High-performance multiqubit system with double-transmon couplers: Toward scalable superconducting quantum computers

Tunable couplers in superconducting quantum computers have enabled fast and accurate two-qubit gates, with reported high fidelities over 99% in various architectures and gate implementation schemes. However, there are few tunable couplers whose performance in multi-qubit systems is clarified, except for the most widely used one: single-transmon coupler (STC). Achieving similar accuracy to isolated two-qubit systems remains challenging due to various undesirable couplings but is necessary for scalability. In this work, we numerically analyze a system of three fixed-frequency qubits coupled via two double-transmon couplers (DTCs) where nearest-neighbor qubits are highly detuned and also next nearest-neighbor ones are nearly resonant. The DTC is a recently proposed tunable coupler, which consists of two fixed-frequency transmons coupled through a common loop with an additional Josephson junction. We find that the DTC can not only reduce undesired residual couplings sufficiently, as well as in isolated two-qubits systems, but also enables implementations of 30-ns CZ gates and individual and simultaneous 10-ns $\pi/2$ pulses with fidelities over 99.99%. For comparison, we also investigate the system where the DTCs are replaced by the STCs. The results show that the DTC outperforms the STC in terms of both residual coupling suppression and gate accuracy in the above systems. From these results, we expect that the DTC architecture is promising for realizing high-performance, scalable superconducting quantum computers.


I. INTRODUCTION
The advent of tunable couplers has dramatically improved the gate performance of superconducting quantum processors [1][2][3].They can substantially suppress undesired residual couplings by adjusting external parameters such as magnetic flux, and also quickly strengthen the couplings to implement fast two-qubit gates.
To overcome this disadvantage of the STC, a new kind of tunable coupler named the double-transmon coupler (DTC) has recently been proposed [28], numerically investigated [29,30], and experimentally realized [31].The DTC consists of two fixed-frequency transmons coupled through a common loop with an addittional Josephson junction [28][29][30][31].It has been numerically [28,29] and experimentally [31] demonstrated that the DTC can make the residual ZZ coupling strength completely zero or negligibly small even for highly detuned qubits, which has not been achieved with STCs.Numerical gate simulations using rigorous superconductingcircuit models without decoherence effect have shown that a fast CZ gate using a tunable longitudinal coupling [28,29] and a fast √ iSWAP gate using a parametric transverse coupling [29] with fidelities over 99.99% can be implemented for highly detuned fixed-frequency transmon qubits.Furthermore, in the experimental work [31], a CZ-gate fidelity of 99.92±0.01%has been realized stably during 12-hour measurement.This is the highest level of two-qubit gate fidelity among superconducting quantum computers ever reported.It has also been reported in Ref. [31] that the coherence times of qubits coupled via a DTC are also at the highest level as transmons (T 1 = 228.6,205.3 µs and T E 2 = 358.9,129.8 µs at the idle point).These results suggest that the degradation of the coherence time due to the noise channels introduced by a DTC may be rather smaller than or at least comparable to that of conventional tunable couplers.
However, all studies of the DTC reported so far have been done only in isolated two-qubit systems [28][29][30][31] and therefore it has not been clear whether the above abilities of the DTC can be maintained in multi-qubit systems like Fig. 1(a).In other words, the scalability of the DTC architecture has not been understood well, unlike the other architectures including the STC [1,2,[22][23][24][25][32][33][34][35][36].Since there are additional error

FIG. 1. (a)
A typical structure of a multiqubit system with the DTCs.Squares and circles represent qubits and coupler transmons, respectively.The colors of qubits (red or blue) indicate which frequency bands (low or high) they belong to.DTCs are represented by two coupler transmons connected via a thin line.(b) A typical layout of the highlighted part of (a) corresponding to the three-qubit system that we study in this work.Its schematic circuit diagram is shown in Fig. 2(a).Its capacitances estimated by electromagnetic simulator ANSYS Q3D [48] are summarized in Table I. sources in multi-qubit systems, such as unwanted interactions between nonadjacent qubits, between a qubit and a nonadjacent coupler, and between couplers, the detailed study on multi-qubit systems with multiple DTCs is highly desirable.
In this paper, we numerically study the system with three qubits coupled via two DTCs as a minimal model for the above purpose [see the highlighted part in Fig. 1(a)].As a result, we find that ZZ coupling between nearest-neighbor (NN) qubits can be reduced as in isolated two-qubit systems.Furthermore, we show that the ZZ coupling between nextnearest-neighbor (NNN) qubits and ZZZ coupling, which do not exist in isolated two-qubit systems, can also be suppressed down to about 1 kHz.We also numerically demonstrate that 30-ns CZ gates and individual and simultaneous 10-ns π/2 pulses can be implemented with fidelities over 99.99%.These results imply that the DTC works well even in multi-qubit systems.Moreover, as a comparison of the above results, we also evaluate the performance of the system where the two DTCs are replaced by the two STCs.The parameters of STCs have been set based on the experiment reporting a highperformance CZ gate [20].The results show that the DTC outperforms the STC in terms of both residual coupling suppression and gate accuracy in the above systems.Based on our findings, the main reason for the difference lies in the fact that the STC architecture typically exhibits a larger stray coupling between the NNN qubits compared to the DTC architecture This paper is organized as follows.In Sec.II, we introduce a theoretical model of the above three-qubit system in Fig. 1 and show the present parameter setting.In Sec.III A, we show the numerical results of the ZZ and ZZZ couplings and also two-qubit and single-qubit gates (CZ gates and individual and simultaneous π/2 pulses, respectively).For the evaluation of gate performance, we use the average gate fidelities [37,38] for the three-qubit system, which can include the effects of spectator errors [23,[39][40][41][42][43][44][45].In Sec.IV, for comparison, we also analyze the system where the DTCs are replaced by the STCs.In Sec.V, we discuss the difference between the results of the DTCs and the STCs.Finally, we summarize our work in Sec.VI.

