Multi-Mode Bus Coupling Architecture of Superconducting Quantum Processor

Changhao Zhao(赵昌昊)$^{1,2}$, Yongcheng He(何永成)$^{1,2}$, Xiao Geng(耿霄)$^{1,2}$, Kaiyong He(何楷泳)$^{1,2}$,\\ Genting Dai(戴根婷)$^{1,2}$, Jianshe Liu(刘建设)$^{1,2}$, and Wei Chen(陈炜)$^{1,2,3\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/1/010301

⇦Chinese Physics Letters, 2023, Vol. 40, No. 1, Article code 010301⇨ Multi-Mode Bus Coupling Architecture of Superconducting Quantum Processor Changhao Zhao (赵昌昊)1,2, Yongcheng He (何永成)1,2, Xiao Geng (耿霄)1,2, Kaiyong He (何楷泳)1,2, Genting Dai (戴根婷)1,2, Jianshe Liu (刘建设)1,2, and Wei Chen (陈炜)1,2,3* Affiliations 1Laboratory of Superconducting Quantum Information Processing, School of Integrated Circuits, Tsinghua University, Beijing 100084, China 2Beijing National Research Center for Information Science and Technology, Beijing 100084, China 3Beijing Innovation Center for Future Chips, Tsinghua University, Beijing 100084, China Received 22 July 2022; accepted manuscript online 6 December 2022; published online 24 December 2022 ^*Corresponding author. Email: weichen@mail.tsinghua.edu.cn Citation Text: Zhao C H, He Y C, Geng X et al. 2023 Chin. Phys. Lett. 40 010301 Abstract Resonators in circuit quantum electrodynamics systems naturally carry multiple modes, which may have non-negligible influence on qubit parameters and device performance. While new theories and techniques are under investigation to deal with the multi-mode effects in circuit quantum electrodynamics systems, researchers have proposed novel engineering designs featuring multi-mode resonators to achieve enhanced functionalities of superconducting quantum processors. Here, we propose multi-mode bus coupling architecture, in which superconducting qubits are coupled to multiple bus resonators to gain larger coupling strength. Applications of multi-mode bus couplers can be helpful for improving iSWAP gate fidelity and gate speed beyond the limit of single-mode scenario. The proposed multi-mode bus coupling architecture is compatible with a scalable variation of the traditional bus coupling architecture. It opens up new possibilities for realization of scalable superconducting quantum computation with circuit quantum electrodynamics systems.

DOI:10.1088/0256-307X/40/1/010301 © 2023 Chinese Physics Society Article Text Circuit quantum electrodynamics (cQED) systems[1,2] are among the most promising physical platforms competing for scalable universal quantum computation. State-of-art multi-qubit cQED systems can be realized with either nearest-neighbor coupling architecture[3-6] or bus coupling architecture.[7-15] In nearest-neighbor coupling architecture, qubits couple to neighboring ones with relatively stronger coupling strengths and tunable couplers help protect qubits from stray couplings,[16-18] thereby fast two-qubit gates can be realized with high fidelity. On the other hand, bus coupling architecture is featured with high connectivity, which provides shortcut to preparation of multi-qubit entangled state or superposition state.[12,14] Although multi-qubit quantum simulation experiments can be realized with bus coupling architecture,[19] those involving universal quantum gates in multi-qubit systems, e.g., demonstration of quantum advantage or quantum supremacy,[4,5,20] have preferred nearest-neighbor coupling architecture for the time being. Common designs of cQED systems incorporate coplanar waveguide resonators or three-dimensional cavity resonators whose basic resonant modes are used for qubit readout or coupling. These resonators naturally carry multiple modes. High-order modes are sometimes not negligible, which may contribute to Purcell decay[21] or virtual photon coupling strength.[9] As cQED systems advance in both scale and complexity, the multiple devices are more likely to interact with each other in unintended manners, and different theories and techniques are being developed for analyses of system parameters to higher precision.[22-27] Multi-mode effects in cQED systems are also being engineered to realize more sophisticated information processing functionalities, using either different modes of a single resonator[28-32] or multiple resonators on a single processor.[33,34] In this Letter, we propose multi-mode bus coupling architecture for superconducting quantum processors, where transmon qubits are coupled via multi-mode bus couplers and the virtual photon coupling strength can be improved compared to the common case where single-mode bus is engaged. With the improved coupling strength, two-qubit iSWAP gate operation time can be shortened while high gate fidelity can be maintained, alleviating the difficulty for realization of fault-tolerant universal two-qubit gate in bus coupling architecture. As an alternative approach to stronger virtual photon coupling besides simple adoption of larger coupling capacitance, multi-mode coupling is especially fitted for the qubit-resonator network architecture, in which bus coupling with at least two modes is naturally embedded. Unfavorable factors for cQED systems are considered in multi-mode coupling architecture, including Purcell effect, thermal population of the bus coupler modes, residual stray couplings, and control pulse imperfections. Our study suggests that multi-mode effect can improve gate performance in bus coupling superconducting quantum processors and may inspire alternative realizations of scalable superconducting quantum computation systems. The Hamiltonian of a system of multiple transmon qubits capacitively coupled to multiple resonator modes can be written as[35] \begin{align} \frac{1}{\hbar}\hat{H}=\,&\sum_k \Big(\omega^{(\mathrm{q})}_k \hat{b}_k^† \hat{b}_k + \frac{\alpha_k}{2} \hat{b}_k^† \hat{b}_k^† \hat{b}_k \hat{b}_k\Big) +\sum_m \omega^{(\mathrm{r})}_m \hat{a}_m^† \hat{a}_m \notag\\ &-\sum_{km} g^{(\mathrm{rq})}_{km} (\hat{a}_m^† - \hat{a}_m) (\hat{b}_k^† - \hat{b}_k).\tag {1} \end{align} Here, $\hat{b}_k$ is the annihilation operator of the $k$-th qubit, while $\omega^{(\mathrm{q})}_k$ and $\alpha_k$ are the angular frequency and anharmonicity respectively; $\hat{a}_m$ and $\omega^{(\mathrm{r})}_m$ are the annihilation operator and angular frequency of the $m$-th harmonic resonator mode; $g^{(\mathrm{rq})}_{km}$ is the direct coupling strength between the $k$-th qubit and the $m$-th resonator mode. When a single resonator mode connects two transmon qubits, with the assumption of strong coupling $g^{(\mathrm{rq})}_k \ll \omega^{(\mathrm{r})}, \omega^{(\mathrm{q})}_k$ and dispersive coupling $g^{(\mathrm{rq})}_k \ll |\varDelta_k| \equiv |\omega^{(\mathrm{q})}_k - \omega^{(\mathrm{r})}|$, the Hamiltonian of the system can be approximated to[35] \begin{align} \frac{1}{\hbar}\hat{H}' &\approx \sum_{k=1,2}\Big(\tilde{\omega}^{(\mathrm{q})}_k \hat{b}_k^† \hat{b}_k + \frac{\alpha_k}{2}\hat{b}_k^† \hat{b}_k^† \hat{b}_k \hat{b}_k\Big) \notag\\ &~~~ + \tilde{\omega}^{(\mathrm{r})} \hat{a}^† \hat{a} + J_1(\hat{b}_1^† \hat{b}_2 + \hat{b}_1 \hat{b}_2^†),\tag {2} \end{align} where the last term describes the effective coupling between the two qubits, and the coupling strength is \begin{align} J_1 = \frac{g^{(\mathrm{rq})}_1 g^{(\mathrm{rq})}_2}{2} \Big(\frac{1}{\varDelta_1} + \frac{1}{\varDelta_2}\Big). \tag {3} \end{align} Equation (2) indicates that two qubits will effectively exchange energy without significantly exciting the bus coupler, which is known as the virtual photon coupling mechanism[36] and has been demonstrated in a variety of experiments.[7,8,12] State swap is enabled by tuning the qubits into resonance, i.e., $\varDelta_1 = \varDelta_2 = \varDelta$, then the effective coupling strength $J_1 = g^{(\mathrm{rq})}_1g^{(\mathrm{rq})}_2/\varDelta$ will be inversely proportional to qubit-bus detuning $\varDelta$. Apparently, iSWAP gate speed can be increased by reducing $|\varDelta|$. However, this also leads to lower gate fidelity, since the assumption of $g^{(\mathrm{rq})}_k \ll |\varDelta_k|$ that guarantees the validity of the dispersive approximation in Eq. (2) would be less stringent, i.e., high-order perturbation terms can have detrimental influence on iSWAP process. Using QuTiP,[37,38] we simulated the iSWAP process in a virtual photon coupling scheme to investigate the relation between iSWAP gate operation time and gate fidelity. The Hamiltonian of the system is in the form of Eq. (1) and the parameters are chosen based on typical superconducting quantum processors. The direct coupling strength between qubit and resonator is fixed at $g^{(\mathrm{rq})}_{km}/2\pi = g^{(\mathrm{rq})}/2\pi = 30$ MHz throughout this work. In the primary model, a single-mode bus resonator R1 at 8.00 GHz mediates the coupling between two transmon qubits Q1 and Q2 at 7.70 GHz, thus the detuning of $\varDelta/2\pi = -300$ MHz satisfies the dispersive coupling condition. Figure 1(a) depicts the process of two-qubit coherent oscillation. The red dashed vertical line indicates the time $t_{\scriptscriptstyle{\rm iSWAP}}$ when iSWAP gate is finished, and Q1, Q2 states are maximally swapped. R1 roughly stays in vacuum state with a population fluctuation of limited scale. Similar ripples are also visible on the cosine-like curves of Q1 and Q2 populations. These fluctuations originate from the off-resonance coupling between the qubits and the resonator, which means that evolution of half a cycle may lead to some deviation from state $|1\rangle$ for Q2. The dashed black curve in Fig. 1(a) is the fitting of Q2 population to cosine function, from which $t_{\scriptscriptstyle{\rm iSWAP}} \approx 83.3$ ns is extracted. Following the definition of worst-case gate fidelity,[39] it can be shown that the average population of Q2 at $t_{\scriptscriptstyle{\rm iSWAP}}$ actually represents an upper limit of iSWAP gate worst-case fidelity under the simulated condition. Such a gate fidelity reflects the quality of the bus coupling architecture, but excludes all other factors (such as control pulse waveform) that may further lower gate fidelity, therefore the gate fidelity in this model is again the upper limit of real iSWAP gate worst-case fidelity. As a result, the simulation in Fig. 1(a) gives an upper limit of iSWAP gate worst-case fidelity $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ of 98.1%.

