Fig. 3we numerically plot the time evolution of populationsfor the ideal state | Φ ideal i = √ ( | RLR · · · R i A + | LRL · · · L i A ) ⊗ | · · · i B , the initial state | Φ i = √ ( | RLR · · · R i A + | P LR · · · R i A ) ⊗| · · · i B , andtwo edge states | l B i and | r B i . Obviously, the populationof | Φ ideal i ( | Φ i ) reaches nearly 1 (0.25) and keeps steadyat the end of evolution time, which proves the feasibilityof Scheme B. The populations of two edge states havethe identical trend with Fig. 3.In order to verify the effectiveness of suppressing thelosses of qutrits in the excited state, we simulate the fi-delity of the ideal GHZ state | Φ ideal i by solving the non-Hermitian Liouville equation. In Fig. 8(b), the fidelitiesfor 11-, 18-, and 25-body GHZ states hold on 98 . . .
2% even though κ/g = 0 .
05. In addition,it is evident that the fidelity in Fig. 8(b) can reach val-ues similar to that without taking account of edge statesin Fig. 5. When γ/g = 0 .
01, the fidelities of 11-, 18-, and 25-body GHZ states maintain 53 . . . γ , the fidelity of theideal state decreases more slightly and displays a morerobust result than Scheme A because of the suppressionof excitation populations in qutrits A and A N duringthe whole evolution. B. Scheme C for suppressing excitation of allqutrits
As shown in Figs. 6(c) and (d), for the qutrit A , thetransition between | e i ↔ | L i is coupled resonantly tothe resonator B with coupling strength J . When N is odd (even), the transition between | e i ↔ | R i ( | L i ) inthe last qutrit A N is coupled resonantly to the resonator B N − with the coupling strength J . Also, one classicalfield drives the transition resonantly between | e i ↔ | P i for qutrit A with Rabi frequency ˜Ω . Under the con-dition of N being odd (even), the other classical fieldwith Rabi frequency ˜Ω drives resonantly the transition | e i ↔ | L i ( | R i ) in qutrit A N . Therefore, the interactionHamiltonian involving the first and last qutrits can bewritten as H ′ = J b | e i h L | + ˜Ω | e i h P | + H . c .,H N ′ = J b N − | e i N h m | + ˜Ω | e i N h n | + H . c ., (18)where m = R ( L ) and n = L ( R ) when N is odd (even).And the interaction Hamiltonian of other cells still keepsconsistent with Eq. (3). The shapes of the couplingstrengths are engineered as Gauss functions˜Ω = J = g exp [ − ( t − τ ) /τ ] , ˜Ω = J = g exp [ − ( t − τ ) /τ ] , where J and J are reverse with respect to Eq. (7).We take N = 25 as an example, and when thechain is initially in the left edge state | l C i = | φ i = | P LR · · · R i A ⊗ | · · · i B . The system evolves inthe finite subspace {| φ n i}| φ i = | P LR · · · R i A ⊗ | · · · i B , | φ i = | eLR · · · R i A ⊗ | · · · i B , | φ i = | LLR · · · R i A ⊗ | · · · i B , | φ i = | LeR · · · R i A ⊗ | · · · i B , ... | φ i = | LRL · · · R i A ⊗ | · · · i B , | φ i = | LRL · · · e i A ⊗ | · · · i B , | φ i = | LRL · · · R i A ⊗ | · · · i B . (19)Then we plot the distribution of the zero energy modeon component states | φ n i in Fig. 7(b). Distinctly, boththe two ends of evolutionary states | φ i and | φ i are0 (a) (b) (c) (d) FIG. 8. (a) and (c) for Scheme B and Scheme C, respectively: Time evolution of population for | Φ i , | Φ ideal i , | ϕ i , | ϕ ideal i ,edge states | l B(C) i and | r B ( C ) i for the size of superconducting qutrit-resonator chain L = 2 N − γ or κ of the superconducting qutrit-resonatorchain for different number of qutrits, where γ ( κ ) in the legend represents decoherence involving only the qutrit (resonator)decay. Parameters for (a) and (b): J ′ = 20 J , ∆ = 400 g , and Ω = 20 g . populated with the maximal distributions in the regionsof g t ∈ [0 , g t ∈ (2200 , N − N + 1 due to the introductions of classicaldrives on the two extremity qutrits of the chain. In Fig. 7,the distribution and spacing of bright and dark fringesare different from that of Scheme A or Scheme B. Thebright fringes in Scheme A or Scheme B are distributedin the excitations of qutrits, while in Scheme C they aredistributed in the excitations of resonators. Therefore,in Scheme C excitations of all qutrits are suppressed.So as to obtain a large-scale GHZ states, the initialstate of the chain is prepared in Eq. (15). The first statecomponent | G i does not evolve, while the second statecomponent | l ′ B i evolves into ( − N | r ′ B i . Hence, the finalstate is | Ψ F i = 1 √ (cid:0) | G i + ( − N | r ′ B i (cid:1) . (20)which has a minus sign difference from Eq. (16). InFig. 