A. Circuit
We consider the three-qubit system shown by the highlighted part in Fig. 1 The three-qubit system consists of seven transmons.Transmons 1, 2, and 3 in Fig. 2(a) are fixed-frequency qubits (Q 1 , Q 2 , and Q 3 ).Neighboring two-qubit pairs, (Q 1 , Q 2 ) and (Q 2 , Q 3 ), are coupled via DTCs L and R, respectively.DTC L (R) consists of two fixed-frequency transmons, C 4 and C 5 (C 6 and C 7 ), coupled through a common loop with an additional Josephson junction, the critical current of which, I c8 (9) , is smaller than that of the transmons, I ci (i ∈ {1, 2, 3, 4, 5, 6, 7}).In the loop of DTC µ (µ ∈ {L, R}), the external magnetic flux Φ ex,µ is applied.Each qubit Q i is coupled to a drive line, where voltage V i is applied via a capacitor C di .We assume that C di is negligible with respect to the other capacitances C i j , where C ii is a capacitance between transmon i and the ground, and C i j (i ̸ = j) is the one between transmons i and j.

B. Hamiltonian of the superconducting circuit model
We assume V i = 0 except for single-qubit gates in Sec.III B 2. Then, the Hamiltonian of the rigorous circuit model of this system is written as follows (see Appendix A for the derivation):  where h is the reduced Planck constant, Θ ex,µ = Φ ex,µ /φ 0 is an angle defined with the external flux Φ ex,µ , φ 0 = h/(2e) is the reduced flux quantum, hW = e 2 M −1 /2 with a capacitor matrix M (M ii = ∑ 7 j=1 C i j and M i j = −C i j for i ̸ = j), and hω C i j = e 2 /(2C i j ) (for i ̸ = j), with the elementary charge e. Operators ni , φi , and hω J i = φ 0 I ci are, respectively, the Cooper-pair number operator, the phase difference operator, and the Josephson energy for the ith Josephson junction.Operators ni and φi satisfy the canonical commutation relation [ φi , n j ] = iδ i, j .
In numerical simulations of superconducting quantum computers, effective models derived by approximating the rigorous circuit ones are widely used [3,[17][18][19][21][22][23].The reason for this is that these models are intuitive and computationally light.However, these effective models may lead to inaccurate results compared to rigorous ones like the above.Therefore, in this study, we use the above rigorous supercon-ducting circuit model without approximations, focusing on demonstrating the performance in an ideal situation in the absence of decoherence as rigorously as possible.In the case of a two-qubit system, it has already been reported that such simulation results are in excellent agreement with experimental ones [31].
The Hamiltonian in Eq. ( 1) is represented by a (2N + 1) 7 × (2N + 1) 7 matrix, where N is a cutoff for the Cooperpair number (see Appendix B).In this work, we choose N = 10 so that the energies converge sufficiently.The matrix size of the Hamiltonian of our three-qubit system (1801088541 × 1801088541) is 85766121 times larger than the one in the two-qubit systems studed in Refs.[28,29] (194481 × 194481).Note that calculations of this size are too heavy to perform in a naive manner, e.g., by directly using QuTip [46,47].Due to these difficulties, there have been no studies that conducted gate simulations for multi-qubit systems coupled via tunable coupler using rigorous circuit models, to the best of our knowledge.We overcome this difficulty introducing the dimension reduction technique in Appendix C. We have confirmed that calculated energies and gate fidelities converge with errors of the orders of sub-kHz and 10 −5 , respectively.Therefore, in this work we evaluate residual couplings up to 1-kHz rounding off the sub-kHz fractions and the gate fidelities up to 4-digit precision rounding off the fifth decimal place.
The eigenfrequencies of the three-qubit system and corresponding eigenstates are denoted by Since we are mainly interested in the qubit subspace, we also use notations Hereafter, we set ω 0,0,0 to 0.