Fig. 1. State swap process between two 7.70 GHz qubits coupled by 1- or 2-mode bus coupler, resulting in varied iSWAP gate operation time and gate fidelities: (a) 1-mode bus coupler with detuning $\varDelta/2\pi \equiv (\omega^{(\mathrm{q})} - \omega^{(\mathrm{r})})/2\pi = -300$ MHz corresponds to 83.3 ns 98.1% iSWAP gate, (b) 1-mode bus coupler with detuning $\varDelta/2\pi = -150$ MHz corresponds to 44.3 ns 93.7% iSWAP gate, (c) 2-mode bus coupler with detuning $\varDelta/2\pi = -600$ MHz corresponds to 81.1 ns 99.0% iSWAP gate, (d) 2-mode bus coupler with detuning $\varDelta/2\pi =-300$ MHz corresponds to 42.4 ns 96.5% iSWAP gate.

Figure 1(b) depicts the average population evolution when qubit-bus detuning $\varDelta/2\pi$ is halved from $-$300 MHz to $-$150 MHz. Here $t_{\scriptscriptstyle{\rm iSWAP}}$ is shortened from 83.3 ns to 44.3 ns, roughly proportional to detuning as expected. However, the system suffers larger fluctuations in resonator and qubit populations, leading to a reduced $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ of 93.7%. Obviously, simply reducing qubit-bus detuning for faster gate speed is confronted with deteriorated gate fidelity.

Fig. 2. Single-mode and multi-mode bus coupling architectures: (a) two transmon qubits (red and green, each being a SQUID shunted by a large capacitor[40]) coupled by a single bus resonator mode (blue, parallel LC circuit), (b) two transmon qubits coupled by multiple bus resonator modes. Each mode corresponds to an LC circuit between the qubits. The circuit diagram is applicable for multi-mode bus coupling with either multiple resonators or a single resonator that carries multiple modes.

