Remote cross-resonance gate between superconducting fixed-frequency qubits

Superconducting quantum computers are a promising candidate for realizing practical large-scale quantum computation in the future [1, 2]. To scale up quantum computer systems, different levels of modularity and interconnects between them have been actively investigated [2–7]. One approach is to connect adjacent quantum chips with sufficiently short distances (∼1 mm), which are the same as the distance between qubits on a single chip [4, 8–10], practically increasing the chip size and possibly allowing mapping of logical qubits encoded using surface codes [11–20]. Quantum chips separated by long distances in a refrigerator are connected through superconducting coaxial cables that pass microwave signals [5, 6, 21–29], millimeter-wave photonic links [30], or acoustic transmission lines as quantum phononic channels [31, 32]. For connecting qubits in different refrigerators or sending quantum information to the so-called quantum internet, quantum information stored in superconducting qubits must be frequency-converted into optical photons [7, 33–40]. However, it is very challenging to overcome large energy differences and achieve high conversion efficiencies. Herein, we focus on interconnects via 0.1–1 m superconducting coaxial cables inside a refrigerator (figure 1(a)). Such medium-range quantum interconnects pave the way not only for scaling up quantum computer systems but also for nonlocal connections on a chip and thus non-two-dimensional (non-2D) quantum error-correcting codes [2, 11].

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Quantum state transfer between two qubits through a superconducting coaxial cable has been investigated using the pitch-and-catch scheme, wherein a photon emitted by a qubit at one end of the communication channel is received by another qubit at the opposite end [3, 21–24]. In 2018, a fidelity of 80% was reported for quantum state transfer using a 0.9 m coaxial cable [22]. Half-quantum state transfer can generate a remote entangled state. A fidelity of 79% was reported for half-quantum state transfer [22]. Recent advances in fidelities have been achieved through a quantum state transfer scheme using the standing-wave modes of cables [5, 6, 25, 26]. A fidelity of 91% was reported for both quantum state transfer and remote entanglement through a 1 m cable [6]. Although the cable length was 0.25 m, the fidelity was increased to 99% by employing an aluminum low-loss coaxial cable and minimizing the dissipation at the junction between the cable and coplanar waveguides (figure 1(a)) [5]. However, the settings for these experiments are limited to the combination of frequency-tunable qubits [41–44] and tunable couplers [45] for matching the qubit frequency to the standing-wave mode of the cable. This can be because a small frequency detuning substantially reduces the transfer efficiency (appendix A).

Fixed-frequency qubits have the advantages of long coherence times and noise immunity [46–54]. Low sensitivity to low-frequency charge and magnetic-flux fluctuations ensures qubit coherence. However, with current manufacturing processes of fixed-frequency qubits, slight frequency shifts are unavoidable [55, 56]. Even if qubits are fabricated at a frequency that matches the standing-wave mode of the cable, as mentioned above, a very small frequency detuning of 5 MHz (0.1% of a typical qubit frequency (approximately 5 GHz)) severely compromises the transfer efficiency. Thus, quantum state transfer using cable standing-wave modes is not applicable to fixed-frequency qubits. A two-qubit gate for fixed-frequency qubits is well known to use the cross-resonance effect [57–65]. Cross-resonance microwave signals enable conditional control between coupled qubits of different frequencies. Recent improvements in control pulses and coupling circuits have increased the fidelity of cross-resonance gates to 99.7%, with gate times of 100–400 ns [60, 62, 63]. In this study, we consider cross-resonance gates between two fixed-frequency qubits connected by a coaxial cable. More specifically, we explore the possibility of cross-resonance gates acting through multiple cable modes with an energy spacing similar to or narrower than qubit frequency detunings suitable for the cross-resonance effect.

The remainder of this paper is organized as follows. In section 2, we present the configuration of the transmission paths and reveal the fundamental properties of remote cross-resonance gates, such as concurrences as measures of entanglement generation and average gate fidelities. Section 3 focuses on the dissipative property of the transmission paths and its effect on the cross-resonance gates. In section 4, we discuss leakage and single-qubit gate properties in remote cross-resonance gate settings. In section 5, we examine the effect of higher energy levels of qubits on remote cross-resonance gates and single-qubit gates. Finally, section 6 presents the conclusions of the study and an outlook for future works.

We first configure the transmission paths and then present the fundamental properties of remote cross-resonance gates via these transmission paths. The qubit frequency dependence of the concurrence in entanglement generation is an important result of this work. We also introduce the average gate fidelity as a more general characteristic of two-qubit gates. In this section, we do not consider any dissipation to focus on the effect of multiple cable modes.

2.1. Transmission path

We consider a transmission path comprising a cable and coplanar waveguides as shown in figure 1(a). Target cable lengths are 0.25, 0.5, and 1 m. Let the frequency of the Mth standing-wave mode of the cable be $\omega_M/(2\pi) = 5$ GHz (figure 1(b)). Here, M is an odd number and is chosen such that the cable length is close to the target cable length. The cable and coplanar waveguide lengths are determined as follows:

$$\begin{align} l_\mathrm{Cable} & = \frac{M-1}{2}\frac{v_\mathrm{Cable}}{\omega_M/\left(2\pi\right)} = \frac{M-1}{2} \lambda_\mathrm{Cable}\nonumber\\[8pt] l_\mathrm{CPW} & = \frac{1}{4}\frac{v_\mathrm{CPW}}{\omega_M/\left(2\pi\right)} = \frac{1}{4} \lambda_\mathrm{CPW}, \end{align} \tag{ 1 }$$

where the microwave speeds in the cable and coplanar waveguide, $v_\mathrm{Cable} = 2.472\times10^8$ m s⁻¹ and $v_\mathrm{CPW} = 1.157 \times10^8$ m s⁻¹, respectively, were determined to reproduce the experimental results [5] and $\lambda_\mathrm{Cable}$ and $\lambda_\mathrm{CPW}$ are the microwave wavelengths in the cable and coplanar waveguide, respectively. The standing-wave mode spacing, namely free spectral range, is obtained as $\omega_\mathrm{FSR} = \omega_M/M$. These characteristics of transmission paths are summarized in table 1.