C. Parameter setting
Parameter setting is shown in Table I.We choose bare transmon frequencies ω i and capacitances C i j as design values.By definition, W i j is uniquely determined by C i j .The anharmonicity of the bare transmon i is roughly given by (−W ii ) [26].The coupling constant between the bare transmons i and j, g i j , is proportinal to W i j (for i ̸ = j) as follows [28,29]: The Josephson frequencies of transmons 1-7, ω Ji , are calculated as [28,29] As for ω J8(9) , we set r J8 (9) , the ratio of ω J8(9) to the average value of ω J4 (6) and ω J5( 7) , to 0.3 [29].The critical current I ci is proportional to the Josephson frequency as Here, we explain our parameter setting.(1) We set the detunings of the NN qubits, (Q 1 , Q 2 ) and (Q 2 , Q 3 ), to be larger than the absolute values of the qubit anharmonicities W ii (i ∈ {1, 2, 3}).Such a parameter regime, called the highly-detuned regime or out-of-straddling regime [21,28], is preferable to suppress microwave crosstalk between the NN qubits, compared to the nearly resonant or in-the-straddling regime [23].(2) From the perspective of suppressing microwave crosstalk, it is desirable to have a larger detuning between the NNN qubits (Q 1 , Q 3 ) as well.However, alternating red-band (lower-frequency) and blue-band (higher-frequency) qubits to keep the NN qubits highly detuned as shown in Fig. 1, the NNN qubits belong to the same-frequency band and thus must nearly resonate.Thus, in this work, we set the detuning between the NNN qubits to a sufficiently small value (10 MHz) compared to the anharmonicity.We will show that even with such a small detuning, crosstalk between the NNN qubits is negligible in this system, but it is not the case for the STCs.The origin of this peformance difference will be discussed in Sec.V. (3) In actual circuits, there are parasitic capacitances, e.g.C 12 , C 13 , and C 56 , which are not shown in Fig. 2.These capasitances may degrade the performance of the DTCs.Therefore, they should not be ignored.Thus, we set all of the capacitances, including the parasitic ones, to correspond to the layout shown in Fig. 1(b).We estimate them by the electromagnetic simulator ANSYS Q3D [48].We will also discuss the case where the NNN qubits are not on the horizontal line but on the diagonal line in Appendix I. (4) We set ω i (i ∈ {4, 5, 6, 7}) and r Ji (i ∈ {8, 9}) such that they lead to small residual ZZ couplings and a high-performance adiabatic CZ gate operation.

A. Idle point, ZZ couplings, and
Here, Θ Id,R(L) is an idle point of the isolated two-qubit subsystem L(R) where the ZZ coupling between Q 1 and Q 2 (Q 2 and Q 3 ) is the closest to 0 (see Appendix D for the details).At the end of this subsection, we will explain the validity that the idle point of the three-qubit system can be chosen as ), are highlighted by being colored and bold, and the ones of the two-qubit system, as functions of Θ ex,L and Θ ex,R , respectively.The qubit frequencies of the three-qubit system, ω ), are highlighted by being colored and bold, and the ones of the two-qubit system, ω µ  ω 0,1,0 , ω 0,0,1 , ω 1,1,0 , and ω 0,1,1 are in good agreement with the corresponding ones in the two-qubit systems.The black thin curves are frequencies of the states other than the computational state.They cannot be ignored to account for leakage errors during gate simulations.The ZZ coupling between Q i and Q j is denoted by ζ i j (i < j).The ones between the NN qubits, ζ 12 and ζ 23 , are expressed as follows in the three-qubit system: We use the same notation , We find that ζ 12 (23) in the three-qubit system as a function of Θ ex,L(R) [the blue (orange) curve] is in good agreement with the one in system L(R).We also found that ζ 23 (12) is almost independent of Θ ex,L(R) in the three-qubit system and its value is close to the one at the idle point in the isolated twoqubit system R(L) shown by the dotted horizontal line in Fig. 3(c) [3(d)].This independence is reasonable because Θ ex,L(R) is the parameter of DTC L(R) only connecting Q 1 and Q 2 (Q 2 and Q 3 ).
So far, we have considered ZZ coupling between NN qubits, which also exists in two-qubit subsystems.Here, we consider the ZZ coupling between the NNN qubits ζ 13 and the ZZZ coupling, which exist only in n(≥ 3)-qubit systems.In the three-qubit system, they are expressed as follows: From Figs.

