Nonadiabatic holonomic quantum computation with dressed-state qubits
Zheng-Yuan Xue, Feng-Lei Gu, Zhuo-Ping Hong, Zi-He Yang, Dan-Wei Zhang, Yong Hu, J. Q. You
aa r X i v : . [ qu a n t - ph ] M a y Nonadiabatic Holonomic Quantum Computation with Dressed-State Qubits
Zheng-Yuan Xue, ∗ Feng-Lei Gu, Zhuo-Ping Hong, Zi-He Yang, Dan-Wei Zhang, Yong Hu, † and J. Q. You ‡ Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, and School of Physicsand Telecommunication Engineering, South China Normal University, Guangzhou 510006, China School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Quantum Physics and Quantum Information Division,Beijing Computational Science Research Center, Beijing 100193, China (Dated: May 26, 2017)Implementing holonomic quantum computation is a challenging task as it requires complicated interactionamong multilevel systems. Here we propose to implement nonadiabatic holonomic quantum computation basedon dressed-state qubits in circuit QED. An arbitrary holonomic single-qubit gate can be conveniently achievedusing external microwave fields and tuning their amplitudes and phases. Meanwhile, nontrivial two-qubitgates can be implemented in a coupled-cavities scenario assisted by a grounding superconducting quantum-interference device (SQUID) with tunable interaction, where the tuning is achieved by modulating the ac fluxthreaded through the SQUID. In addition, our proposal is directly scalable, up to a two-dimensional lattice con-figuration. In the present scheme, the dressed states involve only the lowest two levels of each transmon qubitand the effective interactions exploited are all of resonant nature. Therefore, we release the main difficulties forphysical implementation of holonomic quantum computation on superconducting circuits.
I. INTRODUCTION
The superconducting quantum circuit (SQC) [1–4] is apromising candidate for physical implementation of quantumcomputation due to its flexibility and scalability. However,the noises from the environment severely hinder the perfor-mance of quantum gates. On the other hand, geometric phaseand holonomy depend only on the global property of the evo-lution trajectory and, thus, are insensitive to certain types ofcontrol errors [5–12]. This insensitivity is one of the mainadvantages when implementing quantum computation in ageometric way, as the control lines and devices in a large-scale lattice will inevitably induce local noises and reducethe fidelity of dynamical quantum-gate operations. There-fore, holonomic quantum computation (HQC) [13–17], wherequantum gates are induced by geometric transformations, hasemerged as a potential way for robust quantum computing. Toobtain an adiabatic geometric phase, it requires that the tra-jectory should travel under the adiabatic condition and, con-sequently, the required gate times are on the same order of thecoherent times in typical physical systems [18, 19]. There-fore, increasing research efforts have recently been devotedto nonadiabatic HQC [20–31], and some preliminary quan-tum gates were demonstrated in several experiments [32–36].Nevertheless, due to the complicated interaction needed forimplementing two-qubit gates, up to now only single-qubitholonomic gates have been experimentally demonstrated onSQCs [33]. Existing theoretical investigations of two-qubitholonomic gates usually use multilevel systems and result ina slow dispersive gate construction. This is particularly diffi-cult for SQCs, as the anharmonicity of the energy spectrum ofsuperconducting transmon qubits has been reduced to gain ro-bustness against charge-type /f noises [37, 38]. This small ∗ [email protected] † [email protected] ‡ [email protected] anharmonicity limits the coupling strengths one can exploitand makes the implementation of universal HQC with SQCvery inefficient.Here, we present a practical scheme for nonadiabatic HQCin a circuit QED lattice, where we encode the logical qubitsby dressed states built by transmission line resonators (TLRs)coupled with their transmons [37]. In particular, the arbitrarysingle logical qubit operation can be obtained through theproper ac driving of the transmon qubit. More important, wepropose the nontrivial two-qubit gate through the resonant in-teraction between TLRs of the logical qubits, which can be in-duced by a grounding superconducting quantum-interferencedevice (SQUID) with a single-frequency ac magnetic mod-ulation [39–42]. The distinct merit of our scheme is that itinvolves only the lowest two levels of the transmon qubits andcan result in universal HQC in an all-resonant way, thus lead-ing to fast and high-fidelity gates in a simple setup. There-fore, our proposal opens up the possibility of universal HQCon SQC, which can be immediately tested experimentally asit requires only the current state-of-art technology. The cur-rent proposal is essentially different from our previous scheme[28], where the two-qubit gate is implemented between thelogical qubits defined by the decoherence-free subspace en-coding. In addition, more ac modulations of the groundingSQUID are needed in Ref. [28], and the induced interactionsof the logical qubits are complicated as well. II. THE SYSTEM AND THE LOGICAL QUBIT
We propose to realize the scalable HQC on a circuit QEDlattice shown in Fig. 1(a), which consists of three types ofTLRs differed by their lengths and placed in an interlacedhoneycomb form. At their ends, the TLRs are grounded bySQUIDs with effective inductances much smaller than thoseof the TLRs. The role of the grounding SQUIDs is to es-tablish the well-separated TLR modes on this coupled latticeand to induce the consequent coupling between them [39–42]. (d)(a) g ω c G -+ ~ (b)(c) W W TLR Transmon ~ J0 E ext F FIG. 1. Proposed circuit setup of scalable nonadiabatic HQC. (a)Two-dimensional lattice consisting of three types of TLRs placed inan interlaced honeycomb form. (b) Logic qubit built by the coupledTLR-transmon unit. (c) Energy level and driving configuration forthe single-qubit gates in the dressed-state basis. (d) Coupling of thethree TLRs at their common ends by a grounding SQUID, which isthe building block of the 2D lattice. Through the modulation of theSQUID, the two-qubit gate between two dressed-state qubits can berealized.