Table 1. Cable length ($l_\mathrm{Cable}$), coplanar waveguide length ($l_\mathrm{CPW}$), number of cable standing-wave mode for 5 GHz (M), and free spectral range ($\omega_\mathrm{FSR}$) for target cable lengths of 0.25, 0.5, and 1 m.

$l_\mathrm{Cable}$ (m)	$l_\mathrm{CPW}$ (mm)	M	$\omega_\mathrm{FSR}/(2\pi)$ (GHz)
0.2966	5.8	13	0.3846
0.5438	5.8	23	0.2174
1.0832	5.8	43	0.1163

2.2. Remote cross-resonance Hamiltonian

Two fixed-frequency qubits $\mathrm{Q^A}$ and $\mathrm{Q^B}$ are connected to the transmission path (figure 1(a)). The Hamiltonian for remote cross-resonance gates is given by

$$\begin{align} H_0 & = {\Delta}_\mathrm{A}\sigma_\mathrm{A}^{\dagger}\sigma_\mathrm{A} + {\Delta}_\mathrm{B}\sigma_\mathrm{B}^{\dagger}\sigma_\mathrm{B} + \sum_{m = M_\mathrm{min}}^{M_\mathrm{max}}\left(m-M\right)\omega_\mathrm{FSR}\sigma_m^{\dagger}\sigma_m\nonumber\\ & \quad + \sum_{m = M_\mathrm{min}}^{M_\mathrm{max}} g_\mathrm{A}\sqrt{\omega_m/\omega_M}\left(\sigma_\mathrm{A}\sigma_m^{\dagger}+\sigma_\mathrm{A}^{\dagger}\sigma_m\right)\nonumber\\ & \quad + \sum_{m = M_\mathrm{min}}^{M_\mathrm{max}}\left(-1\right)^m g_\mathrm{B}\sqrt{\omega_m/\omega_M} \left(\sigma_\mathrm{B}\sigma_m^{\dagger}+\sigma_\mathrm{B}^{\dagger}\sigma_m\right)\nonumber\\ H & = H_0+\Omega\left(t\right)\cos\left(\tilde{\omega}_\mathrm{B}t\right)\sigma_x^\mathrm{A}, \end{align} \tag{ 2 }$$

where $\sigma_\mathrm{A}$, $\sigma_\mathrm{B}$, and σ_m are annihilation operators for $\mathrm{Q^A}$, $\mathrm{Q^B}$, and the mth cable mode, $\Delta_\mathrm{A}$ and $\Delta_\mathrm{B}$ are the qubit frequency detunings measured from ω_M , and $\omega_m = m\omega_\mathrm{FSR}$ is the mth cable mode frequency. The couplings between the qubits and cable modes are $g_\mathrm{A}$ and $g_\mathrm{B}$ for the Mth mode and proportional to the square root of the frequency in multimode coupling [6, 66].

The last term of the second equation in equation (2) represents a cross-resonance effect, and $\tilde{\omega}_\mathrm{B}$ is the dressed qubit frequency of $\mathrm{Q^B}$ that is by solving an eigenvalue problem for H₀. The cross-resonance pulse envelope $\Omega(t)$ is schematically presented in figure 1(c). We consider three pulse envelopes: flat pulse envelope, raised-cosine pulse envelope to reduce spectral leakage in the frequency domain [67, 68], and echoed pulse envelope to equalize the difference in pulse duration due to the initial state [59, 60]. We fix the amplitude of the flat parts to $\Omega_0/(2\pi) = 0.1\,\mathrm{GHz}$ and the rise and fall times for the raised-cosine and echoed pulses to $\tau_0 = 100\times2\pi/\tilde{\omega}_\mathrm{B} \sim20\,\mathrm{ns}$ because the cross-resonance effect can be adjusted by the total pulse duration τ.

We write the qubit states as $|\Phi_i\rangle = c_{i,|00\rangle}|00\rangle+c_{i,|01\rangle}|01\rangle+c_{i,|10\rangle}|10\rangle+c_{i,|11\rangle}|11\rangle$, where each ket represents the eigenstate of H₀ in order of $\mathrm{Q^A Q^B}$. The ideal cross-resonance gate is expressed as

$$\begin{align} \left[ZX\right]^{1/2} = \frac{1}{\sqrt{2}} \left( \begin{array}{cccc} 1 & -i & 0 & 0 \\ -i & 1 & 0 & 0 \\ 0 & 0 & 1 & i \\ 0 & 0 & i & 1 \\ \end{array} \right), \end{align} \tag{ 3 }$$

which is combined with two single-qubit gates to form a CNOT gate (i.e. $\mathrm{CNOT} = [ZI]^{-1/2}[ZX]^{1/2}[IX]^{-1/2}$). The CNOT gate results in remote state transfer for the initial state of $|\Phi^\mathrm{I}_1\rangle = |10\rangle$ and forms a Bell state as a maximum entangled state for $|\Phi^\mathrm{I}_+\rangle = (|00\rangle+|10\rangle)/\sqrt{2}$.

2.3. Time evolution simulation

Herein, we present details of time evolution simulations using the case of a 0.25 m coaxial cable. In the Hamiltonian of equation (2), we set M = 13 and $\omega_\mathrm{FSR}/(2\pi) = 0.3846$ GHz from table 1. Let $\Delta_\mathrm{A} = 0.7\omega_\mathrm{FSR}$ and $\Delta_\mathrm{B} = -0.3\omega_\mathrm{FSR}$, then $\omega_\mathrm{A}\gt\omega_M$ and $\omega_\mathrm{B}\lt\omega_M$ as shown in figure 1(b). This is written as $\delta_\mathrm{A} = |\Delta_\mathrm{A}|/$ $\omega_\mathrm{FSR} = 0.7$ and $\delta_\mathrm{B} = |\Delta_\mathrm{B}|/\omega_\mathrm{FSR} = 0.3$ and results in a qubit frequency detuning of $\Delta/(2\pi) = (\omega_\mathrm{A}-\omega_\mathrm{B})/$ $(2\pi) = 0.3846\,\mathrm{GHz}$. Two cable modes above $\omega_\mathrm{A}$ and two below $\omega_\mathrm{B}$ (appendix B) yield the upper and lower cable mode limits $M_\mathrm{max} = 15$ and $M_\mathrm{min} = 11$, respectively. The coupling between the qubits and cable modes is chosen as $g_\mathrm{A}/(2\pi) = g_\mathrm{B}/(2\pi) = g/(2\pi) = 0.03$ GHz. We numerically simulate time evolution using the QuTip framework [69, 70]. The results with a cross-resonance pulse are presented in figure 2.

