Tunable Nonlocal ZZ Interaction for Remote Controlled-Z Gates Between Distributed Fixed-Frequency Qubits

Abstract

Fault-tolerant quantum computing requires large-scale superconducting processors, yet monolithic architectures face increasing constraints from wiring density, crosstalk, and fabrication yield. Modular superconducting platforms offer a scalable alternative, but achieving high-fidelity entangling gates between distant modules remains a central challenge, particularly for highly coherent fixed-frequency qubits. Here, we propose a distributed hardware architecture designed to overcome this bottleneck by employing a pair of double-transmon couplers (DTCs). By synchronously controlling the two DTCs stationed at opposite ends of a macroscopic cable, our scheme strongly suppresses residual static inter-module coupling while enabling on-demand activation of a non-local cross-Kerr interaction with an on/off ratio exceeding $10^6$. Through comprehensive system-level numerical simulations incorporating realistic hardware parameters, we demonstrate that this mechanism can realize a remote controlled-Z (CZ) gate with a fidelity over 99.99\% between fixed-frequency transmons housed in separate packages interconnected by a 25 cm coaxial cable. These results establish a highly viable, hardware-efficient route toward high-performance distributed superconducting processors.

Figures (4)

• Figure 1: Schematic of the distributed architecture for coupling remote fixed-frequency transmon qubits. (a) Schematic of the physical implementation. The transmon qubits are housed in separate packages and are interconnected via a 25-cm coaxial cable. On-chip double-transmon couplers (DTCs) and interface structures (labeled 'To cable') facilitate this inter-module connectivity. (b) Equivalent circuit diagram illustrating two spatially separated fixed-frequency transmon qubits ($\mathrm{Q}_1$, green and $\mathrm{Q}_2$, purple) coupled to a multimode coaxial cable via their respective double-transmon couplers ($\mathrm{DTC}_1$ and $\mathrm{DTC}_2$, blue). Here, $E_{Ji}$ and $C_{i}$ denote the Josephson energy and mode capacitance of transmon $i$, respectively. $E_{J9}\mathrm{(}E_{J10})$ represents the energy of the coupling Josephson element. $\Phi_{\mathrm{ext}}^{(1)}(\Phi_{\mathrm{ext}}^{(2)})$ is the external magnetic flux threading the SQUID loop formed by the three Josephson junctions, and $\varphi_{i}$ indicates the superconducting phase at each node. Note that the cable is modeled by two harmonic modes chosen to lie close to the qubit frequencies. The grey brackets labeled ${L-system}$ and ${R-system}$ represent the local circuit modules within separate packages. Tuning the parameters of the couplers enables the dynamic activation or suppression of the nonlocal interaction.
• Figure 2: Energy spectrum and effective $ZZ$ coupling strength of a subsystem mediated by a DTC. (a) Energy spectrum of a subsystem where qubit $Q_{1}$ is coupled, via a DTC, to two coaxial-cable modes ($m=10\mathrm{~and~}11$), plotted as a function of the applied flux $\Phi_{\mathrm{ext}}^{(1)}$. The system eigenstates are labeled as $|Q_1,Cb_1,Cb_2,Cp_{1A},Cp_{1B}\rangle$ with corresponding eigenenergies $E_{Q_1,Cb_1,Cb_2,Cp_{1A},Cp_{1B}}(\Phi_{\mathrm{ext}})$. Here $Cb_{1}$ and $Cb_{2}$ denote the occupation numbers of the cable modes $m=10$ and $m=11$, respectively, while $Cp_{1A}$ and while $Cp_{1B}$ denote the occupation numbers of the two internal coupler modes (modes A and B) of the DTC. (b) Effective $ZZ$ coupling strengths between $Q_{1}$ and the cable modes $m=10$ (associated with $Cb_1$, solid blue line) and $m=11$ (associated with $Cb_2$, solid red line) as functions of $\Phi_\mathrm{ext}^{(1)}$.
• Figure 3: System spectrum and DTC-mediated effective nonlocal $ZZ$ interaction. (a) Energy spectrum of the full eight-mode system as a function of $\Phi_{\mathrm{ext}}^{(2)}$, with $\Phi_\mathrm{{ext}}^{(1)}=0.5$. The inset provides a magnified view of the avoided crossing region, where the red and green solid curves trace the energy levels corresponding to the $\left|11,0\dots0\right\rangle$ and $\left|02,0\dots0\right\rangle$ states, respectively. (b) Two-dimensional landscape of the effective $ZZ$ coupling strength (plotted logarithmically as $\log_{10}|ZZ|$) under the joint flux modulation of both couplers. The blue and red regions correspond to the idle and fully activated interaction regimes, respectively. (c) Cross-sectional profile of the $ZZ$ coupling strength varying with $\Phi_{\mathrm{ext}}^{(2)}$, demonstrating the independent tunability of the interaction channel even when DTC$_1$ is fixed at its operation point $\Phi_{\mathrm{ext}}^{(1)}=0.5$.
• Figure 4: Dynamics of the remote CZ gate. (a) Simultaneous time-dependent external flux pulses, $\Phi_\mathrm{ext}^{(1)}$ (blue) and $\Phi_\mathrm{ext}^{(2)}$ (orange), applied to $\mathrm{DTC}_1$ and $\mathrm{DTC}_2$. (b) Time evolution of the state population deviations and leakage probabilities on a logarithmic scale. The curves $1-P_{01}$ (blue), $1-P_{10}$ (orange), and $1-P_{11}$ (green) track the swap error and leakage error of the respective computational basis states. The solid red and purple lines denote the transient leakage probabilities from the $\left|11\right\rangle$ state to the non-computational $\left|20\right\rangle$ and $\left|02\right\rangle$ states, respectively. The dynamics indicate that the remote CZ gate is dominated by the interaction between $\left|11\right\rangle$ and $\left|02\right\rangle$, which is responsible for the conditional phase accumulation. Despite significant state hybridization during the peak of the pulse, the populations successfully return to the computational subspace at the end of the gate, yielding a high gate fidelity.