Bare-excitation ground state of a spinless-fermion -- boson model and W-state engineering in an array of superconducting qubits and resonators
aa r X i v : . [ qu a n t - ph ] M a y Bare-excitation ground state of a spinless-fermion – boson model and W -stateengineering in an array of superconducting qubits and resonators Vladimir M. Stojanovi´c ∗ Institut f¨ur Angewandte Physik, Technical University of Darmstadt, D-64289 Darmstadt, Germany (Dated: May 19, 2020)This Letter unravels an interesting property of a one-dimensional lattice model that describesa single itinerant spinless fermion (excitation) coupled to zero-dimensional (dispersionless) bosonsthrough two different nonlocal-coupling mechanisms. Namely, below a critical value of the effectiveexcitation-boson coupling strength the exact ground state of this model is the zero-quasimomentumBloch state of a bare (i.e., completely undressed) excitation. It is demonstrated here how this lastproperty of the lattice model under consideration can be exploited for a fast, deterministic prepara-tion of multipartite W states in a readily realizable system of inductively-coupled superconductingqubits and microwave resonators. Sophisticated quantum-state engineering [1, 2] is a pre-requisite for the development of next-generation quantumtechnologies [3, 4]. In this context, tantalizing progresswas made in recent years by utilizing diverse physicalplatforms [5–7]. In particular, owing to their continu-ously improving scalability and coherence properties su-perconducting (SC) circuits [8–11] and, among them,circuit-QED systems [12, 13], allow accurate prepara-tion of various quantum states of SC qubits and photonsalike [14].The most prominent classes of entangled many-qubit states are maximally-entangled Greenberger-Horne-Zeilinger (GHZ) [15] and W states [16]. An N -qubit W state is the equal superposition of states withexactly one qubit in its “up” state, the remaining onesbeing in their “down” states. In particular, it is knownthat a W state and its GHZ counterpart cannot be trans-formed into each other via local operations and classicalcommunication (LOCC-inequivalence) [17]. W states arealso extremely robust with respect to particle loss, re-maining entangled even if any N − W states. Its point of de-parture is the notion that in one of the relevant regimesthis model – which includes excitation-boson (e-b) cou-plings of Peierls and breathing-mode types – has an un-conventional ground state. Namely, below a critical valueof the effective e-b coupling strength its ground state isthe zero-quasimomentum Bloch state of a bare excita-tion. It is shown here how this property of the modelunder consideration can be exploited for a fast, deter-ministic preparation of N -qubit W states in an array ofinductively-coupled SC qubits and resonators, an analogsimulator of this model [29].The state-preparation protocol proposed here, based on microwave pumping, allows one to obtain multipartite W states within time frames three orders of magnitudesshorter than the currently achievable coherence times ofSC qubits. Unlike the situation in quantum-state control,where typical preparation times scale unfavorably withthe system size, here they do not depend on the numberof qubits at all. What makes this protocol particularlyrobust is the fact that its target state is the ground stateof the system in a parametrically large window of valuesof its main experimental knob – an external dc flux. Model and its ground state .