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We first consider the initial qubit states of $|\Phi^\mathrm{I}_0\rangle = |00\rangle$ and $|\Phi^\mathrm{I}_1\rangle = |10\rangle$. When $[ZX]^{1/2}$ is applied, $|\Phi^\mathrm{I}_0\rangle$ and $|\Phi^\mathrm{I}_1\rangle$ become $|\Phi^\mathrm{F}_0\rangle = (|00\rangle-i|01\rangle)/\sqrt{2}$ and $|\Phi^\mathrm{F}_1\rangle = (|10\rangle+i|11\rangle)/\sqrt{2}$, respectively. Figure 2(a) presents the results with a flat cross-resonance pulse applied continuously for 200 ns. The amplitude of the cross-resonance pulse is set to $\Omega_0/(2\pi) = 0.1\,\mathrm{GHz}$, as described above. The occupation probability of the qubit state is given as $\mathcal{P}_{i,k} = |c_{i,k}|^2$ for each k state. The $\mathcal{P}_0$ for $|\Phi^\mathrm{I}_0\rangle$ and $\mathcal{P}_1$ for $|\Phi^\mathrm{I}_1\rangle$ are shown in the left panels. With increasing time, $\mathcal{P}_{0,|00\rangle}$ decreases from 1, $\mathcal{P}_{0,|01\rangle}$ increases from 0, and $\mathcal{P}_{0,|00\rangle}$ and $\mathcal{P}_{0,|01\rangle}$ intersect at approximately $\mathcal{P}_0 = 0.5$ and 120 ns. Similarly, $\mathcal{P}_{1,|10\rangle}$ and $\mathcal{P}_{1,|11\rangle}$ intersect at approximately $\mathcal{P}_1 = 0.5$ and 140 ns. We define the concurrences

$$\begin{align} \mathcal{C}_0 & = 2|c_{0,|00\rangle}||c_{0,|01\rangle}| \nonumber\\ \mathcal{C}_1 & = 2|c_{1,|10\rangle}||c_{1,|11\rangle}| \end{align} \tag{ 4 }$$

so that they become 1 when $|\Phi_i\rangle = |\Phi^\mathrm{F}_i\rangle$. The right panels of figure 2(a) show $\mathcal{C}_0$ and $\mathcal{C}_1$. The concurrence takes the maximum values of $\mathcal{C}_0 = 96.82\%$ at 120.1 ns and $\mathcal{C}_1 = 96.86\%$ at 139.4 ns, which are near the time of the intersection of each $\mathcal{P}$. These results show that the cross-resonance pulses with parameters currently under consideration can fundamentally act as a $[ZX]^{1/2}$ gate with a pulse duration similar to those of cross-resonance gates between neighboring qubits on a chip [60, 63].

To reduce spectral leakage in the frequency domain, a raised-cosine pulse is employed. We fix the rise and fall time to $\tau_0 = 100\times 2\pi/\tilde{\omega}_\mathrm{B} = 20.5\,\mathrm{ns}$ and find the pulse duration τ when $\mathcal{C}_0$ and $\mathcal{C}_1$ take their maximum values. We obtain the maximum value of $\mathcal{C}_0 = \mathcal{C}_1 = 100.00\%$ at $\tau = 140.4\,\mathrm{ns}$ for $|\Phi^\mathrm{I}_0\rangle$ and $\tau = 159.9\,\mathrm{ns}$ for $|\Phi^\mathrm{I}_1\rangle$. The time evolution during the raised-cosine pulse for each initial state is depicted in figure 2(b).

Further, to equalize the difference in τ due to the initial state, we adopt an echoed cross-resonance pulse. To isolate the precision of remote cross-resonance gates, two single-qubit π rotations for $\mathrm{Q^A}$ are analytically performed during the process. The rise and fall times are again fixed to $\tau_0 = 100\times 2\pi/\tilde{\omega}_\mathrm{B} = 20.5\,\mathrm{ns}$. We find τ when $\mathcal{C}_0$ and $\mathcal{C}_1$ take their maximum values. We obtain $\tau = 169.3\,\mathrm{ns}$ and the maximum $\mathcal{C} = 100.00\%$ for both $|\Phi_0\rangle$ and $|\Phi_1\rangle$. Figure 2(c) depicts the time evolution during the echoed cross-resonance pulse for each initial state $|\Phi_0\rangle$ and $|\Phi_1\rangle$.

We have examined the case where the cross-resonance signal is input from a higher-frequency qubit to a lower-frequency qubit. Here, conversely, the qubit frequency detunings are set to $\Delta_\mathrm{A} = -0.3\omega_\mathrm{FSR}$ and $\Delta_\mathrm{B} = 0.7\omega_\mathrm{FSR}$. We confirm almost the same results with an echoed cross-resonance pulse, $\mathcal{C}_0 = 99.99\%$, $\mathcal{C}_1 = 100.00\%$, and τ = 160.3 ns, as those described above. Thus, we continue our study by focusing on a cross-resonance signal from a higher-frequency qubit.

One important consideration for remote cross-resonance gates is the degree of entanglement generation [5, 6, 22, 25]. We simulate time evolution with the echoed remote cross-resonance pulse determined above for the initial qubit state of $|\Phi^\mathrm{I}_+\rangle = (|00\rangle+|10\rangle)/\sqrt{2}$. When $[ZX]^{1/2}$ is applied to $|\Phi^\mathrm{I}_+\rangle$, we obtain $|\Phi^\mathrm{F}_+\rangle = (|00\rangle-i|01\rangle+|10\rangle+i|11\rangle)/2$, which is an entangled state, unlike $|\Phi^\mathrm{F}_0\rangle$ and $|\Phi^\mathrm{F}_1\rangle$. With a qubit state $|\Phi_+\rangle$, the concurrence as a measure of entanglement is defined by

$$\begin{align} \mathcal{C}_+ = 16|c_{+,|00\rangle}||c_{+,|01\rangle}||c_{+,|10\rangle}||c_{+,|11\rangle}| \end{align} \tag{ 5 }$$

so that this becomes 1 when $|\Phi_+\rangle = |\Phi^\mathrm{F}_+\rangle$. We obtain the concurrence $\mathcal{C}_+ = 100.00\%$. Thus, we examine the qubit frequency dependence of $\mathcal{C}_+$ in the next section.