Our idea to promote both gate speed and gate fidelity is to take advantage of multi-mode effect of the bus coupler. A bus coupler that is implemented with a $\lambda/2$ coplanar waveguide resonator has a series of high-order modes $f_n = n \times f_1$ ($n = 2,\,3,\,\dots$) besides the base mode of frequency $f_1$. Contributions from these high-order modes to the virtual photon coupling strength are only marginal (though may not be negligible[9]) since the modes are usually far off-resonance from the qubits. In order to fully exploit multi-mode effect, we consider using multiple bus resonators whose basic modes are comparatively close to the qubit frequencies. The multiple bus resonators together constitute a multi-mode bus coupler. The corresponding circuit diagram is shown in Fig. 2. Extrapolating from Eq. (3), the effective virtual photon coupling strength $J_{\scriptscriptstyle{M}}$ provided by an $M$-mode bus coupler between Q1 and Q2 is expected to be a summation of the contributions from all bus modes as[41] \begin{align} J_{\scriptscriptstyle{M}} = \sum_{m=1}^{M} \frac{g^{(\mathrm{rq})}_{1,m} g^{(\mathrm{rq})}_{2,m}}{2} \Big( \frac{1}{\varDelta_{1,m}} + \frac{1}{\varDelta_{2,m}} \Big). \tag {4} \end{align} To examine how multiple bus modes affect qubit evolution, two resonators R1 and R2 of identical mode frequencies are incorporated in the QuTiP model to couple Q1 and Q2 with $\varDelta = -300$ MHz. The initial state is $|q_1 q_2 r_1 r_2\rangle = |1000\rangle$ and population evolution of the system is shown in Fig. 1(d). Compared to the single-mode case with the same detuning as in Fig. 1(a), $t_{\scriptscriptstyle{\rm iSWAP}}$ is shortened from 83.3 ns to 42.4 ns. Compared to the single-mode case with half detuning in Fig. 1(b), $t_{\scriptscriptstyle{\rm iSWAP}}$ is similar while the high-frequency ripples on average population curves are suppressed, allowing a higher $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ of 96.5%. Furthermore, Fig. 1(c) depicts population evolution of double-mode coupling with double qubit-resonator detuning. An iSWAP gate time similar to the single-mode normal-detuning case is achieved while $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ can be raised to 99.0%. Motivated by these double-mode improvements, we systematically investigated bus coupling architectures with $M = 1$–4 bus modes. Two qubits Q1 and Q2 are initialized to $|q_1 q_2\rangle = |10\rangle$ and the $M$ bus modes are initialized to vacuum states. QuTiP simulation gives iSWAP gate operation time $t_{\scriptscriptstyle{\rm iSWAP}}$ and worst-case gate fidelity upper limit $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ for different detuning $\varDelta$ and different number of bus modes $M$, as depicted in Fig. 3(a). For a fixed $M$, $t_{\scriptscriptstyle{\rm iSWAP}}$ and $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ are constrained by each other, i.e., higher $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ requires longer $t_{\scriptscriptstyle{\rm iSWAP}}$. When $M$ is increased, however, higher $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ can be achieved without sacrificing gate speed. For example, to push towards quantum error correction threshold of 99% two-qubit gate fidelity,[42] an upgrade from single-mode to double-mode bus coupler can reduce iSWAP gate operation time from 115 ns to 81 ns. Besides the aforementioned worst-case fidelity, we also performed quantum process tomography in QuTiP simulations to obtain the $\chi$ matrix that represents the iSWAP gate operation. Process fidelity $\cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)} \equiv \mathrm{Tr}(\chi^\mathrm{t}\chi^\mathrm{s})$ is plotted versus gate operation time in Fig. 3(b), where $\chi^\mathrm{t}$ corresponds to target iSWAP gate and $\chi^\mathrm{s}$ corresponds to simulated iSWAP gate. Since process fidelity is directly related to average gate fidelity that reflects the quality of a quantum operation for all possible initial states,[43,44] $\cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)}$ is generally higher than $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$. From Fig. 3(b) we can see that multi-mode bus coupling outperforms single-mode bus coupling, especially in the short-gate-time limit. However, it is ambiguous whether three- or four-mode coupling is more beneficial than two.

Fig. 3. The iSWAP gate fidelity versus gate operation time for different numbers of bus modes: (a) upper limit of worst-case fidelities $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$ obtained from cosine-fitting of qubit population curves in state swap processes, (b) process fidelities $\cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)}$ obtained from simulated quantum process tomography. Although $\cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)}$ is generally higher than $\tilde{\cal{F}}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(w)}$, both metrics confirm that multi-mode bus coupling enables higher gate fidelity with shorter gate operation time compared to single-mode case. The error bars in (b) correspond to one-sigma statistical uncertainties of 20 repetitions of QuTiP simulations, where we have assumed that the bus resonator mode frequencies are subject to a uniform random distribution within $\pm10$ MHz range around the target mode frequencies. More descriptions on randomized resonator frequencies as well as other model parameters used in the simulations are given in section S1 of the Supplementary Material.

Fig. 4. Illustration of a scalable qubit-resonator network architecture, where multiple qubits are coupled by multiple bus resonators. The purple boxes represent bus resonator couplers and the gray circles represent qubits. Line connections indicate direct couplings. Q1, Q2, and Q3 are coupled by R1 and R2 simultaneously, which is effectively a double-mode bus coupling situation.

It should also be noted that each additional bus mode means more engineering difficulties. An easier and more intuitive approach to increased virtual photon coupling strength is by directly adopting larger qubit-resonator coupling capacitance. Nevertheless, the advantage of using $M$ resonators lies in that it is an additional method that can be applied together with other ones. It is also realized that double-mode coupling is naturally embedded in the qubit-resonator network architecture illustrated in Fig. 4. Multi-mode coupling with $M > 2$ modes may be a challenge for physical realization, but we expect that an upgrade to double-mode bus coupling at least for a portion of the qubits will be beneficial for two-qubit gate performance. Till now we have discussed how multi-mode bus coupling architecture promotes the effective virtual photon coupling strength between qubits to enable faster speed and higher fidelity of two-qubit gate. On the other hand, all quantum systems suffer from decoherence, residual stray couplings, control imperfections, etc., that limit gate performance. Qubit decoherence affects gate fidelity due to the finite gate operation time during which decoherence inevitably occurs. It is usually separated out from other limiting factors[3,44,45] since it is independent of the gate operation mechanism. Additional complexity in multi-component quantum systems, however, can bring extra qubit decoherence through Purcell effect.[21,46,47] The extra energy relaxation rate $\gamma_\kappa$ of the qubit is called Purcell rate, written as \begin{align} \gamma_\kappa = \Big(\frac{g^{(\mathrm{rq})}}{\varDelta}\Big)^2 \kappa, \tag {5} \end{align} where $\kappa$ is the energy relaxation rate of the bare resonator mode. Equation (5) shows that the Purcell rate is inversely proportional to the qubit-resonator detuning squared. When the qubit is coupled to $M$ resonator modes, the total Purcell rate of the qubit should be the sum of $\gamma_\kappa^{(m)}$ from each single resonator mode: \begin{align} \gamma_{\kappa,\scriptscriptstyle{M}} = \sum_{m=1}^M \gamma_\kappa^{(m)} = \sum_{m=1}^M \Big(\frac{g^{(\mathrm{rq})}_m}{\varDelta_m}\Big)^2 \kappa_m. \tag {6} \end{align} As discussed above, the detuning can be kept to be large in multi-mode bus coupling architecture without sacrificing two-qubit gate speed, therefore multi-mode bus coupling architecture may be exploited to suppress Purcell effect.