– The 1D lattice model un-der consideration describes a single spinless-fermion ex-citation interacting with dispersionless bosons throughtwo different nonlocal coupling mechanisms. The nonin-teracting part of its total Hamiltonian includes the exci-tation kinetic-energy- and free-boson terms: H = − t e X n ( c † n +1 c n + H . c . ) + ~ ω b X n b † n b n . (1)Here c † n ( c n ) creates (destroys) an excitation at site n ( n = 1 , . . . , N ), b † n ( b n ) a boson with frequency ω b at thesame site, while t e is the excitation hopping amplitude.The interacting (e-b) part is given by H e-b = g ~ ω b X n h c † n c n ( b † n − + b n − − b † n +1 − b n +1 )+ ( c † n +1 c n + H . c . )( b † n +1 + b n +1 − b † n − b n ) i , (2)where g is the dimensionless e-b coupling strength. Thefirst term on the right-hand-side (rhs) of the last equa-tion captures the antisymmetric coupling of the excita-tion density at site n with the local boson displacementson the neighboring sites n ± n and n + 1 on the respective boson dis-placements (Peierls-type coupling) [31–33].The coupling Hamiltonian H e-b can be recast in thegeneric momentum-space form H e-b = 1 √ N X k,q γ e-b ( k, q ) c † k + q c k ( b †− q + b q ) . (3)Its corresponding e-b vertex function depends on boththe excitation- and boson quasimomenta ( k and q , re-spectively, here expressed in units of the inverse latticeperiod) and is given by γ e-b ( k, q ) = 2 ig ~ ω b [ sin k + sin q − sin( k + q )] . (4)The ground state of H = H + H e-b undergoes a sharplevel-crossing transition [34] at a critical value λ ce-b ∼ λ e-b ≡ g ~ ω b /t e . For λ e-b < λ ce-b the ground state is the K = 0eigenvalue of the total quasimomentum operator K tot = P k kc † k c k + P q qb † q b q . For λ e-b ≥ λ ce-b , on the other hand,the ground state is twofold-degenerate and correspondsto K = ± K gs ( K gs = 0).A ground state with K = 0 is by no means unusual –in fact, an overwhelming majority of coupled e-b mod-els have such ground states. Yet, the model at handhas the peculiar property that its K = 0 ground statefor λ e-b < λ ce-b is the k = 0 bare-excitation Bloch state | Ψ k =0 i ≡ c † k =0 | i e ⊗| i b , where | i e and | i b are the exci-tation and boson vacuum states. In what follows, it willfirst be demonstrated explicitly that | Ψ k =0 i is an exacteigenstate of H for an arbitrary value of λ e-b . It will sub-sequently be shown numerically (see Fig. 2 below) thatfor λ e-b < λ ce-b this state is the ground state of H .Given that | Ψ k =0 i is an eigenstate of H , to prove thatit is an eigenstate of the total Hamiltonian H it sufficesto show that it is also an eigenstate of H e-b . Indeed, byacting with H e-b [cf. Eq. (3)] on this state and makinguse of the fact that c k c † | i e ≡ δ k, | i e , one obtains H e-b | Ψ k =0 i = 1 √ N X q γ e-b ( k = 0 , q ) c † q | i e ⊗ b †− q | i b . (5)Because here γ e-b ( k = 0 , q ) = 0 for an arbitrary q [cf.Eq. (4)], each term in the sum on the rhs of Eq. (5) van-ishes, implying that H e-b | Ψ k =0 i = 0. Therefore, | Ψ k =0 i is an eigenstate of H e-b (for an arbitrary λ e-b ), the corre-sponding eigenvalue being equal to zero. This concludesthe proof that | Ψ k =0 i is an exact eigenstate of H . Qubit-resonator system .– The analog simulator of themodel under consideration [see Fig. 1(a)] consists of SCqubits ( Q n ) with the energy splitting ε z , microwave res-onators ( R n ) with the photon frequency ω c , and couplercircuits ( B n ) [35] which mediate both qubit-qubit andqubit-resonator interactions in this system. The simula-tor can be realized with transmons [12] ( E sJ /E sC ∼ E sC and E sJ are the single-qubit charging- andJosephson energies) or gatemons [36] ( E sJ /E sC ∼ n -th repeating unit is described by the free Hamiltonian H n = ( ε z / σ zn + ~ ω c b † n b n , where the pseudospin-1 / σ n represent qubit