2.4. Qubit frequency dependence

Let us consider the range of qubit frequencies in which the remote cross-resonance gate operates efficiently. We examine the maximum value of $\mathcal{C}_+$ with an echoed cross-resonance pulse for each $\delta_\mathrm{A}$ and $\delta_\mathrm{B}$ in the ranges $\delta_i \neq 0,1,2\ldots$, where $\delta_i = |\Delta_i|/\omega_\mathrm{FSR}, i = \mathrm{A}$ and $\mathrm{B}$, because $\delta_i = 0,1,2,\ldots$ do not well define the qubit state owing to the same energy levels of the qubit and cable mode (figure 1(b)).

2.4.1. 0.25 m coaxial cable

Figure 3 summarizes the maximum $\mathcal{C}_+$ and echoed pulse duration τ in the range $0.1 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 0.9$ and $0.1 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 0.9$ for a 0.25 m coaxial cable with $g/(2\pi) = 0.03\,\mathrm{GHz}$. We can find $3\times3$ cells with $\mathcal{C}_{+} \gt99.9\%$ in the top-right part ($0.1 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 0.3$ and $0.6 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 0.8$) with $\tau\sim160\,\mathrm{ns}$. The frequency detuning of $\delta_i = 0.1$ corresponds to 38.5 MHz. This means that if we aim for the frequencies of $\delta_\mathrm{A} = 0.2$ and $\delta_\mathrm{B} = 0.7$ to fabricate the qubits, even with ±38 MHz frequency error due to manufacturing, obtaining $\mathcal{C}_{+} \gt99.9\%$ is possible by calibrating the signal frequency and pulse duration. A standard deviation of the frequency distribution of $\sigma_f = 14\,\mathrm{MHz}$ has already been achieved with the current technology [55, 56].

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We can find other $3\times3$ cells with $\mathcal{C}_{+} \gt99.9\%$ in the bottom left ($0.6 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 0.8$ and $0.2 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 0.4$). The results for the center of this area, $\delta_\mathrm{A} = 0.7$ and $\delta_\mathrm{B} = 0.3$, have been shown in the previous section and are summarized in table 2 as an example. The maximum $\mathcal{C}_+$ in the diagonal cells is smaller than the others. We find that this degradation is due to the closeness of the energy levels of the qubit $|11\rangle$ state and the cable state comprising two cable modes $M\pm k$. This is a notable feature in remote cross-resonance gates through cables with multimode coupling.

Table 2. Examples of remote cross-resonance gates with a high concurrence: target cable length, qubit frequency detunings ($\delta_\mathrm{A}$,$\delta_\mathrm{B}$, and Δ), coupling (g), pulse duration (τ), concurrence for the initial qubit state $|\Phi^\mathrm{I}_+\rangle$ ($\mathcal{C}_+$), average gate fidelity ($\bar{\mathcal{F}}$), and concurrence with dissipation for the initial qubit state $|\Phi^\mathrm{I}_+\rangle$ ($\mathcal{C}^{^{\prime}}_+$).

Cable length	$\delta_\mathrm{A}$	$\delta_\mathrm{B}$	$\Delta/(2\pi)$	$g/(2\pi)$	τ	$\mathcal{C}_+$	$\bar{\mathcal{F}}$	$\mathcal{C}^{\prime}_+$
(m)			(GHz)	(GHz)	(ns)
0.25	0.7	0.3	0.3846	0.03	169.3	1.0000	0.9999	0.9999
0.5	1.25	0.75	0.4348	0.03	152.4	0.9998	0.9998	0.9997
	0.35	0.65	0.2174	0.02	147.3	0.9998	0.9998	0.9996
1.0	2.55	1.25	0.4419	0.02	201.9	0.9928	0.9964	0.9859
	2.5	1.2	0.4302	0.01	385.9	0.9964	0.9976	0.9943
	1.55	1.35	0.3372	0.01	400.3	0.9984	0.9989	0.9975

A smaller coupling of $g/(2\pi) = 0.02$ GHz is also examined (appendix C and figure C1(a)). Despite the increase in τ (∼300 ns), the range with $\mathcal{C}_{+} \gt99.9\%$ does not expand. Although some cells with $\mathcal{C}_+\unicode{x2A7D}99\%$ change to $\mathcal{C}_{+} \gt99\%$ and the cell where the qubit state is not defined disappears, it is not beneficial to make the coupling smaller in this case, where high $\mathcal{C}_+$ values are already obtained over a wide range.

2.4.2. 0.5 m coaxial cable

The free spectral range for a 0.5 m coaxial cable is a small value of $\omega_\mathrm{FSR}/(2\pi) = 0.2174$ GHz (table 1). We consider the range $1.1\unicode{x2A7D}\delta_\mathrm{A}\unicode{x2A7D}1.9$ and $0.1\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}0.9$ to include the region $\Delta/(2\pi)\sim0.4$ GHz where high $\mathcal{C}_+$ is obtained for a 0.25 m coaxial cable. The results for $g/(2\pi) = 0.03$ GHz are shown in figure 4(a). As in the case of a 0.25 m coaxial cable, we can find two $2\times2$ cells with $\mathcal{C}_{+} \gt99.9\%$ [($1.2\unicode{x2A7D}\delta_\mathrm{A}\unicode{x2A7D}1.3$ and $0.7\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}0.8$) and ($1.4 \unicode{x2A7D}\delta_\mathrm{A}\unicode{x2A7D}1.5$ and $0.8\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}0.9$)] for $\tau\sim150$ ns. The frequency detuning of $\delta_i = 0.1$ corresponds to 21.7 MHz. This means that if we aim for the frequencies of the center of the areas to fabricate the qubits, obtaining $\mathcal{C}_{+} \gt99.9\%$ is possible even with ±10 MHz frequency error owing to manufacturing. The center of the area $\delta_\mathrm{A} = 1.25$ and $\delta_\mathrm{B} = 0.75$ gives $\mathcal{C}_+ = 99.98\%$, which is included in table 2, and $\delta_\mathrm{A} = 1.45$ and $\delta_\mathrm{B} = 0.85$ also gives $\mathcal{C}_+ = 99.98\%$. We find some $2\times2$ cells with $\mathcal{C}_{+} \gt99.9\%$ in the range $0.1\unicode{x2A7D}\delta_\mathrm{A} \unicode{x2A7D}0.9$ and $1.1\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D} 1.9$ (appendix C and figure C1(b)).