We specify the eigenfrequencies of the three types of TLRs as ( ω c , ω c , ω c ) = ( ω c , ω c + 3 δ c , ω c + δ c ) with ω c / π = 6 GHz and δ c / π = 0 . GHz. Such frequency configurationis for the following application of parametric coupling andcan be experimentally realized through the length selection ofthe TLRs [43–46]. In addition, we introduce for each TLRa transmon qubit with its eigenfrequency tunable through themodulation of its Josephson coupling energy and the TLR-transmon coupling strength which can reach the strong cou-pling region [4]. The logical qubit of our scheme is physicallyformed by the basic building block of the lattice, i.e., eachTLR together with its transmon, as shown in Fig. 1(b). Takingthe particular TLR-transmon unit in Fig. 1(b) as an example,we can describe it by the Jaynes-Cummings Hamiltonian H JC = 12 ~ ω q σ z + ~ ω c a † a + g ( aσ + + a † σ − ) , (1)where ω q is the eigenfrequency of the transmon qubit, σ ± and σ z are the Pauli operators of the transmon qubit, a † and a arethe creation and annihilation operators of the TLR, and g isthe transmon-TLR coupling strength. In the resonant condi-tion ω q = ω c , the first three lowest eigenstates of the systemare | G i = | i q | i c and |±i = ( | i q | i c ± | i q | i c ) / √ witheigenenergies E G = 0 and E ± = ~ ω c ± g , respectively [Fig.1(c)]. Hereafter, we encode the logic qubit by span {| G i , |−i} and exploit | + i as an ancillary state. III. THE UNIVERSAL SINGLE-QUBIT GATES
The single-qubit nonadiabatic holonomic gates can be es-tablished through a two-tone microwave driving H d = 2 f ( t ) σ z + 2 √ f ( t ) σ x , (2)on the transmon qubit, with f ( t ) = ~ Ω cos(2 g t ) , f ( t ) = ~ Ω cos( E + t + ϕ ) , Ω , being the amplitudes of the two tones,and ϕ being a prescribed phase factor. The σ x tone connect-ing the | G i ⇔ | + i transition can be induced by the capaci-tive link of the external ac pulses to the transmon qubit, andthe σ z tone connecting the |−i ⇔ | + i transition can be ac-complished via the modulation of the Josephson energy of thetransmon through its magnetic flux bias [Fig. 1(b)]. The re-duced Hamiltonian in the subspace span {| G i , |−i , | + i} takesthe form of H = H JC + H d (3) = 2 − f ( t ) f ( t ) − f ( t ) E − − f ( t ) f ( t ) − f ( t ) E + . Assuming g ≫ Ω = p Ω + Ω , we can obtain in the rotat-ing frame of H JC the effective Hamiltonian H eff = ~ Ω (cid:20) sin (cid:18) θ (cid:19) e iϕ | G ih + | − cos (cid:18) θ (cid:19) |−ih + | + H.c. (cid:21) , (4)with θ = 2 tan − (Ω / Ω ) . Such a Λ -type energy con-figuration exhibits the bright and dark states of | b i =sin( θ/ e iϕ | G i − cos( θ/ |−i , | d i = cos( θ/ | G i +sin( θ/ e − iϕ |−i , and its dynamics is captured by H eff1 = ~ Ω( | + ih b | + H.c. ) , (5)that is, a resonant coupling between the bright state | b i andthe ancillary state | + i with the dark state | d i being completelydecoupled. The evolution operator U acting on | b i and | d i ,thus, results in | ψ ( t ) i = U ( t ) | d i = | d i , | ψ ( t ) i = U ( t ) | b i = cos(Ω t ) | b i − i sin(Ω t ) | + i . (6)When the condition Ω τ = π is satisfied, the dressedstates undergo a cyclic evolution as | ψ i ( τ ) ih ψ i ( τ ) | = | ψ i (0) ih ψ i (0) | . Under this condition, the time evolution isgiven by U ( τ ) = X i,j =1 h T e i R τ [ A ( t ) − H ] dt i i,j | ψ i (0) ih ψ j (0) | , (7)where T is the time-ordering operator and A i,j ( t ) = i h ψ i ( t ) | ˙ ψ j ( t ) i . Meanwhile, as H i,j ( t ) = h ψ i ( t ) | H | ψ t j ( t ) i =0 is satisfied, there is no transition between the two time-dependent states. Therefore, the induced operation is a nona-diabatic holonomy matrix U ( τ ) = U ( θ, ϕ ) = (cid:20) cos θ sin θe iϕ sin θe − iϕ − cos θ (cid:21) , (8) Ω H t/ π Ω N t/ π |G>|−>|+>F H/N Ω H/N t/ π F H F N (a) (b)(c) FIG. 2. State population and fidelity dynamics of the single-qubit op-erations with the initial state being | G i . The results of the Hadamardgate and the NOT gate situations are shown in (a) and (b), respec-tively. The dynamics of the gate fidelities averaged over inputstates with uniformly distributed θ ′ is plotted in (c), with the detailsaround the top of the curves shown in the inset. in the subspace span {| G i , |−i} . This gate manifests its geo-metric feature by its dependence only on the global propertyof the path but not the traverse detail [20, 21]. In addition, as θ and ϕ can be independently controlled by the two-tone driv-ing H d , Eq. (8), thus, pinpoints the arbitrary synthesization ofnonadiabatic single-qubit HQC gates.The performance of the proposed single-qubit gate U ( θ, φ ) is mainly limited by the decoherence of the TLR-transmoncircuit, the anharmonicity of the