n and the bosonic operators( b n , b † n ) photons in the n -th resonator.The upper and lower loops of B n are threaded by mag-netic fluxes φ un and φ ln , respectively [both are expressed in units of Φ / π , where Φ ≡ hc/ (2 e ) is the flux quantum].In particular, the upper loop is subject to ac-driving withthe flux π cos( ω t ). The other contribution to φ un origi-nates from the modes of resonators n and n + 1 and isgiven by φ n, res = δθ [( b n +1 + b † n +1 ) − ( b n + b † n )], where δθ = [2 eA eff / ( ~ d c )] × ( ~ ω c /C ) / , with A eff being theeffective coupling area, C the resonator capacitance, and d the effective spacing in the resonator [37]. Therefore,unlike the much more common capacitive coupling [38],the qubit-resonator coupling in the system at hand is in-ductive [11]. Similarly, φ ln includes an ac contribution,given by − ( π/
2) cos( ω t ), and a dc part φ dc , the mainexperimental knob in this system.The Josephson energy of B n is given by H Jn = − P i =1 E iJ cos ϕ in , with ϕ in being the respective phasedrops on the three Josephson junctions within B n and E iJ their energies, here chosen such that E J = E J ≡ E J and E J = E Jb = E J . Using the flux-quantizationrules [9], the total Josephson energy P n H Jn can be ex-pressed in terms of the gauge-invariant phase variables ϕ n of SC islands of different qubits. The latter enterthis energy through terms of the type cos( ϕ n − ϕ n +1 ),which in the regime of interest for transmons/gatemons( E sJ ≫ E sC ) can be recast (up to an additive constant) as δϕ (cid:2) σ + n σ − n +1 + σ − n σ + n +1 − ( σ zn + σ zn +1 ) / (cid:3) , where δϕ ≡ (2 E sC /E sJ ) / [38]. FIG. 1: (Color online)(a) Schematic of the qubit-resonatorsystem, whose n -th repeating unit (indicated by the dashedrectangle) comprises SC qubit Q n , resonator R n , and couplercircuit B n . The fluxes from the resonator modes n and n + 1thread the upper loops of the coupler circuit B n , effectivelygiving rise to an indirect inductive qubit-resonator coupling.(b) Pictorial illustration of the effective lattice model of thesystem, with the excitation hopping amplitude t ( φ dc ) andeach lattice site hosting dispersionless bosons with the fre-quency δω = ω c − ω . Further analysis is carried out in the rotating frameof the drive, where δω ≡ ω c − ω is the effective bo-son frequency and the Josephson-coupling term becomestime-dependent. This time dependence can, however,be disregarded due to its rapidly-oscillating character,in line with the rotating-wave approximation (RWA).The remaining part of H Jn can succinctly be written as¯ H Jn = −E Jn ( φ dc , φ n, res ) cos( ϕ n − ϕ n +1 ), where E Jn = E Jb (1 + cos φ dc ) − E J J ( π/ φ n, res , (6)and J m ( x ) are Bessel functions of the first kind whosepresence in this expression stems from the use of theJacobi-Anger expansion [39] in conjunction with theRWA. In what follows, without significant loss of gen-erality E Jb is chosen to be given by 2 E J J ( π/ ϕ n − ϕ n +1 ) in terms ofthe operators σ n implies that the effective interactionbetween adjacent qubits in this system is of XY type.Through the flux φ n, res , the interaction strength acquiresa dependence on the boson displacements u n ∝ b n + b † n whose form is equivalent to that of the XY spin-Peierlsmodel [40]. The spinless-fermion – boson coupling thatresults from this interaction via the JW transformationis nonlocal in nature, in contrast to other examples ofsuch couplings in various solid-state systems [28, 41]. Effective Hamiltonian and its ground states .