8(c), we plot the time evolution of populations forthe initial state | ϕ i = | Φ i , the ideal state | ϕ ideal i = √ ( | RLR · · · R i A − | LRL · · · L i A ) ⊗ | · · · i B , twoedge states | l C i = | l ′ B i and | r C i = | r ′ B i for generat-ing a 25-body GHZ state. As expected, the populationof | ϕ ideal i ( | ϕ i ) is close to unity (0.25) at the time g t = 3600 and remains stable. As for the two edgestates, the population in Scheme C has the same climate as that in Scheme A and Scheme B. The result revealsthat the above theoretical analysis is correct and fea-sible. Figure 8(d) shows the relationship between thefidelity and the decay rate γ or κ . As the decay rates in-crease, the fidelity of the ideal GHZ state decreases. Asopposed to the loss mechanisms of Scheme A and SchemeB, the decay rate κ of resonators has a greater influenceon the fidelity. When κ/g = 0 .
01, the fidelity is reducedto 56 .
1% for the 11-body GHZ state. In contrast amongthe six lines, the fidelity is relatively immune to the decayof qutrits. The fidelity can retain at 95 . . .
6% with γ/g = 0 .
05, which indicates that Scheme Cdisplays the most robust performance to resist the lossesof qutrits in the superconducting qutrit-resonator chainamong the three schemes.In Scheme C, the fidelity of the ideal large-scale GHZstate is affected least by the losses of qutrits, comparedwith Scheme A and Scheme B. However, the cost is thatthe losses of resonators have a greatest influence on thefidelity, which indicates that the Scheme C requires res-onators to hold a long coherence time. While accord-ing to the interaction of Scheme A and Scheme B, itis required that the qutrits have a long coherence time,but without a strict requirement on the quality factorof resonators. Thus, corresponding to the performanceof experimental devices in the superconducting qutrit-resonator chain, we can choose different schemes to real-ize large-scale GHZ states with high fidelity in the case of1
10 30 50 70 90 110 130 1500.20.40.60.81.0
FIG. 9. Fidelity of the GHZ state with the scale of en-tanglement, N , under the feasible coupling strengths (e.g., g / π = 50 MHz, 10 MHz and 1 MHz) and coherence timesof qutrits and resonators in Scheme A. We choose coherencetime of the qutrits and resonators as τ a = τ b = 1 ms. loss mechanisms by adjusting the energy level structureof qutrits and coupling strengths. V. SCALE OF GHZ STATES
In experiment, by designing a π -phase difference acrossthe Josephson junction in circuit to restrain the energyrelaxation induced by quasiparticle dissipation, one canobtain a qutrit with coherence time over 1 ms [55]. Asfor a resonator, the coherence time of the photons in theresonator can be much longer [65]. Up to now, the super-conducting resonator lifetimes between 1 ms and 10 mshave been reported [52–54]. Generally, the typical feasi-ble coupling strength can be modulated in the range of1 MHz to 50 MHz [51], providing a considerable adjusta-bility in experiment.In this section, we discuss an accessible GHZ state scale N by taking into account the experimentally availablecoherence times of qutrits and resonators under the con-dition of feasible coupling strengths. For convenience,we take Scheme A as an example. Figure 9 shows therelationship between fidelities of GHZ states and thescale of entanglement N with feasible g and coherencetimes of qutrits ( τ a = 1 /γ = 1 ms) and resonators τ b = 1 /κ = 1 ms). Obviously, the fidelity exhibits thedecreasing tendency with increasing N . Regarding tofeasible coupling strengths, the fidelity declines at dif-ferent rates. Under the condition of g / π = 50 MHz,fidelities of 10- to 50-body GHZ states keep above 90%.Even if the scale of a GHZ state is N = 150, its fidelitywould still reach 50 . g / π = 10 MHz, thefidelity of 67-body GHZ state stays above 50%. The fi-delity can be over 90% with N less than 23. However, the GHZ state fidelity has the fastest rate of decline in thecase of g / π = 1 MHz, where the fidelity of a GHZ statecan achieve 50% with the maximum entanglement scale N = 20. Under the existing experimental conditions, thehigher fidelity of large-scale GHZ states can be generatedwith the higher value of coupling strengths. VI. EXPERIMENTAL CONSIDERATION ANDPROSPECTIVE IMPROVEMENTA. Device and initial state