transmon, and the leakage ofthe logic qubit subspace and can be numerically simulated byusing the master equation ˙ ρ = i [ ρ , H ] + κ L ( a )+ X j =0 " Γ j L ( σ − j,j +1 ) + Γ j L ( σ zj,j +1 ) , (9)where ρ is the density matrix of the considered system, L ( A ) = 2 Aρ A † − A † Aρ − ρ A † A is the Lindbladianof the operator A , and κ , Γ j , Γ j denote the decay rate ofthe TLR, the decay and dephasing rates of the { j, j + 1 } two-level systems, respectively. Because of the finite anhar-monicity of the transmon, here we include the third levelof the transmon into the numerical simulation by denoting σ − j,j +1 = | j ih j + 1 | , σ zj,j +1 = | j + 1 ih j + 1 | − | j ih j | .Suppose that the qubit is initially prepared in the state | G i .We then evaluate the Hadamard and NOT gates using the fi- delities defined by F H = h ψ f | ρ | ψ f i and F N = h−| ρ |−i ,with | ψ f i = ( | G i − |−i ) / √ or |−i being their correspond-ing target final states. The obtained fidelities are as high as F H = 99 . and F N = 99 . at t = π/ Ω H/N , as shownin Figs. 2(a) and 2(b). The parameters of the logic qubitare set as ω c = ω q = 2 π × GHz, g / π = 300 MHz, Ω H = Ω N = 2 π × MHz, and Γ j = Γ j = κ = 2 π × kHz corresponding to the coherent time of 16 µ s accessi-ble with the current level of technology [4, 47]. The an-harmonicity of the third level is set as π × MHz [33].For the Hadamard gate, we modulate Ω / Ω H ≃ . and Ω / Ω H ≃ . to ensure θ = π/ , while for the NOT gatewe choose Ω = Ω = Ω N / √ . Our numerical results indi-cate that the infidelity is mainly due to the decoherence, thelimitation on the anharmonicity of transmons, and the leakageof the logical qubit subspace.It should be emphasized that our numerical calculation isbased on the full Hamiltonian H in Eq. (3) and does not relyon any further approximation. Moreover, the interactions be-tween the higher levels of the transmon and the TLR mode andthe effects of the two-tone driving in the expanded Hilbert sub-space are taken into account. In addition, for a general initialstate | ψ i = cos θ ′ | G i + sin θ ′ |−i with θ ′ = 0 correspondingto the ground state, we numerically confirm that the fidelitydepends weakly on θ ′ . To fully quantify the performance ofthe implemented gate, we plot in Fig. 2(c) the gate fidelitiesfor input states with θ ′ uniformly distributed over [0 , π ] ,where we find that F GH = 99 . and F GN = 99 . whichare higher than the threshold of surface code error correctionschemes. IV. THE NONTRIVIAL TWO-QUBIT GATE
We next consider the implementation of two-qubit HQCgates between the neighboring logic qubits 1 and 2 in Fig.1(a). This can be achieved by the ancillary of the third logicqubit 3, which shares the same grounding SQUID with thetwo target qubits. Without loss of generality, here we set theTLR-transmon coupling g = g = g = g = 2 π × MHzamong the three logic qubits. When the grounding SQUID isdc biased, the linear coupling between the three TLRs can bereduced to H dc = ~ (cid:16) J a † a + J a † a + J a † a (cid:17) + H . c . = ~ X j J j,j +1 ( | G −i + | G + i ) j,j +1 × ( h− G | + h + G | ) + H . c ., (10)in the dressed-states subspace, with J j,j +1 ≪ δ c being thedc coupling strength induced by the grounding SQUID (seeAppendix A for details). Because of the large detuning δ c , thestatic exchange coupling H dc does not produce a significanteffect. Meanwhile, we can exploit the alternative dynamicmodulation method [43, 44, 48, 49]: The grounding SQUIDcan be regarded as a tunable inductance which can be ac mod-ulated by external magnetic flux oscillating at very high fre-quency [49]. Such ac modulation introduces a small fraction H ac = X j ~ J ac j,j +1 ( t )( a † j a j +1 + H . c . ) , (11)in addition to the irrelevant dc H dc (see Appendix B for de-tails). The modulating frequency of Φ acex ( t ) must be lower thanthe plasma frequency ω p of the grounding SQUID [37], oth-erwise the internal degrees of freedom of the SQUID will beactivated and complex quasiparticle excitations will emerge[39]. In our setup, the condition ω p ≫ δ c is well fulfilled, andthe excitation of the grounding SQUID is highly suppressed.Generally, we may assume that the ac modulation of thegrounding SQUID contains two tones which induce the exci-tation exchange of | − G i , ↔ | G + i , and | − G i , ↔| G + i , by bridging their frequency gaps, respectively. How-ever, with our prescribed TLR frequencies and identical TLR-transmon coupling strength, the two target transitions are ofthe same frequency gap (see Appendix C for details), and,thus, they can be induced by a single frequency ac modula-tion. In the rotating frame of H JC , H ac can then be reducedto H = ~ T ( | − G i , h G + | + | − G i , h G + | ) + H . c ., (12)where T / π ∈ [5 , MHz is the parametric couplingstrength induced by the parametric modulation. The otherallowed transitions in H ac are detuned at least by g andcan thus be safely neglected by the rotating-wave approxi-mation. Similar to the single-qubit case, we can figure outthat the single excitation subspace span {| − GG i , , , | G − G i , , , | GG + i , , } constitutes a three-level system. Whenthe cyclic condition R τ Jdt = π with J = √ T is fulfilled, aholonomic quantum gate U = − , (13)can be induced in the Hilbert subspacespan {| GG i , , | G −i , , | − G i , , | − −i , } . Thecombination of U and U ( θ, ϕ ) , thus, forms a univer-sal set of quantum gates. We note that the minus signfor the element | − −i , h− − | in Eq. (13) comesfrom the holonomic dynamics of another subspacespan {| − − G i , , , | − G + i , , , | G − + i , , } , whichhas the same energy spectrum as that of the two-qubit gatesubspace span {| − GG i , , , | G − G i , , , | GG + i , , } .Within this subspace, the | − −i , state obtains a π phaseduring the implementation of the two-qubit gate in Eq. (13).Similarly, we further verify the performance of the two-qubit gates by taking T / π = 6 MHz. We calculate the statepopulations and fidelity for an initial state | − GG i , , usingthe Hamiltonian H ac in Eq. (11) and plot the fidelity dynamicsof F T = , , h G − G | ρ | G − G i , , with ρ being the time-dependent density matrix of the considered two-qubit system.As shown in Fig. 3, the obtained fidelity is comparable tothat of the single-qubit operations, with a fidelity as high as F T = 99 . . This fidelity is in sharp contrast with the ex-isting implementations and can be interpreted in an intuitive π P −G− P −GG P GG+ P G−G P GGG
FIG. 3. State population and fidelity dynamics of the U gate as afunction of Jt/π with the initial state being | − GG i , , . way: As the interactions exploited in our scheme are resonant,the speed of two-qubit gate is comparable to the case of thesingle-qubit gate, which is distinct from the previous schemesusing dispersive couplings. V. DISCUSSION
Our scheme can be readily scaled up to facilitate the scala-bility criteria of quantum computing. As shown in Fig. 1(a),we can form a 2D array of the logic qubit by placing the TLRsin an interlaced honeycomb lattice. This configuration allowsthe holonomic two-qubit gates to be established between anytwo logic qubits sharing the same grounding SQUID with thethird one serving as ancillary. With regard to the feasibilityof current proposal, we first notice that the elementary gatesinvolve the control of both the SQUIDs of the transmon qubitand the grounding SQUIDs. This is well within the reach ofthe current level of technology, as both the dc and ac flux con-trols have already been achieved in coupled superconductingqubits with both the loop sizes and their distances being atthe range of micrometers [50, 51]. As for the scaled lattice,the individual control, wiring, and readout can be achievedby adding an extra layer on the top of the qubit lattice layer[52–54], and the interlayer connection can be obtained by thecapacitive coupling. In addition, the parametric coupling ex-ploited in our scheme has been demonstrated previously infew-body systems [43–46] and recently in a SQC lattice, witha synthetic gauge field for the microwave photons being ob-served [55]. This experimental progress, thus, partially ver-ifies the feasibility of our scheme. Finally, the fluctuationinduced by the ubiquitous flicker noises in the SQC shouldalso be considered [38]. We notice that the proposed circuitis insensitive to the charge noise as it consists of only linearTLRs, grounding SQUIDs with very small anharmonicity andthe charge-insensitive transmon qubits [37]. For the flux-typeand critical current-type /f noise, their influence is estimatedto be much weaker than the decay effect [40–42], which hasalready been included in our numerical simulations. VI. CONCLUSION
In summary, we propose a scheme of quantum computa-tion with dressed-state qubits in circuit QED using nonadia-batic holonomies. In particular, the single-qubit gates can beachieved through external microwave-driving fields and thetwo-qubit gates can be obtained in a fast resonant way. There-fore, our scheme presents a promising way of realizing robustand efficient HQC in superconducting devices.
ACKNOWLEDGMENTS
This work is supported by the NFRPC (GrantNo. 2013CB921804), the NKRDPC (Grants No.2016YFA0301200 and No. 2016YFA0301803), the NSFC(Grants No. 11104096, No. 11374117 and No. 11604103),NSAF (Grants No. U1330201, and No. U1530401), and theNSF of Guangdong Province (Grant No. 2016A030313436).