– To showthat the effective system Hamiltonian consists of contri-butions akin to H and H e-b [cf. Eqs. (1) and (2)], oneswitches to the spinless-fermion representation using theJordan-Wigner (JW) transformation. The latter reads σ zn = 2 c † n c n − σ + n = 2 c † n e iπ P l 2) (1 + cos φ dc ) . (7)For a typical resonator δθ ∼ . × − [29]. Besides, for δω it is pertinent to take δω/ π = 200 − 300 MHz andalso choose E J such that δϕ E J / π ~ = 100 GHz. λ e-b -20-100 E g s ( λ e - b ) / E u -2 t ( φ dc ) δω / 2 π = 300 ΜΗ z λ e-b = λ e-b c FIG. 2: Ground-state energy of the system with δω/ π =300 MHz as a function of the effective coupling strength λ e-b .For λ e-b < λ ce-b ≈ . 72 (i.e., φ dc < . π ) the ground state ofthe system corresponds to a bare excitation, while for λ e-b ≥ λ ce-b it corresponds to a heavily-dressed (polaronic) one. The ground-state energy of the system, expressed inunits of E u ≡ − δϕ E J , was evaluated throughLanczos-type exact diagonalization [29, 43] and illus-trated (without the constant-energy contribution) inFig. 2. For λ e-b ≥ λ ce-b the system has a polaron-likeground state (strongly boson-dressed excitation), withits energy showing a rather weak dependence on λ e-b .On the other hand, for λ e-b < λ ce-b the ground state cor-responds to | Ψ k =0 i , i.e., a bare excitation with k = 0.Its energy E gs = − t ( φ dc ) is the minimum of a 1Dcosine-shaped dispersion. The energy separation of thisground state from the first excited state exactly equals ~ δω for any φ dc below the critical value. This is consis-tent with the fact that coupled e-b systems with disper-sionless bosons invariably have one-boson continua sepa-rated from their ground states by the single-boson energyand in the weak-coupling regime typically feature onlyone bound state below those continua [44]. W states and their preparation .– Bearing in mind thatJW strings act trivially on | i e , so that c † n | i e ≡ S + n | i e (where S n ≡ ~ σ n / c † k | i e = N − / N X n =1 e − ikn S + n | i e . (8)The last equation is equivalent to | Ψ k i = | W N ( k ) i ⊗ | i b ,where | W N ( k ) i is a “twisted” N -qubit W state. Thus,bare-excitation Bloch states coincide with generalized W states, while in particular | Ψ k =0 i – the ground state ofthe system at hand for λ e-b < λ ce-b – corresponds to theordinary N -qubit W state | W N i = 1 √ N ( | . . . i + | . . . i + . . . + | . . . i ) . (9)In the following, a microwave-pumping based protocolfor the preparation of an N -qubit W state is proposedassuming that the system is initially in the vacuum state | i ≡ | i e ⊗ | i ph . The external driving required for thispurpose is assumed to be represented by the operatorΩ q d ( t ) = ~ β ( t ) √ N N X n =1 (cid:0) σ + n e − iq d n + σ − n e iq d n (cid:1) , (10)where β ( t ) describes its time dependence and the factors e ± iq d n account for the possibility that flip operations ondifferent qubits are applied with a phase difference. Itis important to stress that the general form of driving inEq. (10) allows one to prepare – through different choicesof q d and β ( t ) – various states of the proposed system, in-cluding its strongly boson-dressed ground states realizedwhen φ dc is above the critical value.The transition matrix element of the operator Ω q d ( t )between the initial state | i and the target state | Ψ k =0 i ≡| W N i ⊗ | i ph evaluates to ~ β ( t ) δ q d ,k =0 , which indicatesthat the preparation of this particular target state re-quires only a global driving field [i.e., q d = 0 in Eq. (10)].Thus, in contrast to some other schemes for W -statepreparation [22], the present one does not require a lo-cal qubit control [45, 46]. By assuming that β ( t ) =2 β p cos( ω d t ), where ~ ω d is the energy difference betweenthe two relevant