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The range $0.1 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 0.9$ and $0.1 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 0.9$, where Δ exhibits small values, are also examined. As shown in figure 4(b), reducing the coupling to $g/(2\pi) = 0.02$ GHz leads to $2\times2$ cells with $\mathcal{C}_{+} \gt99.9\%$ and the average values of $\tau\sim150$ ns and $\Delta/(2\pi)\sim0.22$ GHz. The center of the area $\delta_\mathrm{A} = 0.35$ and $\delta_\mathrm{B} = 0.65$ gives $\mathcal{C}_+ = 99.98\%$ for τ = 147.3 ns, which is also included in table 2.

2.4.3. 1 m coaxial cable

The free spectral range for a 1 m coaxial cable is a further small value of $\omega_\mathrm{FSR}/(2\pi) = 0.1163$ GHz (table 1). The range $2.1\unicode{x2A7D}\delta_\mathrm{A} \unicode{x2A7D}2.9$ and $1.1\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}1.9$ includes the region $\Delta/(2\pi)\sim0.4$ GHz where high $\mathcal{C}_+$ is obtained for a 0.25 m coaxial cable. However, the coupling $g/(2\pi) = 0.03$ GHz does not provide well-defined qubit states over the entire range. The results for the range with $g/(2\pi) = 0.02$ GHz are presented in figure 5(a). We find $2\times2$ cells with $\mathcal{C}_{+} \gt99\%$ in the bottom-left part ($2.5 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 2.6$ and $1.2 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 1.3$). The center of the area $\delta_\mathrm{A} = 2.55$ and $\delta_\mathrm{B} = 1.25$ gives $\mathcal{C}_+ = 99.28\%$, and the results are summarized in table 2. We cannot obtain an area with $\mathcal{C}_{+} \gt99.9\%$ as in the cases of 0.25- and 0.5 m cables. It is found that the occupation probability of the cable states with energies corresponding to $\sim2\tilde{\omega}_\mathrm{B}$ or $\sim3\tilde{\omega}_\mathrm{B}$ gradually increases during the cross-resonance pulse. This can be the most important characteristic with regard to remote cross-resonance gates through cables with multimode coupling. However, if we aim for the frequencies of $\delta_\mathrm{A} = 2.55$ and $\delta_\mathrm{B} = 1.25$ to fabricate the qubits, even in case of manufacturing errors of $\pm5\,\mathrm{MHz}$, it is possible to obtain $\mathcal{C}_{+} \gt99\%$, which is in sharp contrast to quantum state transfer, where a detuning of 5 MHz substantially reduces the efficiency, as mentioned earlier. The frequency shift of 5 MHz corresponds to 0.1% of the qubit frequency ∼5 GHz and matches the standard deviation of the frequency distribution that is required to achieve a 1000-qubit computer [55, 56]. We also find two $2\times2$ cells with $\mathcal{C}_{+} \gt99\%$ in the range $1.1 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 1.9$ and $2.1 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 2.9$ (appendix C and figure C1(c)).

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Unlike in the case of a 0.25 m coaxial cable, it is very beneficial to reduce the coupling to $g/(2\pi) = 0.01$ GHz. Although τ increases, we can find much wider areas with $\mathcal{C}_{+} \gt99\%$ as shown in figure 5(b). For example, there are $3\times3$ cells ($2.4 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 2.6$ and $1.1 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 1.3$) with $\mathcal{C}_{+} \gt99.5\%$ for $\tau\sim400\,\mathrm{ns}$. This corresponds to allowing manufacturing errors of $\pm11\,\mathrm{MHz}$ for $\mathcal{C}_{+} \gt99.5\%$. The center of the area $\delta_\mathrm{A} = 2.5$ and $\delta_\mathrm{B} = 1.2$ gives $\mathcal{C}_+ = 99.64\%$, which is included in table 2. The range $1.1 \unicode{x2A7D} \delta_\mathrm{A} \unicode{x2A7D} 1.9$ and $1.1 \unicode{x2A7D} \delta_\mathrm{B} \unicode{x2A7D} 1.9$ for $g/(2\pi) = 0.01\,\mathrm{GHz}$ are also examined, and the results are presented in figure 5(c). Numerous $2\times2$ cells with $\mathcal{C}_{+} \gt99\%$ are obtained. Among them, the area centered on $\delta_\mathrm{A} = 1.55$ and $\delta_\mathrm{B} = 1.35$ leads to $\mathcal{C}_{+} \gt99.5\%$. The results on the center of $\mathcal{C}_+ = 99.84\%$ and τ = 400.3 ns are also included in table 2.

To summarize the results thus far, as shown in table 2, there are qubit frequencies of $\mathcal{C}_{+} \gt99.9\%$ for 0.25- and 0.5 m cables even with manufacturing errors of ±38 MHz and ±10 MHz, respectively, and $\mathcal{C}_{+} \gt99\%$ for a 1 m cable even with ±5 MHz. In the case of a 1 m cable, by reducing the coupling between the qubits and cable, although τ increases, the lower limit of $\mathcal{C}_+$ and allowed manufacturing error can be 99.5% and ±11 MHz. These are in contrast to quantum state transfer, where a small detuning of 5 MHz substantially reduces the efficiency (appendix A).