Appendix A: The dc mixing induced by the grounding SQUID
In this appendix, we derive in detail the coupling betweenthe logic qubits through the detailed analysis of a three-qubitunit cell of the proposed circuit lattice. During this investiga-tion, we also estimate the parameters of the proposed circuitbased on recently reported experimental data [43–46, 49] andpropose their representative values, as listed in Table I. Theinfluence from the other part of the lattice is temporarily mini-mized by setting the grounding SQUIDs at the individual endsof the three TLRs with infinitesimal effective inductances.We assume the common grounding SQUID of theunit cell has an effective Josephson energy of E J = E J cos( π Φ ext /φ ) with E J being its maximal Josephsonenergy, Φ ext the external flux bias, and φ = h/ e the fluxquanta. In the first step, let us assume that only a dc flux bias Φ dcex is added. Physically, a certain TLR can hardly ”feel” theother two TLRs as the currents from them will flow mostly tothe ground through the SQUID due to its very small induc-tance [39, 40]. The SQUID can then be regarded as a low-voltage shortcut of the three TLRs and, thus, allows the def-inition of individual TLR modes in this unit cell; see Refs.[28, 42] for details. Meanwhile, the eigenmodes are well sep-arated in the corresponding TLRs, indicating the one-to-onecorrespondence between the TLRs and the eigenmodes. Fur-thermore, these eigenmodes can well be approximated by the λ/ mode of the TLRs with the nodes located at the nodes,which is consistent with the described shortcut boundary con-dition. In addition, the eigenmodes can be quantized as H = X m ~ ω cm ( a † m a m + 12 ) , (A1)where ω cm , a † m , and a m are the frequency, creation, and anni-hilation operators of the m th eigenmode.Here, we temporarily stop to check the role played by thegrounding SQUID. First, the gauge-invariant phase difference TABLE I. Representative parameters of the proposed circuit selectedbased on recently-reported experiments.TLRs parametersunit inductance l = 4 . × − H · m − [43–45]unit capacitance c = 1 . × − F · m − [43–45]lengths L = 10 . L = 8 . L = 9 .
57 mm [43, 44, 49]SQUIDsmaximal critical currents I J0 = 46 µ A [43, 49, 56, 57]dc flux bias points Φ dcex = 0 . [43, 44]effective critical currents I J = 10 µ A junction capacitances C J = 0 . [56, 57]ac modulation amplitudes Φ = 1 . Φ = 1 . [43]Eigenmodes and couplingeigenfrequencies ω c / π = 6 GHz ω c / π = 7 . GHz ω c / π = 6 . GHz [43, 44, 49]uniform decay rate κ/ π = 10 kHz [4]hopping strengths T / π = T / π = 6 MHz of the SQUID can be written as φ J = X m φ m ( a m + a † m ) , (A2)where φ m = f α,m ( x = L α ) p ~ / ω cm c is the rms node fluxfluctuation of the m th mode across the SQUID with ( φ , φ , φ ) /φ = (3 . , . , . × − . (A3)Such small fluctuation of φ J verifies the linearized treatmentof the grounding SQUID in the quantization of the eigen-modes and indicates that the eigenmodes can be regarded asthe individual λ/ modes of the TLRs slightly mixed by thegrounding SQUID with small inductance.We then proceed to estimate to what extent the groundingSQUID mixes the individual λ/ modes of the TLRs, whichis due to the dc Josephson coupling E dc = − E J cos (cid:18) φ J φ (cid:19) ≈ (cid:18) φ J φ (cid:19) E J cos (cid:18) Φ dcex φ (cid:19) = ~ X m,n J dc mn ( a † m + a m )( a † n + a n ) , (A4)with Φ dcex being the external dc magnetic flux, and the couplingstrength between two eigenmodes is J dc mn = φ m φ n φ E J cos (cid:18) Φ dcex φ (cid:19) . (A5) J dc mn can then be regarded as the dc mixing between the indi-vidual λ/ modes induced by the static bias of the groundingSQUID. As J dc mn ≃ π × MHz < δ c / , (A6)the grounding SQUID can slightly mix the original modes ofthe TLRs.We can also estimate the higher fourth-order nonlinear termof − E J cos( φ J /φ ) as E ≈ (cid:18) φ j φ (cid:19) E J cos (cid:18) Φ dcex φ (cid:19) ∼ − J dc mn , (A7)i.e., 6 orders of magnitude smaller than the second-order termsreserved in Eq. (A4) and, thus, verifies the validity of keepingonly the second-order terms in Eq. (A4).In addition, we can observe that J mn scales versus E J as J dc mn ∝ E − J with increasing E J . This can be inter-preted by the role of the grounding SQUID. Because of thelow-inductance shortcut boundary condition, the node flux φ J across the grounding SQUID scales as E − ; thus, the cou-pling energy E J cos( φ J /φ ) ≈ − φ J / L J ∝ E − J . Thisscaling behavior provides an efficient way of suppressing theunwanted cross talk on the lattice: One can isolate a part ofthe lattice (e.g., a few logic qubits) by simply tuning up theJosephson energies of the grounding SQUIDs it shares withthe other parts. Appendix B: Parametric coupling between the eigenmodes