Benefiting from the rapid development in circuit-QEDtechnologies, the circuit-QED system provides us an ex-cellent experimental platform to realize large-scale GHZstates proposed in our work. We can construct a circuit-QED system via arranging the transmission line res-onator and the superconducting qutrits in the space, asshown in Fig. 1(b). According to the existing circuit-QED technology, the qubit-resonator chain systems withon the order of 10–20 qubits have been demonstrated [66–71]. Simultaneously, a chain of 72 superconducting res-onators coupled via transmons can be realized [72]. Thescale of qutrit-resonator chain manufacturing in a metalchip is generally from micron to millimeter [41, 73–75].In general, the flux qutrit is a superconducting circuitmade up of Josephson junctions or/and capacitance [76].While the flux qutrit can be operated at any applied ex-ternal flux through the flux qutrit loop. The resonatoris composed by a linear inductance in parallel with acapacitance, which can be fabricated from a NbN filmdeposited on a sapphire substrate [77]. The coupleris replaced by two Josephson junctions with a SQUIDloop to realize tunable coupling: the magnetic fluxthat threads this loop determines its effective inductance E J ( φ ) ∼ E J (0) cos(2 πφ/ Φ ) [78]. Josephson junctionscan be designed by the metallic chip (Al/AlO x /Al) usingthe electron-beam lithography [74, 79]. In experiment,the circuit-QED device is usually operated in a single-shot liquid He with a base millikelvin temperatures andheavily filtered cryogenic microwave lines [41, 75].In the above schemes, specified initial states are neededto prepare in Eqs. (9) and (15) to realize N -body GHZstates. These specified initial states denote A in the su-perposition ( | R i + | e ( P ) i ) / √ B in | i , A in | L i , · · · , A N in | R i ( | L i ) when N is odd (even). In experiment,the preparation of the initial states can be conductedthrough two steps: (i) As reported recently in Ref. [80],the qutrit holding a Λ-type structure can be cooled toone of its two lowest energy eigenstates ( | R i or | L i ) byresonator decay through spontaneous Raman scatteringso as to obtain the state | RLR · · · R ( L ) i A N . (ii) A π/ A to create a superpositionstate of ( | R i + | e ( P ) i ) / √ / √ | RLR · · · R ( L ) i A N + | P ( e ) LR · · · R ( L ) i A N ) ⊗ | · · · i B N − .2 B. Tunable couplings
In order to achieve high-fidelity GHZ states, tunablecouplings among qutrits and resonators are required.Tunable couplers of both varieties have been realized inseveral experiments: between two tunable qubits [82, 83],between a qubit and a resonator [84, 85], and betweenresonators [86, 87]. Particularly, a direct tunable cou-pler is realized by a tunable circuit element between thequtrit and the resonator, e.g., a flux-biased direct-currentSQUID to generate strong resonant and nonresonant tun-able interactions between a qubit and a lumped-elementresonator [84].In this work, we adopt a direct tunable coupler re-placed by SQUID between qutrits and resonators. Thecoupling strengths J and J can be tunable throughadopting controlled voltage pulses generated by an AWGto tune the flux threading the SQUID loop [41]. For ex-ample, the resonator B N is coupled resonantly to tran-sitions | e i ↔ | m i ( m ∈ { R, L } ). The coupling strength J can be expressed by [88, 89] J = 1 L r ω B N µ ~ h e | Φ | m i Z S B B N ( ~r, t ) · d S , (21)where S is the surface bounded by the loop of SQUID, ω B N the resonator frequencies of B N . Accordingly, B B N ( ~r, t ) is the the magnetic components of the B N resonator mode. For a standingwave resonators, B B N ( ~r, t ) = µ p /V N cos k N z N ( k N is the wave num-ber of B N resonator, V N and z N are the B N resonatorvolume and the B N resonator anxis). In this case, amodulating field can be added when a qutrit works onits optimal frequency point [90], and thus does not causethe shortening of qutrit coherent times. C. Accelerating adiabatic process