Here, let us discuss the effect of residual cable states on the results. In this study, we evaluate a concurrence immediately after a cross-resonance pulse. However, the residual cable states may affect the qubit state after the cross-resonance pulse is turned off. We evaluate $C_+$ 200 ns after a cross-resonance pulse for one parameter set for each cable length of the remote cross-resonance gates listed in table 2 ($\delta_\mathrm{A} = 0.7$, $\delta_\mathrm{B} = 0.3$, and $g/(2\pi) =$ 0.03 GHz for a 0.25 m cable, $\delta_\mathrm{A} = 0.35$, $\delta_\mathrm{B} = 0.65$, and $g/(2\pi) =$ 0.02 GHz for a 0.5 m cable, and $\delta_\mathrm{A} = 1.55$, $\delta_\mathrm{B} = 1.35$, and $g/(2\pi) =$ 0.01 GHz for a 1 m cable). We find that $C_+$ change by -0.05%, -0.001%, and +0.01% for 200 ns after an echoed cross-resonance pulse for 0.25-, 0.5-, and 1 m cables, respectively. For the same parameter sets with a flat cross-resonance pulse, when the pulse is turned off after maximum $C_+$ values is obtained, $C_+$ increase by 0.68%, 0.16%, and 0.10% for 200 ns for 0.25-, 0.5-, and 1 m cables, respectively. The effect of residual cable states observed for flat cross-resonance pulses seems to be suppressed for echoed cross-resonance pulses. Thus, the effect of residual cable states is less significant in the discussion using echoed raised-cosine pulses.

2.5. Average gate fidelity

We determine the dissipative properties of the transmission path and perform time evolution simulations for remote cross-resonance gates using these dissipative properties.

3.1. Relaxation time

The Q value of the mth cable standing-wave mode Q_m is given as follows:

$$\begin{align} 1/Q_m & = 1/Q_\mathrm{Cable} + 1/Q_\mathrm{Loss}\nonumber\\ Q_\mathrm{Loss} & = \frac{\omega_m L}{\mathrm{cos}^2\left(2\pi l_\mathrm{CPW}/\lambda_\mathrm{CPW}^m\right)R}, \end{align} \tag{ 6 }$$

where $Q_\mathrm{Cable}$ is the cable specific value, which is $1.2\times10^6$ for a 0.25 m coaxial cable [5], and $Q_\mathrm{Loss}$ is determined based on the ratio of $l_\mathrm{CPW}$ and $\lambda_\mathrm{CPW}^m$. Here, $l_\mathrm{CPW}$ is presented in table 1 and $\lambda_\mathrm{CPW}^m = 2\pi v_\mathrm{CPW}/\omega_m$ is the wavelength of the mth mode in the coplanar waveguide, where $v_\mathrm{CPW} = 1.157\times10^8$ m s⁻¹ [5] and $\omega_m = m\omega_\mathrm{FSR}$. The equivalent inductance L is given by

$$\begin{align} L \approx \frac{1}{2}\left(2\mathcal{L}_\mathrm{CPW}l_\mathrm{CPW}+\mathcal{L}_\mathrm{Cable} l_\mathrm{Cable}\right), \end{align} \tag{ 7 }$$

where $\mathcal{L}_\mathrm{CPW} = 402$ nH m⁻¹ and $\mathcal{L}_\mathrm{Cable} = 216$ nH m⁻¹ are the specific inductances of the coplanar wave guide and coaxial cable, respectively [5]. The resistance between the coplanar waveguide and coaxial cable R is determined to be 0.0749 $\mathrm{n}{\Omega}$ by reproducing the experimental results [5]. Figure 6 shows the Q value and relaxation time of each standing-wave mode for each cable length. The relaxation time of the mth mode T_m is obtained by $Q_m = \omega_m T_m$.

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3.2. Time evolution simulation

Herein, we discuss leakage and single-qubit gates in settings for remote cross-resonance. For the leakage, we examine the time evolution of the initial qubit states of $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$ without any driving pulse, i.e. using the H₀ in equation (2). In view of the results of remote cross-resonance gates, the coupling is set to $g/(2\pi) = 0.03$, 0.02, and 0.01 GHz for 0.25-, 0.5-, and 1 m coaxial cables, respectively. The range $0.1\unicode{x2A7D}\delta_\mathrm{A}\unicode{x2A7D}0.9$ and $0.1\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}0.9$ is examined for each case. When the dissipation is not considered, the occupation probabilities for all initial qubit states remain 100.00% in 200 ns for all cable lengths.

Next, we consider the dissipation of the transmission path. We do not consider the decoherence due to qubits to isolate the cable dissipation effect. Even when the dissipation is considered, the occupation probability of the $|00\rangle$ initial qubit state remains 100.00% for 200 ns. Other results are presented in appendix D, figures D1–D3. For 0.25- and 0.5 m cables, the occupation probabilities for $|01\rangle$ and $|10\rangle$ in the area corresponding to the remote cross-resonance gates in table 2 are above 99.9%, while those for $|11\rangle$ are less than 99.9%. For a 1 m cable, the occupation probabilities are between 99.5% and 99.9% for all initial qubit states. These results mean that although reducing the coupling leads to considerable fidelities for longer cable lengths, tunable couplers can be useful even when connecting different frequency qubits, as used in previous quantum-state-transfer experiments [5, 6, 25, 26].

For single-qubit gates, we consider a π rotation around the x axis of $\mathrm{Q^A}$. In equation (2), $\tilde{\omega}_\mathrm{B}$ is replaced by $\tilde{\omega}_\mathrm{A}$ and a raised-cosine pulse envelope is used for $\Omega(t)$. The amplitude of the flat part of the signal is set to $\Omega_0 = 0.005\tilde{\omega}_\mathrm{A}$ and the raised-cosine duration is set to $\tau_0 = 100\times2\pi/\tilde{\omega}_\mathrm{A}$. The final state fidelities are averaged for the initial qubit states of $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$. The pulse duration is optimized for the maximum averaged state fidelity for each $\delta_\mathrm{A}$ and $\delta_\mathrm{B}$. For a 0.25 m cable, the range $0.1\unicode{x2A7D}\delta_\mathrm{A}\unicode{x2A7D}0.9$ and $0.1\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}0.9$ is examined. The results are presented in figure 7. We find that the fidelities obtained for $\delta_\mathrm{A} = 0.5$ are ∼99.8%, which are lower than the others of $\gt$99.9%. This is also a notable feature of qubits connected through a cable with multimode coupling. When considering the dissipation of the transmission path, we obtain almost the same results, as shown for remote cross-resonance gates in the previous section.