Fig. 5. Qubit Purcell decay induced by the bus resonator modes. By fitting qubit populations to exponential decay functions (dashed black curves), Purcell decay rates are extracted as 97.1 kHz for 1-mode bus (blue circle), 49.4 kHz for 2-mode bus (orange square), 33.1 kHz for 3-mode bus (green downward triangle), 25.0 kHz for 4-mode bus (red upward triangle).

According to Eq. (4), single-mode virtual photon coupling strength between the two qubits is $J_1 = g^2/\varDelta$. If $M$-mode bus coupler is implemented with the detuning increased by a factor of $M$,[48] the effective qubit coupling strength $J_{\scriptscriptstyle{M}} = M \times g^2/(M\varDelta) = J_1$ would be approximately unaltered. Then the Purcell rates of the coupling qubits read \begin{align} \gamma_{\kappa,\scriptscriptstyle{M}} = M \times \Big(\frac{g^{(\mathrm{rq})}}{M \times \varDelta}\Big)^2 \kappa = \frac{\gamma_{\kappa,1}}{M}, \tag {7} \end{align} suppressed by a factor of $1/M$ compared to single-mode case. As a verification, four QuTiP models are simulated, involving a single qubit at $\omega^{(\mathrm{q})}/2\pi = 7.70$ GHz coupled to $M = 1$–4 resonator modes, whose frequencies can be determined with Eq. (S1) in the Supplementary Material. Simulated Purcell decay processes are demonstrated in Fig. 5. The colored dots represent the energy relaxation process of the qubit for different numbers of bus modes, and the dashed black curves are numerical fits to exponential decay functions. The Purcell rate $\gamma_{\kappa,\scriptscriptstyle{M}}$ thus obtained are in good conformity with Eq. (7).

Fig. 6. Purcell rates in multi-mode bus coupling architecture: (a) Purcell rates versus qubit frequencies. Different mark styles and line colors correspond to different numbers of bus modes. The marks represent Purcell rates obtained by fitting qubit population decays in QuTiP simulations to exponential functions. The dashed curves represent theoretical values calculated with Eq. (6) and the solid vertical lines represent the corresponding bus mode frequencies. (b) Purcell rates versus the number of bus modes. Each curve represents a case where the qubit is fixed at certain frequency while the number of bus mode and the bus frequencies are changed according to Eq. (S1) in the Supplementary Material.

To turn down the coupling between qubits, we should tune them to idle frequencies below the coupling frequency of $\omega^{(\mathrm{q})}/2\pi = 7.70$ GHz. Here we investigate Purcell effect of the idle qubit in multi-mode coupling scenario. For the same bus modes defined by Eq. (S1), qubit idle frequency $\omega^{(\mathrm{i})}/2\pi$ is tuned from 7.70 GHz down to 6.10 GHz. The Purcell rates extracted for varied $\omega^{(\mathrm{i})}/2\pi$ and $M$ are shown in Fig. 6. In Fig. 6(a), Purcell rates extracted from QuTiP simulations (colored markers) are plotted against qubit frequencies. They agree well with the theoretical values calculated with Eq. (6) (dashed lines). For $\omega^{(\mathrm{i})}/2\pi \gtrsim 7.30$ GHz, increased number of bus modes help suppress Purcell rates of the qubit. For $\omega^{(\mathrm{i})}/2\pi \lesssim 6.90$ GHz, however, increased number of bus modes leads to larger Purcell rates. In Fig. 6(b), Purcell rates are plotted against the number of bus modes. Purcell rate and the number of bus modes are inversely proportional to each other when the qubit is at the coupling frequency of $\omega^{(\mathrm{q})}/2\pi = 7.70$ GHz, as indicated by the red straight line with negative slope in double-logarithm axes. For qubit frequencies below 7.70 GHz, the suppression of Purcell effect due to larger detuning is partly counteracted by the increased number of bus resonator modes. Since the Purcell rates of coupling qubits are approximately one order of magnitude higher than those of the idle qubits, and that two-qubit gates usually need longer operation time than single-qubit ones, the global coherence of the superconducting quantum processor may still benefit from multi-mode coupling effect.

Fig. 7. The iSWAP gate process infidelity $1 - \cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)}$ versus average thermal population $n_\mathrm{th}$ of the resonator modes. Each subfigure is characterized by the coupling frequency $\omega^{(\mathrm{q})}/2\pi$ chosen for simulation, while the $M = 1$–3 bus mode frequencies are determined by Eq. (S1) in the Supplementary Material. Therefore, as qubit-resonator detuning $|\varDelta|$ is in increasing sequence from (a) to (d), infidelity in low-$n_\mathrm{th}$ limit is in decreasing sequence.

In an environment with finite temperature, the bus coupler modes are usually slightly populated to thermal equilibrium state, deviating from the assumption that the coupler modes should be in vacuum states. We simulated iSWAP gate under finite temperature conditions and extracted by quantum process tomography the process ‘infidelities’ $1 - \cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)}$, which are plotted against average thermal population $n_\mathrm{th}$ in Fig. 7. In common cQED experiment, device temperature is roughly below 30 mK while resonator level spacing is around 5–10 GHz, hence by Boltzmann distribution the average thermal population is $n_\mathrm{th} \approx N_1/N_0 = \exp[-(E_1 - E_0)/k_{\scriptscriptstyle{\rm B}}T] \lesssim 10^{-3}$. Figure 7 covers $n_\mathrm{th} = 10^{-4} \sim 10^{-1}$, from which we can see that multi-mode coupling is more susceptible to resonator thermal population for $n_\mathrm{th} \gtrsim 10^{-2}$, while multi-mode coupling has lower infidelity (thus higher fidelity) than single-mode coupling for $n_\mathrm{th} \lesssim 10^{-3}$. In low-$n_\mathrm{th}$ limit, multi-mode coupling is outstanding in suppression of infidelity for the smaller detunings used in Figs. 7(a) and 7(b) compared to the larger detunings used in Figs. 7(c) and 7(d). On the other hand, $M = 2$ or $M = 3$ does not clearly distinguish themselves from each other, which is in conformity with the results shown in Fig. 3(b). Based on the above simulations, we conclude that under common cQED experiment conditions, thermal population of the bus coupler would not be a limiting factor for iSWAP gate performance in the multi-mode bus coupling mechanism. Coupling between qubits can be categorized into $XY$ and $ZZ$ types, which are also known as transverse coupling and longitudinal coupling, respectively.[49] The former involves energy exchange between qubits and is represented by $\hat{\sigma}_x\hat{\sigma}_x$ or $\hat{\sigma}_y\hat{\sigma}_y$ coupling terms in the Hamiltonian, while the latter involves energy level shift of one qubit induced by the other and is represented by $\hat{\sigma}_z\hat{\sigma}_z$ terms. In multi-mode bus coupling architecture discussed heretofore, we have primarily focused on $XY$ coupling to construct iSWAP gate. When idle qubits are added to the system, they are supposed to be well off-resonance from the coupling ones, yet still bear some residual $XY$ coupling that may affect the iSWAP gate in progress. To see how the residual $XY$ coupling affects gate performance in multi-mode coupling architecture, one idle qubit I1 at varied frequency of $\omega^{(\mathrm{i})}/2\pi = 6.70$ GHz $\sim 7.60$ GHz is included in the simulation model where two qubits Q1 and Q2 at $\omega^{(\mathrm{q})}/2\pi =7.70$ GHz are coupled via the $M$-mode bus coupler defined with Eq. (S1) from the Supplementary Material. I1 is initialized to $|\psi^{(\mathrm{i})}_0\rangle = (|0\rangle+|1\rangle)/\sqrt{2} \equiv |+\rangle$, and Q1–Q2 energy exchange process is observed to be distorted periodically by the residual $XY$ coupling from I1.[50]