states, in the RWA these states are Rabi-coupled with the effective Rabi frequency β p [47, 48].Thus, starting from the state | i , the desired state N -qubit W state will be prepared within a time interval ofduration τ prep = π ~ / (2 β p ), which does not depend on N .Taking the pumping amplitude to be β p / (2 π ~ ) =10 MHz, one finds τ prep ≈ 25 ns, which is three ordersof magnitude shorter than typical coherence times of SCqubits (e.g., for transmons T ∼ − µ s [10]). Thusthe proposed protocol should not be affected by a lossof coherence in the system. At the same time, the ob-tained τ prep is sufficiently long that a leakage outside ofthe computational subspace of a single qubit can be ne-glected. Namely, due to the multilevel character of SCqubits, a finite anharmonicity α ≡ E − E (where E ij is the energy difference between qubit states j and i ) isrequired. In order to avoid such a leakage, the minimalpulse duration of t p ∼ ~ / | α | is necessary. For transmons( α ∼ − 200 MHz), even a few-nanoseconds-long pulse isfrequency selective enough that such a leakage is negligi-ble [13]. The obtained τ prep ∼ 25 ns suffices even in thecase of gatemons, whose typical anharmonicity is by afactor of two smaller than that of transmons [49].Importantly, the large energy separation ~ δω betweenthe target state and the lowest-lying excited state of thesystem ensures that the proposed W -state preparationwill not be hampered by an inadvertent population of un-desired states. For instance, for δω/ π = 200 (300) MHzthis energy separation is equal to 2 E u (3 E u ), which rep-resents a significant fraction of the energy difference be- tween the initial and target states (cf. Fig. 2).The proposed protocol is deterministic in nature andgenerates W -type entanglement of all the qubits in par-allel. Moreover, in contrast to the typical situation inquantum-state control, where state-preparation times of-ten scale unfavorably with the system size, here τ prep doesnot depend on the system size at all. Finally, becausethey represent ground states of the system, multipartite W states prepared by this protocol can be expected tobe extremely robust.Besides allowing W -state preparation, the proposedsystem features an XY -type qubit-qubit interaction,which opens the possibility for a universal quantum com-putation [50, 51]. Because the strength of this interac-tion depends dynamically on the boson degrees of free-dom (photons), this system bears a formal similarity tocertain trapped-ion systems in which the role of bosonsis played by collective motional modes (phonons) [52].Compared to its trapped-ion counterparts, this systemhas an added advantage that it merely involves disper-sionless bosons of one single frequency, which circumventsthe spectral crowding problem resulting from the quasi-continuous character of phonon spectra in large trapped-ion chains [53]. Robustness to losses and feasibility .– It is pertinent tobriefly address the robustness of the system at hand topossible deleterious effects of losses. To this end, it isworthwhile to first note that qubit-state flips and dis-placements of the resonator modes are the two leadingsources of decoherence in this system. In addition to thevery long T times of transmon (gatemon) qubits, thedamping time of microwave photons in coplanar waveg-uide resonators can reach the same order of magnitudeas T , with the corresponding quality factor being largerthan 10 [54]. Besides, the relevant excitation- and pho-ton energy scales in this system ( δω , gδω , t / ~ ), ex-pressed in frequency units, are all of the order of several2 π × 100 MHz. Thus, they far exceed the decoherencerates whose state-of-the-art values in this type of systemsare γ ∼ . 