B. Gate performance
To evaluate gate performance, we calculate an average gate fidelity F. This is defined by averaging gate fidelities over uniformly distributed initial states and calculated by the following formula [37,38]: where d = 2 n for an n-qubit system, Ûid is an ideal (target) gate operation matrix, and Û′ is an implemented gate operation matrix determined by numerical results (see Appendices E and F).
To implement Ûπ/2,k , we fix Θ ex,µ to Θ Id,µ and apply a driving voltage to Q i for i ∈ k.This operation is described by the following Hamiltonian (see Appendix A for the derivation): (22) where α i is a dimensionless parameter controlled by V i (see Appendix A).Applying optimized α i (t) (see Appendix G 2), we can implement individual and simultaneous 10-ns π/2 pulses with average gate fidelities over 99.99% as summarized in Table III.
In this section, we have shown that DTCs can not only reduce undesired residual couplings to several kHz but also enable implementations of fast CZ gates and individual and simultaneous π/2 pulses with average fidelities over 99.99%, Drive line 3 (b) Two-qubit system L (c) Two-qubit system R  in the three-qubit system as well as in isolated two-qubit systems.These results indicate that the DTC architecture is scalable for highly detuned fixed-frequency qubits.

IV. COMPARISON WITH STC ARCHITECTURE
For comparison, here we consider the system where the DTCs in Fig. 2(a) are replaced by STCs.We compare the DTC architecture with the STC one with respect to the following: 1. Residual ZZ and ZZZ couplings.

A. Circuit
The circuit diagram is shown in Fig. 4(a).This system consists of five transmons.STC L (R) consists of a tunablefrequency transmon C 4 (C 5 ) and a capacitor C 12 (C 23 ).The tunable-frequency transmon C i (i ∈ {4, 5}) consists of two capacitors, C iiA and C iiB , and a dc SQUID including two Josephson junctions with critical currents I ciA and I ciB .Operator φiν and hω Jiν = φ 0 I ciν are, respectively, the phase difference operator and the Josephson energy corresponding to I ciν (ν ∈ {A, B}).In the following, we assume C iiA = C iiB and use notations C ii = C iiA +C iiB , I ci = I ciA + I ciB , and ω Ji = ω JiA + ω JiB .

B. Hamiltonian of the superconducting circuit model
We assume V i = 0 except for single-qubit gates.The Hamiltonian of this system is then written as follows (see Appendix A for the derivation): where hW = e 2 M −1 /2 with a capacitor matrix M (M ii = ∑ 5 j=1 C i j and M i j = −C i j for i ̸ = j), and we have removed φ4B(5B) using the constraint φ 4B(5B) = φ 4A(5A) − Φ ex,L(R) and simply written φ4A(5A) as φ4 (5) .The matrix representation of the Hamiltonian can be obtained in the same manner as the DTC architecture.We again choose the Cooper-pair number cutoff N = 10 for sufficient convergence of energies.
The eigenfrequencies of the three-qubit system and corresponding eigenstates are denoted by ω , respectively.Similarly to the DTCs, we also use notations of qubit frequencies Note that almost all previous theoretical analyses of the STC reported in the literature have been based on the effective model [3,[17][18][19][21][22][23].Unlike them, we use the circuit model for higher accuracy, as well as the DTCs.To the best of our knowledge, this is the first estimation of ZZ coupling between NNN qubits and ZZZ coupling and also the first gate simulation based on the circuit model for systems with the STCs.

C. Comparing conditions
The preferable parameter setting for the comparison between the STC and DTC architectures is non-trivial.This is because the STC and the DTC obey different operating principles, resulting in different parameter values desired for high performance.Here, in order to make the comparison as fair as possible, we set the parameters as shown in Table IV , considering the following points.(1) We set ω i and C ii for i ∈ {1, 2, 3} to be the same as in Table I, which means that qubit frequencies and anharmonicities are almost the same as in the DTC case.(2) Under this condition, we set C i j that exists in the two-qubit subsystems and I c4B(c5B) /I c4A(c5A) to the almost same values as the corresponding ones in Ref. 20, and we also set ω 4 (5) to the almost same value as the coupler idle frequency in Ref. 20. Reference 20 experimentally implemented a high-performance CZ gate (with a gate time of 38 ns and a fidelity of 97.9%), using a parameter setting close to the above condition (1) , where qubits frequencies are 5.038 and 5.400 GHz and both qubits capacitances are 77.8 fF.Moreover, they also showed that the magnitude of the residual ZZ coupling can be suppressed to 60-80 kHz by an STC, which is known to be one of the most ZZ-coupling suppressed results for highly detuned qubits where a fast CZ gate can be implemented.We will show that our numerical simulation reproduces their results well.(3) As for the rest C i j that does not exist in the two-qubit subsystems (e.g.C 13 ,C 15 , and C 45 ), we choose feasible values estimated by electromagnetic simulator ANSYS Q3D [48] assuming the scaled-up design of the circuit in Ref. 20 (see Fig. 5).IV. two-qubit subsystems L(R) Θ Id,R(L) ≡ 0(0) (see Appendix D).The qubit frequencies of the three-qubit system, ω