Fig. 8. Distortion time of coupling qubits versus idle qubit frequency for different numbers of bus modes. The exceptional drop of $T_\mathrm{d}$ near $\omega^{(\mathrm{i})}/2\pi = 7.5$ GHz is due to the fact that all transmons in the simulated models have been assigned $\alpha_k/2\pi = -0.22$ GHz, thus one idle qubit I1 is close to resonance with the 1–2 level spacing of Q1 and Q2.

We define distortion time $T_\mathrm{d}$ as the time interval within which Q1–Q2 state swap amplitude is modulated from maximum to minimum. For $M = 1$–4, $T_\mathrm{d}$ is plotted versus $\omega^{(\mathrm{i})}/2\pi$ in Fig. 8. We can see that $T_\mathrm{d}$ is generally decreasing as $\omega^{(\mathrm{i})}/2\pi$ approaches $\omega^{(\mathrm{q})}/2\pi = 7.70$ GHz, and more coupling modes also lead to shorter $T_\mathrm{d}$. It is noted that the four curves in Fig. 8 are not much separated from each other, which means that increased $M$ is a less significant factor than insufficient idle qubit detuning in terms of causing unwanted $XY$ coupling, and the decrease of $T_\mathrm{d}$ with increased $M$ can be compensated for by raising the detuning between I1 and Q1–Q2 by a few hundreds of MHz. At present, common coherence time of aluminum-based transmon qubits on multi-qubit superconducting quantum processor is on the order of a few tens of microseconds,[4,5,51] so residual $XY$ coupling from idle qubits would be acceptable if $T_\mathrm{d}$ is not definitely shorter. For the qubit and bus parameters used in Fig. 8, when $\omega^{(\mathrm{i})}/2\pi$ is below 7.00 GHz, i.e., the detuning between I1 and Q1–Q2 is above 0.7 GHz, there would be no severe interference. If alternative coupling strengths or detunings are applied between the coupling qubits and the bus coupler, the proper detuning range for idle qubits can be scaled accordingly to maintain sufficient isolation.

Fig. 9. Residual $ZZ$ coupling strengths $|\zeta|/2\pi$ between Q1 and Q2. Different Q1 frequencies are adopted in (a)–(d), while Q2 frequency $\omega^{(\mathrm{q})}_2/2\pi$ is swept from 6.40 GHz to 7.90 GHz. For $M = 1$–4 bus mode(s), residual $ZZ$ coupling strengths are obtained with either numerical diagonalization of the system Hamiltonian (separated markers labeled ‘diag’) or fourth-order perturbation method (dashed curves labeled ‘pert’).

As for $ZZ$ coupling between two qubits Q1 and Q2, we calculated $\zeta_\mathrm{diag} \equiv \tilde{\omega}^{(\mathrm{q})}_{01} + \tilde{\omega}^{(\mathrm{q})}_{10} - \tilde{\omega}^{(\mathrm{q})}_{11}$ from numerical diagonalization of the system Hamiltonian. Here $\tilde{\omega}^{(\mathrm{q})}_{mn}$ corresponds to the level spacing between dressed state $|\tilde{q}_1\tilde{q}_2\tilde{r}\rangle = |\tilde{m}\tilde{n}\tilde{0}\rangle$ and ground state $|000\rangle$, with $|\tilde{r}\rangle = |\tilde{0}\rangle$ representing the $M$ dressed bus resonator modes all in their ground states. Figure 9 shows the $\zeta_\mathrm{pert}$ obtained with extrapolation[52] of fourth-order perturbation method[53] together with $\zeta_\mathrm{diag}$, in which we can see that $\zeta_\mathrm{diag}$ and $\zeta_\mathrm{pert}$ loosely agree with each other and both of them diverge when certain energy level is tuned close to resonance with another. Contrary to the predictions given by $\zeta_\mathrm{pert}$, multi-mode coupling generally results in larger $\zeta_\mathrm{diag}$ than single-mode coupling, especially when 0–1 energy level spacing of one qubit is in-between 1–2 and 0–1 energy level spacings of the other qubit, as manifested by the straddling regimes to the left of the peak in Fig. 9(a), between the peaks in Figs. 9(b) and 9(c) and to the right of the peak in Fig. 9(d). Similar to the case of residual $XY$ coupling, multi-mode bus coupler inevitably leads to enhanced residual $ZZ$ coupling when it provides stronger virtual photon coupling strength. Since residual $ZZ$ coupling is one of the major obstacles for accurate manipulation of multi-qubit superconducting quantum processors, more efforts shall be taken to deal with this problem.[17,45,54,55] Another limitation to be considered on iSWAP gate performance in multi-mode coupling mechanism comes from control pulse imperfections. In practice, dispersion and bandwidth of microwave transmission lines can only allow possibly distorted tuning pulses with finite ramp time. Limited sampling rates of waveform generation instruments does not permit arbitrarily accurate pulse control either. Although techniques such as quantum optimal control[56-61] have been developed to remedy control imperfections, we can still try basic control signals with multi-mode coupling to learn how it competes with single-mode coupling. Q1 and Q2 are tuned from their idle frequencies to coupling frequency $\omega^{(\mathrm{q})}/2\pi$ near an $M$-mode bus coupler. In Fig. 10, the infidelities, obtained with quantum process tomography of the simulated control processes of iSWAP gate operations, are plotted against $t_\mathrm{ramp}$. In general, multi-mode coupling improves gate fidelity for faster-ramping pulses, while $M > 2$ does not show advantage over the $M = 2$ case. Comparison between the two subfigures once again confirms that multi-mode coupling mechanism is more appealing for smaller qubit-resonator detuning, i.e., stronger virtual photon coupling strength. In terms of pulse ramp shapes, although Gauss error function form seems to be better for $M > 1$ yet worse for $M = 1$, the difference between the two simulated types is not significant.