The parametric coupling between the three logic qubitsoriginates from the dependence of E J on the total externalmagnetic flux Φ ext = Φ dcex + Φ acex ( t ) , E J = E J cos (cid:18) Φ ext φ (cid:19) ≈ E J cos (cid:18) Φ dcex φ (cid:19) − E J Φ acex ( t )2 φ sin (cid:18) Φ dcex φ (cid:19) , (B1)where we assume that a small ac fraction Φ acex ( t ) is added to Φ ext with | Φ acex ( t ) | ≪ (cid:12)(cid:12) Φ dcex (cid:12)(cid:12) . We first consider the case ofomitting the transmons (e.g., by tuning them far off resonantwith their TLRs) and assume that Φ acex ( t ) is composed of twotones Φ acex ( t ) = Φ cos( ω t ) + Φ cos( ω t ) , (B2)where the ω tone is exploited to induce the ⇔ hopping,and the ω tone is used for the ⇔ hopping. By repre-senting φ J as the form shown in Eq. (A2), we obtain the accoupling from the second term of Eq. (B1) H ac = E J0 Φ acex ( t )4 φ sin (cid:18) Φ dcex φ (cid:19) "X m φ m (cid:0) a m + a † m (cid:1) . (B3)In the rotating frame of H , the induced parametric photonhopping between the TLRs can be further written as H eff2 = e itH H ac e − itH ≃ ~ (cid:16) T , a † a + T , a † a (cid:17) + H . c ., (B4) (cid:90) (cid:90) (cid:16) (cid:14) (cid:16) (cid:14) (cid:90) (cid:16) (cid:14) FIG. 4. The allowed transitions of the three coupled TLRs systemwith the two target transitions are indicated by solid arrows. where T m,n are the effective hopping strengths proportionalto the corresponding Φ mn in Eq. (B2), and other fast-oscillating terms can be omitted due to the rotating-wave ap-proximation. The amplitudes of the two tones can be selectedin the range [Φ , Φ ] = Φ [1 . , . such that thecoupling strength T m,n / π ∈ [5 ,
10] MHz can be induced[43–46].
Appendix C: The two-qubit gates
The described parametric coupling scheme is not influ-enced much by the inclusion of the transmons. We recall thatthe dressed states |−i of the logical qubits are half-TLR plushalf-transmon excitation; therefore, the parametric hopping ofthese states can be directly induced by the parametric couplingof their photonic component. In this situation, we just need toadjust the two-tone pulse to fill the gaps between the transi-tions of ⇔ and ⇔ and enlarge the amplitudes of thetones by twice, as the dressed states contain only half-TLRcomponents. Explicitly, when transmons are loaded into eachof the TLRs, the energy spectrum splits. However, the para-metric coupling can still induce relevant transitions. We nowpresent an example with two TLRs. We still set the parame-ters of the first TLR-transmon unit as ω c, = 2 π × GHz and g = g = 2 π × MHz. The third ancillary TLR-transmonunit is designed to be ω c, = 2 π × . GHz and g = g . Bythese settings, the energy spectrum of the two-cavity systemis shown in Fig. 4. Similar to the discussion above, the twoTLRs are coupled in an exchanged manner as H = ~ J ac ( t ) a † a + H.c. (C1) ≡ ~ J ac ( t )2 ( | G −i + | G + i ) , ( h− G | + h + G | ) + H.c. , which means that the four transitions indicated by red lines,both solid and dashed, are allowed. However, as J dc13 ≪ g ,direct transition is not allowed due to the existence of the en-ergy mismatch. To see this, we transform H in Eq. (C1)into the interaction picture with respect to H = ~ X j =1 ( ω − ,j |−i j h−| + ω + ,j | + i j h + | ) . (C2) The transformed Hamiltonian is H = ~ J ac ( t )2 ( | G −i , h + G | e igt + | G −i , h− G | e igt + | G + i , h + G | e igt + | G + i , h− G | e igt ) + H.c. . To induce the transition of | − G i , ↔ | G + i , , we set J ac ( t ) = 4 T cos(6 gt ) . In this case, other allowed transi-tions will be detuned at least by g .For the two-qubit-gate purpose, we set the parameters of thesecond TLR-transmon unit to be ω c, = 2 π × . GHz, g = g , and J ac ( t ) = 4 T cos(6 gt ) , which lead to the transition of |− G i , ↔ | G + i , . Therefore, we need only to ac modulatethe grounding SQUID with a single frequency, i.e., J ac ( t ) =4 T cos(6 gt ) . [1] Y. Makhlin, G. Sch¨on, and A. Shnirman, Quantum-state engi-neering with Josephson-junction devices, Rev. Mod. Phys. ,357 (2001).[2] J. Clarke and F. K. Wilhelm, Superconducting quantum bits,Nature (London) , 1031 (2008).[3] J. Q. You and F. Nori, Atomic physics and quantum optics usingsuperconducting circuits, Nature (London) , 589 (2011).[4] M. H. Devoret and R. J. Schoelkopf, Superconducting cir-cuits for quantum information: An outlook, Science , 1169(2013).[5] P. Solinas, P. Zanardi, and N. Zangh`ı, Robustness of non-Abelian holonomic quantum gates against parametric noise,Phys. Rev. A , 042316 (2004).