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For 0.5- and 1.0 m cables, we investigate one parameter set for each length of the remote cross-resonance gates listed in table 2. The case of $\delta_\mathrm{A} = 0.35$, $\delta_\mathrm{B} = 0.65$, and $g/(2\pi) = 0.02$ GHz for a 0.5 m cable gives average fidelities of 99.99% and 99.97% without and with dissipation, respectively. For a 1 m cable, we examine the case of $\delta_\mathrm{A} = 1.55$, $\delta_\mathrm{B} = 1.35$, and $g/(2\pi) = 0.01\,\mathrm{GHz}$. We find that average fidelities without and with dissipation only go up to 99.19% and 99.18%, respectively. In case of a 1 m cable with a narrower mode space, it seems difficult to avoid increasing the occupation probability of states near the energies of $\tilde{\omega}_\mathrm{A}, 2\tilde{\omega}_\mathrm{A},3\tilde{\omega}_\mathrm{A},\ldots$, which comprise the cable modes or their hybridized states with qubit excited states. This means that the use of tunable couplers is increasingly needed.

Finally, we examine the effect of higher energy levels of qubits on the results. We have treated the qubits as a two level system ($|0\rangle$ and $|1\rangle$) as represented in equation (2). Here, we consider a higher state $|2\rangle$ of each qubit. If the anharmonicity of $\mathrm{Q_A}$ and $\mathrm{Q_B}$ are $\alpha_\mathrm{A}$ and $\alpha_\mathrm{B}$, the energy levels of $|2\rangle$ can be written as $\omega_\mathrm{2A} = \omega_M+2\Delta_\mathrm{A}-\alpha_\mathrm{A}$ and $\omega_\mathrm{2B} = \omega_M+2\Delta_\mathrm{B}-\alpha_\mathrm{B}$, respectively. For transmon qubits, α is usually 0.1–0.3 GHz, which can be designed independently of the qubit frequency [46]. We again focus on the remote cross-resonance gates presented in table 2.

Figure 8 shows the concurrence $C_{+}$ considering the $|2\rangle$ state of $\mathrm{Q_A}$ and $\mathrm{Q_B}$ when $\alpha/(2\pi)$ is changed from 0.1 to 0.3 GHz. For a 0.25 m cable, $C_+$ has two dips for the change in $\alpha_\mathrm{A}$ and does not change with respect to $\alpha_\mathrm{B}$ (figures 8(a) and (b)). The first dip at $\alpha_\mathrm{A}/(2\pi) = 0.13$ GHz is caused by the states where the sum of the numbers of two cable modes constituting the state is $2M+1$, such as $M+(M+1)$ or $(M-1)+(M+2)$. The second dip results from the states consisting of a qubit state and a cable mode, such as $|10\rangle+M$ or $|01\rangle+(M+1)$. Thanks to the equal spacing of the cable modes and the qubit frequency detuning equal to that spacing ($\Delta = \omega_\mathrm{FSR}$), the energy levels appear in clusters at intervals, and when $\omega_\mathrm{2A}$ is within the intervals, $C_+$ is not significantly impaired (figure 8(a)). The maximum $C_+$ value of 100.0% is obtained around 0.22 GHz and matches the result without higher energy states. The energy levels at $\alpha_\mathrm{A}/(2\pi) = \alpha_\mathrm{B}/(2\pi) = 0.2$ GHz are presented for each cross-resonance parameter set in figure 8 as a reference (crosses). Note that the energy levels vary with α.

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The results for one parameter set of a 0.5 m cable and for a 1.0 m cable can be understood in a similar manner (figures 8(c) and (f)). For a 0.5 m cable, there is a dip at $\alpha_\mathrm{A}/(2\pi) = 0.14$ GHz due to the $2M+2$ cluster, and a wide interval with the $|10\rangle+M$ cluster on the right. The same $C _+$ values as the result without higher energy states are obtained over a wide range in the interval (figure 8(c)). For a 1 m cable, the dip at $\alpha_\mathrm{A}/(2\pi) = 0.16$ GHz is caused by the $|10\rangle+M$ cluster. High $C _+$ values are obtained within the intervals with the $2M+2$ cluster on the left and with the $2M+1$ cluster on the right (figure 8(f)). However, the maximum $C+$ values are lower by 0.1% than the result without higher energy states, which can be attributed to the narrow intervals. There is a $|10\rangle+(M-1)$ cluster further to the right.

The other parameter set of a 0.5 m cable presents very different results (figures 8(d) and (e)). The qubit $|11\rangle$ state appearing at $\alpha_\mathrm{A}/(2\pi) = 0.21$ GHz forms a deep and wide dip for the change in $\alpha_\mathrm{A}$. There is a 2M cluster on the left and a $|10\rangle+(M-1)$ cluster on the right, each forming an additional dip. Even in this case, $C_+$ does not change with respect to $\alpha_\mathrm{B}$. The maximum $C_+$ of 99.83% is obtained at $\alpha_\mathrm{A}/(2\pi) = 0.11$ GHz and decreased by 0.15% from the result without higher energy states. Thus, the higher energy levels of qubits do not seem to have a significant effect on the summary in section 2.4. However, further selection and optimization can be done considering the higher energy levels and anharmonicity of qubits.

Next, we consider the effect of higher energy levels of qubits on the results for single-qubit gates described in section 4. Figure 9 shows the average gate fidelities of π rotations around the x axis of $\mathrm{Q_A}$ when $\alpha_\mathrm{A}/(2\pi)$ is changed from 0.1 to 0.3 GHz. The fidelity does not change with respect to $\alpha_\mathrm{B}$. The maximum fidelity values of 99.92%, 99.93%, and 99.08% are obtained at $\alpha_\mathrm{A}/(2\pi) = 0.3$ GHz for 0.25-, 0.5-, and 1 m cables, respectively. The decreases of 0.05%, 0.06%, and 0.11% from the results without higher energy states mainly occupy the $|2\rangle$ state of $\mathrm{Q_A}$ rather than increases in cable states. This is consistent with the fact that the fidelity increases nearly monotonically, and there is no structure reflecting the energy levels like remote cross-resonance gates. The accuracy improvement methods already implemented in single-qubit gates within chip can be also effective in this case.