Fig. 10. The iSWAP gate infidelity $1 - \cal{F}_{\scriptscriptstyle{\rm iSWAP}}^\mathrm{(p)}$ versus control pulse ramp time $t_\mathrm{ramp}$ for different numbers of bus coupler modes and different ramp shapes. Two qubits are tuned from their off-resonance idle frequencies to coupling frequencies of (a) $\omega^{(\mathrm{q})}/2\pi = 7.70$ GHz and (b) $\omega^{(\mathrm{q})}/2\pi = 7.85$ GHz, at which they are coupled by $M$-mode ($M = 1$–4) bus coupler defined with Eq. (S1), respectively. The inset in (b) illustrates qualitatively the tuning process of the two qubits: the upward- or downward-ramping process each takes a time interval of $t_\mathrm{ramp}$; for each $t_\mathrm{ramp}$ value, $t_\mathrm{hold}$ is varied to find the proper operation time that enables iSWAP gate to the best process fidelity, which makes one data point in the two subfigures. Ramp shapes in Gauss error function (solid curves labeled erf) or cosine function (dashed curves labeled cos) forms are compared as well.

In conclusion, our study suggests that multi-mode bus coupling architecture can provide stronger virtual photon coupling strengths. This new architecture helps promote two-qubit iSWAP gate performance to achieve faster gate speed and higher gate fidelity. As an alternative approach besides directly increasing qubit-resonator coupling strength, multi-mode coupling with $M = 2$ bus modes is found naturally embedded in a scalable qubit-resonator network, where its advantage can be best exploited with least additional engineering complexity. Various unfavorable factors for cQED systems are considered to compare multi-mode coupling to traditional single-mode coupling. It is revealed that (a) Purcell effect can be suppressed for coupling qubits; (b) thermal population under common cQED experiment conditions will not affect multi-mode coupling; (c) intensified residual stray couplings, especially the $ZZ$ type, may be alleviated with better frequency allocation, and will probably need more sophisticated circuit designs and control protocols as well; (d) fast control pulse ramp rate is important for achieving high iSWAP gate fidelity in multi-mode bus coupling architecture. We expect that multi-mode coupling effect will facilitate gate performance in multi-qubit bus coupling superconducting quantum processors, and may inspire alternative realizations of scalable universal quantum computation in cQED systems. Acknowledgments. The authors thank Rutian Huang, Qing Yu, Xinyu Wu and Liangliang Yang for their helpful discussions. This work was partially supported by the National Natural Science Foundation of China (Grant No. 60836001), and the Key R&D Program of Guangdong Province (Grant No. 2019B010143002). References Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computationStrong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamicsSuperconducting quantum circuits at the surface code threshold for fault toleranceQuantum supremacy using a programmable superconducting processorStrong Quantum Computational Advantage Using a Superconducting Quantum ProcessorQuantum walks on a programmable two-dimensional 62-qubit superconducting processorCoupling superconducting qubits via a cavity busCoherent quantum state storage and transfer between two phase qubits via a resonant cavityMultimode mediated qubit-qubit coupling and dark-state symmetries in circuit quantum electrodynamicsComputing prime factors with a Josephson phase qubit quantum processorExperimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits10-Qubit Entanglement and Parallel Logic Operations with a Superconducting CircuitConstruction of two-qubit logical gates by transmon qubits in a three-dimensional cavityGeneration of multicomponent atomic Schrödinger cat states of up to 20 qubitsTunable coupling between Xmon qubit and coplanar waveguide resonator*Qubit Architecture with High Coherence and Fast Tunable CouplingTunable Coupling Scheme for Implementing High-Fidelity Two-Qubit GatesHigh-Fidelity, High-Scalability Two-Qubit Gate Scheme for Superconducting QubitsEmulating Many-Body Localization with a Superconducting Quantum ProcessorQuantum computational advantage via 60-qubit 24-cycle random circuit samplingControlling the Spontaneous Emission of a Superconducting Transmon QubitJosephson-junction-embedded transmission-line resonators: From Kerr medium to in-line transmonBlack-Box Superconducting Circuit QuantizationBlackbox quantization of superconducting circuits using exact impedance synthesisConvergence of the multimode quantum Rabi model of circuit quantum electrodynamicsQuantum networks in divergence-free circuit QEDEnergy-participation quantization of Josephson circuitsCavity Quantum Electrodynamics with Separate Photon Storage and Qubit Readout ModesBeyond Strong Coupling in a Multimode CavityMulti-mode ultra-strong coupling in circuit quantum electrodynamicsMode Structure in Superconducting Metamaterial Transmission-Line ResonatorsSeamless High-

Q

Microwave Cavities for Multimode Circuit Quantum ElectrodynamicsHigh-Contrast Qubit Interactions Using Multimode Cavity QEDRandom access quantum information processors using multimode circuit quantum electrodynamicsCircuit quantum electrodynamicsEfficient Scheme for Two-Atom Entanglement and Quantum Information Processing in Cavity QEDQuTiP: An open-source Python framework for the dynamics of open quantum systemsQuTiP 2: A Python framework for the dynamics of open quantum systemsCharge-insensitive qubit design derived from the Cooper pair boxSurface codes: Towards practical large-scale quantum computationA simple formula for the average gate fidelity of a quantum dynamical operationError matrices in quantum process tomographyTunable Coupler for Realizing a Controlled-Phase Gate with Dynamically Decoupled Regime in a Superconducting CircuitFast reset and suppressing spontaneous emission of a superconducting qubitGate-based superconducting quantum computingSuperconducting Qubits: Current State of PlayDemonstration of two-qubit algorithms with a superconducting quantum processorHigh-Contrast

Z Z

Interaction Using Superconducting Qubits with Opposite-Sign AnharmonicitySuppression of Static

Z Z

Interaction in an All-Transmon Quantum ProcessorOptimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithmsOptimizing entangling quantum gates for physical systemsImplementing quantum logic gates with gradient ascent pulse engineering: principles and practicalitiesSearching for quantum optimal controls under severe constraintsHybrid Quantum-Classical Approach to Quantum Optimal ControlUniversal gates for protected superconducting qubits using optimal control