[6] P. Solinas, M. Sassetti, T Truini, and N. Zangh`ı, On the stabilityof quantum holonomic gates, New J. Phys. , 093006 (2012).[7] M. Johansson, E. Sj¨oqvist, L. M. Andersson, M. Ericsson, B.Hessmo, K. Singh, and D. M. Tong, Robustness of nonadiabaticholonomic gates, Phys. Rev. A , 062322 (2012).[8] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J.Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband,and D. J. Wineland, Experimental demonstration of a robust,high-fidelity geometric two ion-qubit phase gate, Nature (Lon-don) , 412 (2003).[9] J. Du, P. Zou, and Z. D. Wang, Experimental implementation ofhigh-fidelity unconventional geometric quantum gates using anNMR interferometer, Phys. Rev. A , 020302 (2006).[10] P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. G¨oppl, J.M. Gambetta, D. I. Schuster, L. Frunzio, R. J. Schoelkopf, andA. Wallraff, Observation of Berry’s phase in a solid-state qubit,Science , 1889 (2007).[11] M. Pechal, S. Berger, A. A. Abdumalikov, J. M. Fink, J. A.Mlynek, L. Steffen, A. Wallraff, and S. Filipp, Geometric phaseand nonadiabatic effects in an electronic harmonic oscillator,Phys. Rev. Lett. , 170401 (2012).[12] S. Gasparinetti, S. Berger, A. A. Abdumalikov, M. Pechal, S.Filipp, and A. J. Wallraff, Measurement of a vacuuminducedgeometric phase, Sci. Adv. , e1501732 (2016).[13] P. Zanardi and M. Rasetti, Holonomic quantum computation,Phys. Lett. A , 94 (1999).[14] J. Pachos, P. Zanardi, and M. Rasetti, Non-Abelian Berry con-nections for quantum computation, Phys. Rev. A , 010305(1999).[15] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Geometricquantum computation using nuclear magnetic resonance, Na- ture (London) , 869 (2000).[16] L.-M. Duan, J. I. Cirac, and P. Zoller, Geometric manipulationof trapped ions for quantum computation, Science , 1695(2001).[17] V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.-B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret, andL. Jiang, Holonomic quantum control with continuous variablesystems, Phys. Rev. Lett. , 140502 (2016).[18] X.-B. Wang and M. Keiji, Nonadiabatic conditional geometricphase shift with NMR, Phys. Rev. Lett. , 097901 (2001).[19] S.-L. Zhu and Z. D. Wang, Implementation of universal quan-tum gates based on nonadiabatic geometric phases, Phys. Rev.Lett. , 097902 (2002).[20] E. Sj¨oqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M.Johansson, and K. Singh, Non-adiabatic holonomic quantumcomputation, New J. Phys. , 103035 (2012).[21] G. F. Xu, J. Zhang, D. M. Tong, E. Sj¨oqvist, and L. C. Kwek,Nonadiabatic holonomic quantum computation in decoherence-free subspaces, Phys. Rev. Lett. , 170501 (2012).[22] V. A. Mousolou and E. Sj¨oqvist, Non-Abelian geometric phasesin a system of coupled quantum bits, Phys. Rev. A , 022117(2014).[23] J. Zhang, L.-C. Kwek, E. Sj¨oqvist, D. M. Tong, and P. Zanardi,Quantum computation in noiseless subsystems with fast non-Abelian holonomies, Phys. Rev. A , 042302 (2014).[24] Z.-T. Liang, Y.-X. Du, W. Huang, Z.-Y. Xue, and H. Yan, Nona-diabatic holonomic quantum computation in decoherence-freesubspaces with trapped ions, Phys. Rev. A , 062312 (2014).[25] J. Zhou, W.-C. Yu, Y.-M. Gao, and Z.-Y. Xue, Cavity QED im-plementation of non-adiabatic holonomies for universal quan-tum gates in decoherence-free subspaces with nitrogen-vacancycenters, Opt. Express , 14027 (2015).[26] Z.-Y. Xue, J. Zhou, and Z. D. Wang, Universal holonomic quan-tum gates in decoherence-free subspace on superconducting cir-cuits, Phys. Rev. A , 022320 (2015).[27] Y. Wang, J. Zhang, C. Wu, J. Q. You, and G. Romero, Holo-nomic quantum computation in the ultrastrong-coupling regimeof circuit QED, Phys. Rev. A , 012328 (2016).[28] Z.-Y. Xue, J. Zhou, Y.-M. Chu, and Y. Hu, Nonadiabatic holo-nomic quantum computation with all-resonant control, Phys.Rev. A , 022331 (2016).[29] E. Herterich and E. Sj¨oqvist, Single-loop multiple-pulse nona-diabatic holonomic quantum gates, Phys. Rev. A , 052310(2016). [30] P. Z. Zhao, G. F. Xu, and D. M. Tong, Nonadiabatic geomet-ric quantum computation in decoherence-free subspaces basedon unconventional geometric phases, Phys. Rev. A , 062327(2016).[31] V. A. Mousolou, Universal non-adiabatic geometric manipula-tion of pseudo-spin charge qubits, EPL , 10006 (2017).[32] G. Feng, G. Xu, and G. Long, Experimental realization of nona-diabatic holonomic quantum computation, Phys. Rev. Lett. ,190501 (2013).[33] A. A. Abdumalikov, Jr., J. M. Fink, K. Juliusson, M. Pechal, S.Berger, A.Wallraff, and S. Filipp, Experimental realization ofnon-Abelian non-adiabatic geometric gates, Nature (London) , 482 (2013).