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We have proposed remote cross-resonance gates to realize high-precision quantum state transfer and remote entanglement between superconducting fixed-frequency qubits. The Hamiltonian for two qubits connected through a multimode cable has been constructed, and time evolution simulations have been performed. For a 0.25 m cable, there are qubit frequencies to exhibit the concurrence in remote entanglement generation $\mathcal{C}_{+} \gt99.9\%$ by calibrating the signal frequency and pulse duration, even with ±38 MHz qubit frequency error due to manufacturing. A 0.5 m cable also allows manufacturing errors of ±10 MHz for $\mathcal{C}_{+} \gt99.9\%$. We have also found that the average gate fidelity, which is a more general characteristic of two-qubit gates, is higher than 99.9%. These high-precision quantum interconnects are promising not only for scaling up quantum computer systems but also for nonlocal connections on a chip and may open new avenues such as non-2D quantum error-correcting codes.

For a 1 m cable with a narrower mode spacing, remote cross-resonance gates with $\mathcal{C}_{+} \gt99.5\%$ can be achieved by reducing the coupling between the qubits and cable. This is in sharp contrast to quantum state transfer, where a small qubit frequency detuning of 5 MHz notably decreases the efficiency. Although the leakage shows considerable fidelities, it seems difficult to avoid reduction in the fidelity for single-qubit gates on qubits connected a 1 m cable.

The proposed remote cross-resonance gates should first be experimentally tested. In this context, we must introduce the accuracy improvement methods already implemented in single-qubit gates and cross-resonance gates within chips. It is also necessary to seriously consider utilization of tunable couplers between qubits and a coaxial cable even with frequency detunings between the cable mode and the qubits.

We would like to thank members of the Quantum Hardware project in Quantum Laboratory for their daily knowledge sharing and discussions.

All data that support the findings of this study are included within the article (and any supplementary files).

We show the effect of a qubit frequency detuning Δ on quantum state transfer via the standing-wave modes of a cable. Two qubits $\mathrm{Q^A}$ and $\mathrm{Q^B}$ are connected using a 1 m cable through tunable couplers. Time evolution simulations are performed for the H₀ in equation (2) with M = 43, $M_\mathrm{Min} = 41$, $M_\mathrm{Max} = 45$, $\omega_\mathrm{FSR}/(2\pi) = 0.1163$ GHz, and $g_\mathrm{A}/(2\pi) = g_\mathrm{B}/(2\pi) = 0.003$ GHz. After preparing the excited state $|1\rangle$ for $\mathrm{Q^A}$, the tunable couplers are turned on until the occupation probability of state $|1\rangle$, $P_{|1\rangle}$ of $\mathrm{Q^B}$ reaches its maximum. The time evolution of $P_{|1\rangle}$ for each qubit and the Mth cable mode is depicted in figure A1. The final $P_{|1\rangle}$ of $\mathrm{Q^B}$ corresponds to the transfer efficiency. The qubit frequency detunings are set to $\Delta_\mathrm{A}/(2\pi) = \Delta/(4\pi)$ and $\Delta_\mathrm{B}/(2\pi) = -\Delta/(4\pi)$ for the maximum transfer efficiency. We find that a small detuning of $\Delta/(2\pi) = 5$ MHz considerably reduces the efficiency of quantum state transfer.

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Herein, we discuss the effect of the cable mode limits $M_\mathrm{Max}$ and $M_\mathrm{Min}$ in equation (2) on the results. For 0.25-, 0.5- and 1.0 m cables, we investigate one parameter set for each length of the remote cross-resonance gates listed in table 2. Figure B1 shows the concurrence $C_{+}$ defined in equation (5) for each cable length when the number of cable modes above $\omega_\mathrm{A}$ and below $\omega_\mathrm{B}$ is changed from 1 to 4. The total pulse duration τ is optimized for the maximum $C_{+}$ for each number of cable modes. It can be seen that the $C_{+}$ values are nearly constant for the number of cable modes greater than 1. These results show that two cable modes above $\omega_\mathrm{A}$ and two below $\omega_\mathrm{B}$ are reasonable to use for all cable lengths in this study.

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The primary results for qubit frequency dependence of remote cross-resonance gates are presented in the main text. Here are some additional results. Figure C1(a) shows the results of $g/(2\pi) = 0.02$ GHz for a 0.25 m cable. We cannot find $3\times3$ cells with $\mathcal{C}_{+} \gt99.9\%$ as shown in figure 3. Figure C1(b) shows the results of $g/(2\pi) = 0.03$ GHz for a 0.5 m cable. We find six $2\times2$ cells with $\mathcal{C}_{+} \gt99.9\%$. Figure C1(c) presents the results of $g/(2\pi) = 0.02$ GHz for a 1 m cable. We find $2\times2$ cells with $\mathcal{C}_{+} \gt99\%$.

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We examine the time evolution of the initial states of $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$ for 200 ns without any driving pulse, i.e. using the H₀ in equation (2) for the leakage properties. The coupling is set to $g/(2\pi) = 0.03, 0.02$, and $0.01\,\mathrm{GHz}$ for 0.25-, 0.5-, and 1 m coaxial cables, respectively. The range $0.1\unicode{x2A7D}\delta_\mathrm{A}\unicode{x2A7D}0.9$ and $0.1\unicode{x2A7D}\delta_\mathrm{B}\unicode{x2A7D}0.9$ are examined for each case. We present the results for the initial states of $|01\rangle$, $|10\rangle$, and $|11\rangle$ with cable dissipation in figures D1–D3 because the occupation probabilities without dissipation for all initial qubit states and those with dissipation for the $|00\rangle$ initial qubit state remain 100.00% for 200 ns. For 0.25- and 0.5 m cables, the occupation probabilities for $|01\rangle$ and $|10\rangle$ in the area corresponding to the remote cross-resonance gates in table 2 are above 99.9%, while those for $|11\rangle$ are less than 99.9%. For a 1 m cable, the occupation probabilities are between 99.5% and 99.9% for all initial qubit states. These results mean that reducing the coupling leads to considerable fidelities for longer cable lengths.

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