[1]	Blais A, Huang R S, Wallraff A, Girvin S M, and Schoelkopf R J 2004 Phys. Rev. A 69 062320
[2]	Wallraff A, Schuster D, Blais A, Frunzio L, Huang R, Majer J, Kumar S, Girvin S M, and Schoelkopf R J 2004 Nature 431 162
[3]	Barends R, Kelly J, Megrant A, Veitia A, Sank D, Jeffrey E, White T C, Mutus J, Fowler A G, Campbell B, Chen Y, Chen Z, Chiaro B, Dunsworth A, Neill C, O'Malley P, Roushan P, Vainsencher A, Wenner J, Korotkov A N, Cleland A N, and Martinis J M 2014 Nature 508 500
[4]	Arute F, Arya K, Babbush R, Bacon D, Bardin J C, Barends R, Biswas R, Boixo S, Brandao F G S L, Buell D A, Burkett B, Chen Y, Chen Z, Chiaro B, Collins R, Courtney W, Dunsworth A, Farhi E, Foxen B, Fowler A, Gidney C, Giustina M, Graff R, Guerin K, Habegger S, Harrigan M P, Hartmann M J, Ho A, Hoffmann M, Huang T, Humble T S, Isakov S V, Jeffrey E, Jiang Z, Kafri D, Kechedzhi K, Kelly J, Klimov P V, Knysh S, Korotkov A, Kostritsa F, Landhuis D, Lindmark M, Lucero E, Lyakh D, Mandrà S, McClean J R, McEwen M, Megrant A, Mi X, Michielsen K, Mohseni M, Mutus J, Naaman O, Neeley M, Neill C, Niu M Y, Ostby E, Petukhov A, Platt J C, Quintana C, Rieffel E G, Roushan P, Rubin N C, Sank D, Satzinger K J, Smelyanskiy V, Sung K J, Trevithick M D, Vainsencher A, Villalonga B, White T, Yao Z J, Yeh P, Zalcman A, Neven H, and Martinis J M 2019 Nature 574 505
[5]	Wu Y L, Bao W S, Cao S, Chen F, Chen M C, Chen X, Chung T H, Deng H, Du Y, Fan D, Gong M, Guo C, Guo C, Guo S, Han L, Hong L, Huang H L, Huo Y H, Li L, Li N, Li S, Li Y, Liang F, Lin C, Lin J, Qian H, Qiao D, Rong H, Su H, Sun L, Wang L, Wang S, Wu D, Xu Y, Yan K, Yang W, Yang Y, Ye Y, Yin J, Ying C, Yu J, Zha C, Zhang C, Zhang H, Zhang K, Zhang Y, Zhao H, Zhao Y, Zhou L, Zhu Q, Lu C Y, Peng C Z, Zhu X, and Pan J W 2021 Phys. Rev. Lett. 127 180501
[6]	Gong M, Wang S, Zha C, Chen M C, Huang H L, Wu Y, Zhu Q, Zhao Y, Li S, Guo S, Qian H, Ye Y, Chen F, Ying C, Yu J, Fan D, Wu D, Su H, Deng H, Rong H, Zhang K, Cao S, Lin J, Xu Y, Sun L, Guo C, Li N, Liang F, Bastidas V M, Nemoto K, Munro W J, Huo Y H, Lu C Y, Peng C Z, Zhu X, and Pan J W 2021 Science 372 948
[7]	Majer J, Chow J M, Gambetta J M, Koch J, Johnson B, Schreier J A, Frunzio L, Schuster D I, Houck A A, Wallraff A, Blais A, Devoret M H, Girvin S M, and Schoelkopf R J 2007 Nature 449 443
[8]	Sillanpää M A, Park J I, and Simmonds R W 2007 Nature 449 438
[9]	Filipp S, Göppl M, Fink J M, Baur M, Bianchetti R, Steffen L, and Wallraff A 2011 Phys. Rev. A 83 063827
[10]	Lucero E, Barends R, Chen Y, Kelly J, Mariantoni M, Megrant A, O'Malley P, Sank D, Vainsencher A, Wenner J, White T, Yin Y, Cleland A N, and Martinis J M 2012 Nat. Phys. 8 719
[11]	Takita M, Cross A W, Córcoles A D, Chow J M, and Gambetta J M 2017 Phys. Rev. Lett. 119 180501
[12]	Song C, Xu K, Liu W, Yang C P, Zheng S B, Deng H, Xie Q, Huang K, Guo Q, Zhang L, Zhang P, Xu D, Zheng D, Zhu X, Wang H, Chen Y A, Lu C Y, Han S, and Pan J W 2017 Phys. Rev. Lett. 119 180511
[13]	Cai H, Liu Q C, Zhao C H, Zhang Y S, Liu J S, and Chen W 2018 Chin. Phys. B 27 084207
[14]	Song C, Xu K, Li H, Zhang Y R, Zhang X, Liu W, Guo Q, Wang Z, Ren W, Hao J, Feng H, Fan H, Zheng D, Wang D W, Wang H, and Zhu S Y 2019 Science 365 574
[15]	Li H K, Li K M, Dong H, Guo Q J, Liu W X, Wang Z, Wang H H, and Zheng D N 2019 Chin. Phys. B 28 080305
[16]	Chen Y, Neill C, Roushan P, Leung N, Fang M, Barends R, Kelly J, Campbell B, Chen Z, Chiaro B, Dunsworth A, Jeffrey E, Megrant A, Mutus J Y, O'Malley P J J, Quintana C M, Sank D, Vainsencher A, Wenner J, White T C, Geller M R, Cleland A N, and Martinis J M 2014 Phys. Rev. Lett. 113 220502
[17]	Yan F, Krantz P, Sung Y, Kjaergaard M, Campbell D L, Orlando T P, Gustavsson S, and Oliver W D 2018 Phys. Rev. Appl. 10 054062
[18]	Xu Y, Chu J, Yuan J, Qiu J, Zhou Y, Zhang L, Tan X, Yu Y, Liu S, Li J, Yan F, and Yu D 2020 Phys. Rev. Lett. 125 240503
[19]	Xu K, Chen J J, Zeng Y, Zhang Y R, Song C, Liu W, Guo Q, Zhang P, Xu D, Deng H, Huang K, Wang H, Zhu X, Zheng D, and Fan H 2018 Phys. Rev. Lett. 120 050507
[20]	Zhu Q L, Cao S R, Chen F S, Chen M C, Chen X, Chung T H, Deng H, Du Y, Fan D, Gong M, Guo C, Guo C, Guo S, Han L, Hong L, Huang H L, Huo Y H, Li L, Li N, Li S, Li Y, Liang F, Lin C, Lin J, Qian H, Qiao D, Rong H, Su H, Sun L, Wang L, Wang S, Wu D, Wu Y, Xu Y, Yan K, Yang W, Yang Y, Ye Y, Yin J, Ying C, Yu J, Zha C, Zhang C, Zhang H, Zhang K, Zhang Y, Zhao H, Zhao Y, Zhou L, Lu C Y, Peng C Z, Zhu X, and Pan J W 2022 Sci. Bull. 67 240