[34] C. Zu, W. B. Wang, L. He, W. G. Zhang, C. Y. Dai, F. Wang, andL. M. Duan, Experimental realization of universal geometricquantum gates with solid-state spins, Nature (London) , 72(2014).[35] S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasub-ramanian, Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin, Nat. Commun. , 4870 (2014).[36] C. G. Yale, F. J. Heremans, B. B. Zhou, A. Auer, G. Burkard,and D. D. Awschalom, Optical manipulation of the berry phasein a solid-state spin qubit, Nat. Photonics , 184 (2016).[37] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schus-ter, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R.J. Schoelkopf, Charge-insensitive qubit design derived from theCooper pair box, Phys. Rev. A , 042319 (2007).[38] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, /f noise: Implications for solid-state quantum information, Rev.Mod. Phys. , 361 (2014).[39] S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson,P. Delsing, and E. Solano, Dynamical casimir effect entanglesartificial atoms, Phys. Rev. Lett. , 093602 (2014).[40] Y.-P. Wang, W. Wang, Z.-Y. Xue, W.-L. Yang, Y. Hu, and Y.Wu, Realizing and characterizing chiral photon flow in a circuitquantum electrodynamics necklace, Sci. Rep. , 8352 (2015).[41] Y.-P. Wang, W.-L. Yang, Y. Hu, Z.-Y. Xue, and Y. Wu, Detect-ing topological phases of microwave photons in a circuit quan-tum electrodynamics lattice, npj Quantum Inf. , 16015 (2016).[42] Z.-H. Yang, Y.-P. Wang, Z.-Y. Xue, W.-L. Yang, Y. Hu, J.-H.Gao, and Y. Wu, Circuit quantum electrodynamics simulatorof flat band physics in a Lieb lattice, Phys. Rev. A , 062319(2016).[43] E. Zakka-Bajjani, F. Nguyen, M. Lee, L. R. Vale, R. W. Sim-monds, and J. Aumentado, Quantum superposition of a singlemicrowave photon in two different colour states, Nat. Phys. ,599 (2011).[44] F. Nguyen, E. Zakka-Bajjani, R. W. Simmonds, and J. Au-mentado, Quantum interference between two single photons ofdifferent microwave frequencies, Phys. Rev. Lett. , 163602(2012). [45] M. S. Allman, J. D. Whittaker, M. Castellanos-Beltran, K. Ci-cak, F. da Silva, M. P. DeFeo, F. Lecocq, A. Sirois, J. D. Teufel,J. Aumentado, and R. W. Simmonds, Tunable resonant and non-resonant interactions between a phase qubit and lc resonator,Phys. Rev. Lett. , 123601 (2014).[46] A. J. Sirois, M. A. Castellanos-Beltran, M. P. DeFeo, L. Ran-zani, F. Lecocq, R. W. Simmonds, J. D. Teufel, and J. Aumen-tado, Coherent-state storage and retrieval between supercon-ducting cavities using parametric frequency conversion, Appl.Phys. Lett. , 172603 (2015).[47] M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gud-mundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gus-tavsson, Coherence and decay of higher energy levels of a su-perconducting transmon qubit, Phys. Rev. Lett. , 010501(2015).[48] K. Fang, Z. Yu, and S. Fan, Realizing effective magnetic fieldfor photons by controlling the phase of dynamic modulation,Nat. Photonics , 782 (2012).[49] C. M.Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R.Johansson, T. Duty, F. Nori, and P. Delsing, Observation of thedynamical Casimir effect in a superconducting circuit, Nature(London) , 376 (2011).[50] J. H. Plantenberg, P. C. de Groot, C. J. Harmans, and J. E.Mooij, Demonstration of controlled-not quantum gates on a pairof superconducting quantum bits, Nature (London) , 836(2006).[51] S. H. W. van der Ploeg, A. Izmalkov, Alec Maassen van denBrink, U. H¨ubner, M. Grajcar, E. Il’ichev, H.G. Meyer, andA. M. Zagoskin, Controllable coupling of superconducting fluxqubits, Phys. Rev. Lett. , 057004 (2007).[52] R. Barends et al ., Superconducting quantum circuits at the sur-face code threshold for fault tolerance, Nature (London) ,500 (2014).[53] T. Brecht, W. Pfaff, C. Wang, Y. Chu, L. Frunzio, M. H. De-voret, and R. J. Schoelkopf, Multilayer microwave integratedquantum circuits for scalable quantum computing, npj Quan-tum Inf. , 16002 (2016).[54] Z. K. Minev, K. Serniak, I. M. Pop, Z. Leghtas, K. Sliwa, M.Hatridge, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret,Planar multilayer circuit quantum electrodynamics, Phys. Rev.Appl. , 044021 (2016).[55] P. Roushan et al ., Chiral groundstate currents of interactingphotons in a synthetic magnetic field, Nat. Phys. , 146 (2017).[56] Y. Yu, S. Han, X. Chu, S.-I. Chu, and Z. Wang, Coherent tem-poral oscillations of macroscopic quantum states in a Josephsonjunction, Science , 889 (2002).[57] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Rabioscillations in a large Josephson-junction qubit, Phys. Rev. Lett.89