Idle point, ZZ couplings, and ZZZ coupling
), are highlighted by being colored and bold, and the ones of the two-qubit system, ω µ , 1} and µ ∈ {L, R}) are shown by scatter plots.All of them are needed for calculations of ZZ and ZZZ couplings below.Similarly to the DTC architecture, we find that the corresponding qubit frequencies of the three-qubit and two-qubit systems are in good agreement with each other.The black thin curves represent the states other than the computational ones.They are necessary for accounting leakage errors during gate simulations.
Figure 6(c) [6(d)] shows the ZZ and ZZZ couplings.Solid curves represent the ones of the three-qubit system, and scatter plots represent the ones of the two-qubit system.We find that ζ 12 and ζ 23 , are almost unchanged from the ones of the isolated two-qubit systems and ζ 13 and ζ ZZZ are negligibly small (about 10 and 0 kHz, respectively).These results show that STCs can suppress the residual couplings even in three-qubit systems.Therefore, we can take the idle point of the threequbit system to (Θ ex,L , Θ ex,R ) = (Θ Id,L , Θ Id,R ).However, as in the two-qubit system, their magnitudes shown in Table V are about an order of magnitude larger than the correspond- as functions of Θ ex,L and Θ ex,R , respectively.The qubit frequencies of the three-qubit system, ω ), are highlighted by being colored and bold, and the ones of the two-qubit system, ω µ  ing ones of the DTC architecture.These results show that the DTC is superior to the STC in terms of the residual couplings in three-qubit systems, as well as in two-qubit systems.

Gate performance
Gate simulation results are summarized in Table VI.We find that 30-ns CZ gates with average fidelities of 99.99% cannot be achieved in the three-qubit system.This is in contrast to the corresponding fidelities of the two-qubit subsystems, which exceeds 99.99%.The fidelity degradation of the CZ gate due to the scale-up from two qubits to three is about 0.2%-0.6%,which is roughly 10 times larger than the corre-sponding one in the DTC architecture.We also find that 10-ns π/2-pulses with average fidelities over 99.99% cannot be implemented except for the Q 2 .The infidelities of the π/2 pulses for Q 1 and Q 3 , are about 0.2%.This is also roughly 10 times larger than the corresponding ones of the system of the DTC.
From these results, we conclude that the DTC exhibits higher scalability than the STC, as the tunable coupler for highly detuned fixed-frequency qubits.

V. DISCUSSION
As demonstrated in the above two sections, the DTCs can implement more accurate gate operations in the three-qubit system than the STCs.This difference mainly comes from the larger parasitic coupling of the NNN qubits g 13 in the STC architecture than that in the DTC architecture (see Tables I  and IV).Since tunable couplers can, in principle, only cancel couplings between NN qubits coupled via them, and have no mechanism to cancel the other couplings, g 13 is always active.It means that a larger |g 13 | will lead to more serious errors in the Q 1 -Q 3 subspace, such as microwave crosstalk.In fact, these errors are more pronounced for the STC than for the DTC (see Appendix H).The above difference in g 13 /(2π) (0.13 MHz and 1.62 MHz for the DTC and the STC architecture, respectively) arises from the structure of the coupler and the coupling cancellation mechanism.Since the STCs utilize the relatively large direct capacitances between the NN qubits, C 12 and C 23 , for canceling residual couplings [3], NN qubits tend to be close to each other.This makes NNN qubits close as well, and thus the capacitance between the NNN qubits C 13 tends to be large (0.03 fF).On the other hand, the DTC does not require C 12 and C 23 , allowing for greater distances between NN qubits and hence between NNN qubits, leading to a smaller C 13 (0.003 fF).This difference in C 13 is a major reason for the difference in g 13 .
Note that even if C 13 is zero, there is still an effective capacitance, and hence the g 13 is not zero: 0.04 MHz and 0.71 MHz for the DTC and the STC architecture, respectively.The one in the STC still remains about a half of the original value.
In an ideal limit where all parasitic capacitances are zero, g 13 /(2π) of the DTC and STC become 0.00 MHz and 0.38 MHz, respectively.This means that in the DTC architecture, g 13 arises through parasitic capacitances and vanishes in the ideal situation.On the other hand, even in this ideal situation, g 13 in the STC architecture is about a half of that before taking the limit.
The remaining g 13 in the STC architecture is determined by two factors.One of them is the direct couplings between NN qubits, namely C 12 and C 23 .In the STC architecture, they are large for canceling residual couplings as mentioned above, and thus this contribution does not vanish even in the above ideal situation with no parasitic capacitances.On the other hand, the DTC does not rely on C 12 and C 13 , that is, C 12 and C 23 are parasitic capacitances for the DTC architecture, and hence the contribution is negligible.
The other factor is the indirect couplings through a coupler transmon.This contribution can be estimated by considering the case where C 12 and C 23 are zero, resulting in 0.11 MHz for the STC architecture.In the DTC architecture, corresponding contribution is negligible because the coupling between the two transmons inside DTC L(R) is nearly disconnected due to the small parasitic capacitance C 45 (C 67 ).
In summary, the higher performance of the DTC architec-ture than the STC one mainly comes from the fact that the DTC has the advantage that it can reduce the NNN coupling, g 13 , compared to the STC from various points.