[21]	Houck A A, Schreier J A, Johnson B R, Chow J M, Koch J, Gambetta J M, Schuster D I, Frunzio L, Devoret M H, Girvin S M, and Schoelkopf R J 2008 Phys. Rev. Lett. 101 080502
[22]	Bourassa J, Beaudoin F, Gambetta J M, and Blais A 2012 Phys. Rev. A 86 013814
[23]	Nigg S E, Paik H, Vlastakis B, Kirchmair G, Shankar S, Frunzio L, Devoret M H, Schoelkopf R J, and Girvin S M 2012 Phys. Rev. Lett. 108 240502
[24]	Solgun F, Abraham D W, and DiVincenzo D P 2014 Phys. Rev. B 90 134504
[25]	Gely M F, Parra-Rodriguez A, Bothner D, Blanter Y M, Bosman S J, Solano E, and Steele G A 2017 Phys. Rev. B 95 245115
[26]	Parra-Rodriguez A, Rico E, Solano E, and Egusquiza I L 2018 Quantum Sci. Technol. 3 024012
[27]	Minev Z K, Leghtas Z, Mundhada S O, Christakis L, Pop I M, and Devoret M H 2021 npj Quantum Inf. 7 131
[28]	Leek P J, Baur M, Fink J M, Bianchetti R, Steffen L, Filipp S, and Wallraff A 2010 Phys. Rev. Lett. 104 100504
[29]	Sundaresan N M, Liu Y, Sadri D, Szőcs L J, Underwood D L, Malekakhlagh M, Türeci H E, and Houck A A 2015 Phys. Rev. X 5 021035
[30]	Bosman S J, Gely M F, Singh V, Bruno A, Bothner D, and Steele G A 2017 npj Quantum Inf. 3 46
[31]	Wang H, Zhuravel A P, Indrajeet S, Taketani B G, Hutchings M D, Hao Y, Rouxinol F, Wilhelm F K, LaHaye M D, Ustinov A V, and Plourde B L T 2019 Phys. Rev. Appl. 11 054062
[32]	Chakram S, Oriani A E, Naik R K, Dixit A V, He K, Agrawal A, Kwon H, and Schuster D I 2021 Phys. Rev. Lett. 127 107701
[33]	McKay D C, Naik R, Reinhold P, Bishop L S, and Schuster D I 2015 Phys. Rev. Lett. 114 080501
[34]	Naik R K, Leung N, Chakram S, Groszkowski P, Lu Y, Earnest N, McKay D C, Koch J, and Schuster D I 2017 Nat. Commun. 8 1904
[35]	Blais A, Grimsmo A L, Girvin S M, and Wallraff A 2021 Rev. Mod. Phys. 93 025005
[36]	Zheng S B and Guo G C 2000 Phys. Rev. Lett. 85 2392
[37]	Johansson J R, Nation P D, and Nori F 2012 Comput. Phys. Commun. 183 1760
[38]	Johansson J R, Nation P D, and Nori F 2013 Comput. Phys. Commun. 184 1234
[39]	Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[40]	Koch J, Yu T M, Gambetta J M, Houck A, Schuster D, Majer J, Blais A, Devoret M H, Girvin S M, and Schoelkopf R J 2007 Phys. Rev. A 76 042319
[41]	Numerical verification of this extrapolation is given by Fig. S1 in the Supplementary Material.
[42]	Fowler A G, Mariantoni M, Martinis J M, and Cleland A N 2012 Phys. Rev. A 86 032324
[43]	Nielsen M A 2002 Phys. Lett. A 303 249
[44]	Korotkov A N 2013 arXiv:1309.6405 [quant-ph]
[45]	Li X, Cai T, Yan H, Wang Z, Pan X, Ma Y, Cai W, Han J, Hua Z, Han X, Wu Y, Zhang H, Wang H, Song Y, Duan L, and Sun L 2020 Phys. Rev. Appl. 14 024070
[46]	Purcell E M 1995 Spontaneous Emission Probabilities at Radio Frequencies, in Confined Electrons and Photons: New Physics and Applications, edited by Burstein E and Weisbuch C (Boston, MA: Springer) NATO ASI Series vol 340 p 839
[47]	Reed M D, Johnson B R, Houck A A, DiCarlo L, Chow J M, Schuster D I, Frunzio L, and Schoelkopf R J 2010 Appl. Phys. Lett. 96 203110
[48]	See Eq. (S1) in the Supplementary Material.
[49]	Kwon S, Tomonaga A, Lakshmi B G, Devitt S J, and Tsai J S 2021 J. Appl. Phys. 129 041102
[50]	See Sec. S3 in the Supplementary Material.
[51]	Kjaergaard M, Schwartz M E, Braumüller J, Krantz P, Wang J I J, Gustavsson S, and Oliver W D 2020 Annu. Rev. Condens. Matter Phys. 11 369
[52]	See Sec. S4 in the Supplementary Material.
[53]	DiCarlo L, Chow J M, Gambetta J M, Bishop L S, Johnson B R, Schuster D I, Majer J, Blais A, Frunzio L, Girvin S M, and Schoelkopf R J 2009 Nature 460 240
[54]	Zhao P, Xu P, Lan D, Chu J, Tan X, Yu H, and Yu Y 2020 Phys. Rev. Lett. 125 200503
[55]	Zhao P, Lan D, Xu P, Xue G, Blank M, Tan X, Yu H, and Yu Y 2021 Phys. Rev. Appl. 16 024037
[56]	Khaneja N, Reiss T, Kehlet C, Schulte-Herbrüggen T, and Glaser S J 2005 J. Magn. Reson. 172 296
[57]	Müller M M, Reich D M, Murphy M, Yuan H, Vala J, Whaley K B, Calarco T, and Koch C P 2011 Phys. Rev. A 84 042315
[58]	Rowland B and Jones J A 2012 Philos. Trans. R. Soc. A 370 4636
[59]	Riviello G, Tibbetts K M, Brif C, Long R, Wu R B, Ho T S, and Rabitz H 2015 Phys. Rev. A 91 043401
[60]	Li J, Yang X, Peng X, and Sun C P 2017 Phys. Rev. Lett. 118 150503
[61]	Abdelhafez M, Baker B, Gyenis A, Mundada P, Houck A A, Schuster D, and Koch J 2020 Phys. Rev. A 101 022321