01 MHz [10]. Finally, in this system thermalexcitations – which at temperatures typical for such SC-qubit setups ( T ∼ 100 mK) have characteristic energiesof a few GHz – can be safely neglected. Therefore, theloss mechanisms do not pose obstacles to realizing theproposed system. Conclusions .– The present paper proposes a schemefor a fast, deterministic creation of a large-scale W -typeentanglement [55] in a system of inductively-coupled su-perconducting qubits and microwave resonators. Themechanism behind this scalable entanglement resource– which allows one to engineer W states with the prepa-ration times independent of the system size – stemsfrom the unconventional ground-state properties of a one-dimensional model describing a nonlocal coupling of aspinless fermion to zero-dimensional bosons. The feasi-bility and robustness of the underlying state-preparationprotocol – which only requires a global driving field – isdemonstrated with realistic system parameters.This study can be viewed as being complementary tothat of Ref. 22, where the preparation of W states of pho-tons – rather than qubits – was proposed. The commondenominator of these two proposals is that they both relyon superconducting systems and an in-situ tunability of ahopping amplitude, albeit being based on completely dif-ferent physical mechanisms. These schemes are far morescalable than the conventional ones in which resonator-mediated qubit-qubit interactions are utilized to control-lably entangle multiple qubits; such an approach was re-cently used to prepare a GHZ state of 10 superconduct-ing qubits [7] – the largest entanglement demonstrated sofar in solid-state architectures. Thus, the need to demon-strate the envisioned W -state preparation is compelling. Acknowledgments .– The author dedicates this paperto the memory of P. W. Anderson (1923 - 2020). Thisresearch was supported by the Deutsche Forschungsge-meinschaft (DFG) – SFB 1119 – 236615297. ∗ Electronic address: [email protected][1] A. M. Zagoskin, Quantum Engineering: Theory and De-sign of Quantum Coherent Structures (Cambridge Uni-versity Press, Cambridge, UK, 2011).[2] Y. Makhlin, G. Sch¨on, and A. Shnirman, Quantum-stateengineering with Josephson-junction devices, Rev. Mod.Phys. , 357 (2001).[3] J. P. Dowling and G. J. Milburn, “Quantum Technology:The Second Quantum Revolution,” Phil. Trans. R. Soc.Lond. A , 16551674 (2003).[4] W. P. Schleich et. al. , Quantum Technology: from re-search to application, Appl. Phys. B , 130 (2016).[5] F. Haas, J. Volz, R. Gehr, J. Reichel, and J. Est`eve,Entangled states of more than 40 atoms in an opticalfiber cavity, Science , 180 (2014).[6] N. Friis et. al. , Observation of Entangled States of a FullyControlled 20-Qubit System, Phys. Rev. X , 021012(2018).[7] C. Song et. al. , 10-Qubit Entanglement and ParallelLogic Operations with a Superconducting Circuit, Phys.Rev. Lett. , 180511 (2017).[8] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybridquantum circuits: Superconducting circuits interactingwith other quantum systems, Rev. Mod. Phys. , 623(2013).[9] For an introduction, see U. Vool and M. Devoret, Intro-duction to quantum electromagnetic circuits, Int. J. Circ.Theor. Appl. , 897 (2017).[10] See, e.g., G. Wendin, Quantum information process-ing with superconducting circuits: a review, Rep. Prog.Phys. , 106001 (2017).[11] For a recent review, see P. Krantz, M. Kjaergaard, F.Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, AQuantum Engineer’s Guide to Superconducting Qubits,Appl. Phys. Rev. , 021318 (2019).[12] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I.Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design de-rived from the Cooper pair box, Phys. Rev. A , 042319(2007).[13] For an introduction, see S. M. Girvin, in Lecture Notes onStrong Light-Matter Coupling: from Atoms to Solid-StateSystems (World Scientific, Singapore, 2013), pp. 155-206.[14] See, e.g., B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E.Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H.Devoret, and R. J. Schoelkopf, Deterministically Encod-ing Quantum Information Using 100-Photon Schr¨odingerCat States, Science , 607 (2013).[15] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell’sTheorem, Quantum Theory, and Conceptions of the Uni-verse (Kluwer Academic, Dordrecht, 1989), pp. 73-76.[16] W. D¨ur, G. Vidal, and J. I. Cirac, Three qubits can beentangled in two inequivalent ways, Phys. Rev. A ,062314 (2000).[17] M. A. Nielsen, Conditions for a Class of EntanglementTransformations, Phys. Rev. Lett. , 436 (1999).[18] M. Koashi, V. Buˇzek, and N. Imoto, Entangled webs:Tight bound for symmetric sharing of entanglement,Phys. Rev. A , 050302(R) (1999).[19] J. Joo, Y.-J. Park, S. Oh, and J. Kim, Quantum telepor-tation via a W state, New J. Phys. , 136 (2003).[20] H. H¨affner et. al. , Scalable multiparticle entanglement oftrapped ions, Nature (London) , 643 (2005).[21] R. McConnell, H. Zhang, J. Z. Hu, S. ´Cuk, and V.Vuleti´c, Entanglement with negative Wigner functionof almost 3,000 atoms heralded by one photon, Nature(London) , 439 (2015).[22] A. A. Gangat, I. P. McCulloch, and G. J. Milburn, De-terministic Many-Resonator W Entanglement of NearlyArbitrary Microwave States via Attractive Bose-HubbardSimulation, Phys. Rev. X , 031009 (2013).[23] C. Li and Z. Song, Generation of Bell, W ,and Greenberger-Horne-Zeilinger states via exceptionalpoints in non-Hermitian quantum spin systems, Phys.Rev. A , 062104 (2015).[24] F. Reiter, D. Reeb, and A. S. Sørensen, Scalable Dissi-pative Preparation of Many-Body Entanglement, Phys.Rev. Lett. , 040501 (2016).[25] Y.-H. Kang, Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia,Fast preparation of W states with superconducting quan-tum interference devices by using dressed states, Phys.Rev. A , 052311 (2016).[26] J. Chen, H. Zhou, C. Duan, and X. Peng, PreparingGreenberger-Horne-Zeilinger and W states on a long-range Ising spin model by global controls, Phys. Rev.A , 032340 (2017).[27] B. Fang, M. Menotti, M. Liscidini, J. E. Sipe, and V.O. Lorenz, Three-Photon Discrete-Energy-Entangled W State in an Optical Fiber, Phys. Rev. Lett. , 070508(2019).[28] See, e.g., F. M. Souza, P. A. Oliveira, and L. Sanz, Quan-tum entanglement driven by electron-vibrational modecoupling, Phys. Rev. A , 042309 (2019), and refer-ences therein.[29] V. M. Stojanovi´c and I. Salom, Quantum dynamics ofthe small-polaron formation in a superconducting analogsimulator, Phys. Rev. B , 134308 (2019).[30] C. Slezak, A. Macridin, G. A. Sawatzky, M. Jarrell, andT. A. Maier, Spectral properties of Holstein and breath-ing polarons, Phys. Rev. B , 205122 (2006). [31] V. M. Stojanovi´c, P. A. Bobbert, and M. A. J. Michels,Nonlocal electron-phonon coupling: Consequences forthe nature of polaron states, Phys. Rev. B , 144302(2004).[32] V. M. Stojanovi´c and M. Vanevi´c, Quantum-entanglement aspects of polaron systems, Phys. Rev. B , 214301 (2008).[33] K. Hannewald, V. M. Stojanovi´c, and P. A. Bobbert,A note on temperature-dependent band narrowing inoligo-acene crystals, J. Phys: Condens. Matter , 2023(2004).[34] For a general discussion of such transitions, see V.M. Stojanovi´c, Entanglement-spectrum characterizationof ground-state nonanalyticities in coupled excitation-phonon models, Phys. Rev. B , 134301 (2020).