VI. SUMMARY
In this paper, we have analyzed the system of three fixedfrequency-transmon qubits coupled via two DTCs, by numerical simulation using the circuit model, where the NN qubits are highly detuned and NNN qubits are nearly resonant.As a result, we have found that the DTC can suppress the residual ZZ couplings and ZZZ coupling, as well as in isolated twoqubit systems.Moreover, we have succeeded in implementing 30-ns CZ gates and not only individual but also simultaneous 10-ns π/2 pulses with average fidelities over 99.99%.The degradation of the CZ gate fidelities due to the increasing number of qubits from two to three is negligible in the 4-digit precision.
For comparison, additional simulations of the system where DTCs were replaced by STCs have been performed.Then, we have found that both suppressing residual couplings to several kHz and achieving two-qubit and single-qubit gates with fidelities of 99.99% are highly challenging for the STCs.Due to the structure of STCs, the NNN-qubit coupling strength g 13 tends to be larger than DTCs, and hence causes more severe errors.
From these results, we have concluded that the DTC architecture is more promising than the STC architecture for realizing high-performance, scalable superconducting quantum computers.We expect that our results will be confirmed experimentally in the near future.
By canonical quantization of it, we obtain the following quantized Hamiltonian: Equation ( 23) is obtained when α = 0.The Hamiltonian of the isolated two qubit systems ĤL(R) can also be derived in a similar manner.

Appendix B: Matrix representation of the Hamiltonian
The matrix representations of the operators ni , cos φi , and sin φi in the basis of ni eigenfunctions are given as follows: where N is a cutoff for the Cooper-pair number.Each operator is expressed as a (2N + 1) × (2N + 1) matrix.

Appendix C: Dimension reduction technique
In this section, we introduce our numerical method to reduce the computational cost.Here we only explain the case of the three-qubit system with two DTCs.Other systems can also be treated similarly.

Diagonalization
The Hamiltonian in Eq. ( 1) can be divided as follows.
Here, Ĥ123 , Ĥ45 , and Ĥ67 correspond to the qubit, DTC-L, and DTC-R subspaces , respectively, and Ĥint is the interaction Hamiltonian between them.The qubit-subspace Hamiltonian Ĥ123 is expressed by (2N + 1) 3 × (2N + 1) 3 matrix.We diagonalize it using SciPy [49] to obtain the eigenenergy matrix ê123 and the energy eigenstate matrix V123 satisfiyng the following relation: where the diagonal components of ê123 are the eigenenergies of Ĥ123 in the ascending order.We introduce the cutoff N 123 to restrict the matrix size of ê123 and V123 to N 123 × N 123 .Using V123 , we can obtain n′ i for i ∈ {1, 2, 3}, which is the ni represented by the V123 basis as follows: This procedure reduces the matrix size of ni from (2N + 1) 3 × (2N + 1) 3 to N 123 × N 123 .In this work, we set N 123 to 120.Similarly, we reduce the matrix size of ni for i ∈ {4, 5, 6, 7} as follows: where the eigenstate matrices V45 and V67 satisfy the following equations: The diagonal components of the eigenenergy matrices ê45 and ê67 are the eigenenergies of Ĥ45 and Ĥ67 , respectively, in the ascending order.Here, we introduce the cutoff N 45 and N 56 to restrict the matrix size of ( ê45 , V45 ) and ( ê67 , V67 ) to N 45 × N 45 and N 67 × N 67 , respectively.In this work, we set both N 45 and N 67 to 25.
Using them, we obtain the Hamiltonian with reduced matrix size as follows: The size of these matrices is (N 123 × N 45 × N 67 ) × (N 123 × N 45 × N 67 ) = 75000 × 75000.By diagonalizing Ĥ′ , we can obtain the energy eigenvalues of the total system.We have set the dimensions as N 123 = 120 and N 45 = N 67 = 25 for sufficient convergence of eigenfrequencies with sub-kHz accuracy.By this method, the matrix size of the Hamiltonian for diagonalizations is greatly reduced from (2N + 1) 7