Over the past decade, techniques of shortcuts to adi-abaticity (STA) [91, 92] receive a lot of attention, be-cause STA can accelerate adiabatic processes but remainthe robustness of adiabatic processes. Recently, a fastquantum state transfer from the left edge state to theright edge state in a topological SSH chain with next-to-nearest-neighbor (NNN) interaction was presented,which provides the simplest one-dimensional lattice withprotected edge state [50]. The idea of this approach isbased on an engineering of NNN interactions betweenthe sites of the chain, which exactly cancels nonadia-batic couplings due to an imperfect condition of adi-abatic evolution. Inspired by counterdiabatic drivingmethods [93, 94], the quantum chain without limitationof the adiabaticity constraint, combined with a dynami-cal control of NNN interactions, is thus governed by the following Hamiltonian H ( t ): H ( t ) = H ( t ) + H c ( t ) ,H ( t ) = N − X n =1 t ( t ) | B n ih A n | + t ( t ) | A n +1 ih B n | + H . c .,H c ( t ) = N − X n =1 iα n ( t ) | A n +1 ih A n | + H . c ., (22)where H c ( t ) is the control Hamiltonian literally can-celing the nonadiabatic couplings. The time-dependentNNN hoppings α n ( t ) is only to cancel nonadiabatic tran-sitions from the time-dependent eigenvector | φ ( t ) i = N P Nn =1 ( − t t ) n − | A n i , where N is the normalizationconstant.Analogously, with the help of such a method for speed-ing up the adiabatic state transfer in the SSH model, it ispossible to find a control Hamiltonian, added to the ini-tial Hamiltonian, that literally cancels the nonadiabaticcouplings in the topological model. Therefore, it is ofgreat potential to realize fast preparation of a large-scaleGHZ state by engineering the NNN interaction betweenqutrits, which may constrain the impact of systematicdecoherence and thus enhance the fidelity of generatinglarge-scale GHZ states.In addition, an alternative method, Floquet-engineering STA (FESTA) [95, 96], may also beapplied in speeding up the generation of a large-scaleGHZ state. FESTA is an effective succedaneum ofcounterdiabatic driving methods, which does not requirean additional control Hamiltonian but is combined witha periodic driving component to oscillate the initialHamiltonian [95]. This Floquet-engineering periodicdriving component will form an effective connectionbetween transferred two states so as to offset exactlynonadiabatic couplings [97]. FESTA has been demon-strated for quantum state transfer in spin chains in arecent experiment [98]. Therefore, it is of great potentialto use FESTA in our model for accelerating the adiabaticgeneration of GHZ states by replacing the adiabaticpulses with the Floquet-engineering oscillating pulses. VII. CONCLUSION
In conclusion, we have proposed a model of a supercon-ducting qutrit-resonator chain, and the topological edgestate with zero energy is analytically derived. Along thistopological zero-energy mode, a state transfer from oneextremity qutrit to the other of the chain can be imple-mented, accompanied with state flips of all intermediatequtrits. Three schemes are shown for generating large-scale GHZ states that are protected by the topologicalzero-energy mode and thus hold great robustness againstdisorder of the qutrit-resonator coupling strengths. Theeffect of losses induced by decay of qutrits and resonatorson entanglement fidelity is investigated, and the results3indicate that the three schemes meet different perfor-mance requirements of experimental devices. Further-more, we study the accessible entanglement scale whentaking into account the experimentally available coher-ence times of qutrits and resonators, and find that withthe maximum of the qutrit-resonator coupling strengths g / π = 50 MHz, it is possible to achieve a 50-bodyGHZ state with the fidelity F > .
9, and a 150-bodyGHZ state with
F > .
5. Finally, we discuss the experi-mental consideration of generating GHZ states, includingthe physical implementation of the model, preparation ofthe initial states, and tunable couplings in circuit-QEDchain, and also show the potential to accelerate adiabatic process of generating GHZ states. The present work isexpected to be helpful for promoting the experimentalimprovement of large-scale GHZ states.
ACKNOWLEDGEMENTS
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