[35] For other types of tunable couplers, see M. R. Geller, E.Donate, Y. Chen, C. Neill, P. Roushan, and J. M. Mar-tinis, Tunable coupler for superconducting Xmon qubits:Perturbative nonlinear model, Phys. Rev. A , 012320(2015).[36] T. W. Larsen, F. Kuemmeth, T. S. Jespersen,P. Krogstrup, J. Nyg˚ard, and C. M. Marcus,Semiconductor-Nanowire-Based Superconducting Qubit,Phys. Rev. Lett. , 127001 (2015).[37] T. P. Orlando and K. A. Delin, Introduction to AppliedSuperconductivity (Addison-Wesley, Reading, MA, 1991).[38] F. Mei, V. M. Stojanovi´c, I. Siddiqi, and L. Tian, Ana-log superconducting quantum simulator for Holstein po-larons, Phys. Rev. B , 224502 (2013).[39] M. Abramowitz and I. A. Stegun, Handbook of Mathe-matical Functions (Dover Publications, New York, 1972).[40] See, e.g., L. G. Caron and S. Moukouri, Density MatrixRenormalization Group Applied to the Ground State ofthe XY Spin-Peierls System, Phys. Rev. Lett. , 4050(1996), and references therein.[41] See, e.g., A. P´alyi, P. R. Struck, M. Rudner, K. Flens-berg, and G. Burkard, Spin-Orbit-Induced Strong Cou-pling of a Single Spin to a Nanomechanical Resonator,Phys. Rev. Lett. , 206811 (2012), and referencestherein.[42] P. Coleman, Introduction to Many-Body Physics (Cam-bridge University Press, Cambridge, UK, 2015).[43] J. K. Cullum and R. A. Willoughby, Lanczos Algo-rithms for Large Symmetric Eigenvalue Computations (Birkh¨auser, Boston, 1985).[44] See, e.g., V. M. Stojanovi´c, M. Vanevi´c, E. Demler, andL. Tian, Transmon-based simulator of nonlocal electron- phonon coupling: A platform for observing sharp small-polaron transitions, Phys. Rev. B , 144508 (2014).[45] R. Heule, C. Bruder, D. Burgarth, and V. M. Stojanovi´c,Controlling qubit arrays with anisotropic XXZ Heisen-berg interaction by acting on a single qubit, Eur. Phys.J. D , 41 (2011).[46] V. M. Stojanovi´c, Feasibility of single-shot realizations ofconditional three-qubit gates in exchange-coupled qubitarrays with local control, Phys. Rev. A , 012345(2019).[47] See, e.g., M. Russ, D. M. Zajac, A. J. Sigillito, F. Borjans,J. M. Taylor, J. R. Petta, and G. Burkard, High-fidelityquantum gates in Si/SiGe double quantum dots, Phys.Rev. B , 085421 (2018).[48] See, e.g., D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans,J. M. Taylor, G. Burkard, and J. R. Petta, Resonantlydriven CNOT gate for electron spins, Science , 439(2018).[49] A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F.Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J.Nyg˚ard, K. D. Petersson, and C. M. Marcus, Anhar-monicity of a superconducting qubit with a few-modeJosephson junction, Phys. Rev. B , 060508(R) (2018).[50] N. Schuch and J. Siewert, Natural two-qubit gate forquantum computation using the XY interaction, Phys.Rev. A , 032301 (2003).[51] D. M. Abrams, N. Didier, B. R. Johnson, M. P. daSilva, and C. A. Ryan, Implementation of the XY interaction family with calibration of a single pulse,arXiv:1912.04424.[52] M. L. Wall, A. Safavi-Naini, and A. M. Rey, Boson-mediated quantum spin simulators in transverse fields: XY model and spin-boson entanglement, Phys. Rev. A , 013602 (2017).[53] K. A. Landsman, Y. Wu, P. H. Leung, D. Zhu, N. M.Linke, K. R. Brown, L. Duan, and C. Monroe, Two-qubitentangling gates within arbitrarily long chains of trappedions, Phys. Rev. A , 022332 (2019).[54] Z. Wang, S. Shankar, Z. K. Minev, P. Campagne-Ibarcq,A. Narla, and M. H. Devoret, Cavity Attenuators forSuperconducting Qubits, Phys. Rev. Appl. , 014031(2019).[55] F. Fr¨owis, P. Sekatski, N. Gisin, W. D¨ur, and N. San-gouard, Macroscopic quantum states: Measures, fragility,and implementations, Rev. Mod. Phys.90