Gate simulation
We explain the gate simulation method.As an example, we cosider the case of CZ gate for Q 1 and Q 2 .
From the results of the diagonalization of the total Hamiltonian explained in the previous subsection, we identify the idle point.Fixing Θ ex,R to Θ Id,R , we again diagonalize Ĥ′ for Θ ex,L = Θ n = 0.05nπ (n = 0, 1, • • • , 20) to obtain the eigenenergy matrix ê0 (Θ n ) and the energy eigenstate matrix V0 (Θ n ) satisfiyng the following relation: where the diagonal components of ê0 (Θ n ) are the eigenenergies of Ĥ′ (Θ n ) in the ascending order.We introduce the cutoff N 0 to restrict the matrix size of ê0 (Θ n ) and V0 (Θ n ) to N 0 × N 0 .
In this work, we set N 0 to 1000.Using V0 (Θ n ), we calculate the following matrices: and express the Hamiltonian at the magnetic flux The matrix size of Ĥ′′ is N 0 × N 0 = 1000 × 1000.When Θ ex,L (t) ∈ R m , we calculate the time evolution of the state vector represented by the P0 (Θ m ) basis, using the above Hamitonian Ĥ′′ and QuTiP [46,47].At the particular time when the range including Θ ex,L changes from R m to R m+1 or to R m−1 , we operate the following basis transformation matrices respectively, on the state vector.
Appendix D: Isolated Two-Qubit Subsystems L and R

DTC architecture
The Hamiltonian of the system µ ∈ {L, R} is denoted by Ĥµ (see Appendix A and Ref. 28 for the help of the derivation).Its eigenfrequencies and corresponding eigenstates are denoted by ω Eigenfrequencies of the computational states in the system L (R), ω 1,1 , as functions of Θ ex,L(R) are shown by scatter plots in Fig. 3 These values should be as small as possible at the idle point to suppress conditional phase rotation which causes quantum crosstalk.As shown in Fig. 3(c) [3(d)] by the scatter plot, the minimum value of |ζ 12(23) |/(2π) is about 2(2) kHz at Θ ex,L(R) = 0.65π(0.65π)≡ Θ Id,L(R) .Hence, we can choose Θ Id,L and Θ Id,R as idle points of system L and R, respectively.

STC architecture
Similarly to the DTC architecture, we define ĤL(R) .
Its eigenfrequencies and corresponding eigenstates are denoted by ω L , respectively.We also use Eigenfrequencies of the computational states in the system L (R), (a).A typical device layout and a schematic circuit diagram are shown in Fig. 1(b) and Fig. 2(a), respectively.We also introduce the isolated two-qubit subsystems L and R shown in Figs.2(b) and 2(c), respectively, corresponding to the left and right sides of the three-qubit system.

FIG. 2 .
FIG. 2. (a) Circuit diagram of the three qubit system corresponding to the highlighted part of Fig. 1.(b) Circuit diagram of the two-qubit system L, which is the subsystem of (a).(c) Similar diagram to (b) for the right side of (a).The colors of Q i (red or blue) indicate which frequency bands (low or high) they belong to.
1} and µ ∈ {L, R}) (see Appendix D), are shown by scatter plots.All of them are needed for calculations of ZZ and ZZZ couplings below.We find that ω 1,0,0 ,

Fig. 2 .
FIG. 3. Energy levels and ZZ coupling strengths of the systems in Fig. 2. (a) and (b) show eigenfrequenciesω Q 1 ,Q 2 ,Q 3 ,C 4 ,C 5 ,C 6 ,C 7as functions of Θ ex,L and Θ ex,R , respectively.The qubit frequencies of the three-qubit system, ωQ 1 ,Q 2 ,Q 3 (Q i ∈ {0, 1}), are highlighted by being colored and bold, and the ones of the two-qubit system, ω 1} and µ ∈ {L, R}) (see Appendix D), are shown by colored scatter plots.Eigenfrequencies ω Q 1 ,Q 2 ,Q 3 ,C 4 ,C 5 ,C 6 ,C 7 other than the above are represented by black thin curves.(c) and (d) represent ZZ and ZZZ couplings as functions of Θ ex,L and Θ ex,R , respectively.Solid curves represent the ones of the three-qubit system, and scatter plots represent ζ 12(23) in the system L(R).The orange (blue) dotted horizontal line is ζ Id,23(12) in the system R(L).Note that Θ ex,R in the left column and Θ ex,L in the right column are fixed to Θ Id,R and Θ Id,L , respectively.
ζ 12 and ζ 23 , for the corresponding ZZ couplings in systems L and R, respectively (see Appendix D for their definitions).The scatter plot in Fig. 3(c)[3(d)] show ζ 12(23) in system L(R).The above Θ Id,L(R) has been identified as the point where |ζ 12(23) | takes the smallest value.

FIG. 4 .
FIG. 4. (a) Circuit diagram of the system with three qubits coupled via two STCs.(b) Circuit diagram of the two-qubit system L, which is the subsystem of (a).(c) Similar diagram to (b) for the right side of (a).The colors of Q i (red or blue) indicate which frequency bands (low or high) they belong to.

FIG. 6 .
FIG. 6.Energy levels and ZZ coupling strengths of the systems in Fig. 4. (a) and (b) show eigenfrequenciesω Q 1 ,Q 2 ,Q 3 ,C 4 ,C 5as functions of Θ ex,L and Θ ex,R , respectively.The qubit frequencies of the three-qubit system, ωQ 1 ,Q 2 ,Q 3 (Q i ∈ {0, 1}), are highlighted by being colored and bold, and the ones of the two-qubit system, ω 1} and µ ∈ {L, R}) (see Appendix D), are shown by colored scatter plots.Eigenfrequencies ω Q 1 ,Q 2 ,Q 3 ,C 4 ,C 5 other than the above are represented by black thin curves.(c) and (d) represent ZZ and ZZZ couplings as functions of Θ ex,L and Θ ex,R , respectively.Solid curves represent the ones of the three-qubit system, and scatter plots represent ζ 12(23) in the system L(R).The Orange (blue) dotted horizontal line is ζ Id,23(12) in the system R(L).Note that Θ ex,R in the left column and Θ ex,L in the right column are fixed to Θ Id,R and Θ Id,L , respectively.
(a)[3(b)] where we have set ω L(R) 0,0 to 0 (outside the graph scale).The ZZ couplings, ζ 12 in the system L and ζ 23 in the system R are calculated as follows: as functions of Θ ex,L(R) are shown by scatter plots in Fig. 6(a)[6(b)] where we have set ω L(R) 0,0 to 0 (outside the graph scale).The ZZ couplings, ζ 12 in the system L and ζ 23 in the system R are calculated by Eq. (D1) and Eq.(D2), respectively.As shown in Fig. 6(c)[6(d)] by the scatter plot, the minimum value of |ζ 12(23) |/(2π) is about 64(64) kHz at Θ ex,L(R) = 0(0) ≡ Θ Id,L(R) .Hence, we choose Θ Id,L and Θ Id,R as idle points of system L and R, respectively.The residual couplings are summarized in Table V.They are in good agreement with the result of Ref. 20: about −60 kHz.Their magnitudes are roughly 30 times larger than the ones of the corresponding subsystems with the DTC.

FIG. 9 .
FIG. 9. (a) A three-qubit system of diagonal structure (highlighted part).(b)-(e) ZZ and ZZZ couplings around the idle point as functions of Θ ex,L .Here, Θ ex,R is fixed to Θ Id,R .(f) The average gate fidelities of 30-ns CZ gates and 10-ns π/2 pulses, implemented by optimized pulses, as functions of κ.

TABLE I .
Parameter setting for the DTC circuits in Fig.2.Bold values are design values.The others are calculated using them.

TABLE II .
Residual ZZ couplings and ZZZ coupling in the DTC architecture.

TABLE III .
Average gate fidelities of 30-ns CZ gates and 10-ns individual and simultaneous π/2 pulses.They are implemented by optimized pulses in the DTC architecture.
3(c) and 3(d), we found that |ζ 13 | and |ζ ZZZ | take negligibly small values of about 0-1 kHz in the wide range of the external flux.Thus, we conclude that ζ 12 and ζ 23 are almost unchanged from the ones of the isolated two-qubit systems and also ζ 13 and ζ ZZZ are negligible.Therefore, it is reasonable to take the idle point of the three-qubit system to (Θ ex,L , Θ ex,R ) = (Θ Id,L , Θ Id,R ), as mentioned above.We represent ζ 12 , ζ 23 , ζ 13 , and ζ ZZZ at the idle point simply as ζ Id,12 , ζ Id,23 , ζ Id,13 , and ζ Id,ZZZ , respectively.Their values are summarized in Table II.
2. Single-qubit gate: π/2 pulse Next, we consider the single-qubit gate as well.Our targets are individual and simultaneous implementations of 10ns π/2 pulses.It leads to the following Ûid :

TABLE IV .
Parameter setting for the STC circuits in Fig.4.Bold values are design values.The others are calculated using them.

TABLE V .
Residual ZZ couplings and ZZZ coupling in the STC architecture.

TABLE VI .
Average gate fidelities of 30-ns CZ gates and 10-ns individual and simultaneous π/2 pulses.They are implemented by optimized pulses in the STC architecture.