Microwave-activated high-fidelity three-qubit gate scheme for fixed-frequency superconducting qubits

TL;DR

Addressing spectral crowding and ZZ crosstalk in fixed-frequency superconducting qubits, the paper presents a microwave-activated three-qubit gate that leverages a third-order nonlinearity to mediate a resonant $|001\rangle \leftrightarrow |110\rangle$ exchange under the large-detuning condition $|\Delta| \gg g$. A phase-compensation protocol and continuous-wave Stark control suppress both dynamic and static phase errors, enabling a high-fidelity gate with average fidelity exceeding $99.9\%$ in simulations and robust performance under decoherence. The analysis shows long-range ZZ coupling is effectively suppressed in the large-detuning regime, and the gate integrates with existing all-microwave two-qubit gate architectures, enabling direct constructions like iFredkin. Together, these results offer a hardware-efficient path to scalable multi-qubit operations on fixed-frequency superconducting platforms in the NISQ era.

Abstract

Scalable superconducting quantum processors require balancing critical constraints in coherence, control complexity, and spectral crowding. Fixed-frequency architectures suppress flux noise and simplify control via all-microwave operations but remain limited by residual ZZ crosstalk. Here we propose a microwave-activated three-qubit gate protocol for fixed-frequency transmon qubits in the large-detuning regime ($|Δ| \gg g$), leveraging the third-order nonlinear interaction to coherently exchange $\ket{001} \leftrightarrow \ket{110}$ states. By incorporating a phase-compensated optimization protocol, numerical simulations demonstrate a high average gate fidelity exceeding $99.9\%$. Systematic error analysis identifies static long-range ZZ coupling as the dominant error source in multi-qubit systems, which can be suppressed via operations in the large-detuning regime ($\sim 1$ GHz). The protocol maintains process fidelities exceeding $98\%$ under decoherence, while demonstrating intrinsic robustness to fabrication-induced parameter variations and compatibility with existing all-microwave two-qubit gate architectures. This hardware-efficient strategy advances scalable quantum computing systems by improving coherence properties, reducing spectral congestion, and expanding the experimental toolkit for error-resilient quantum operations in the noisy intermediate-scale quantum era.

Figures (11)

• Figure 1: Circuit diagram and microwave-activated scheme. (a) Circuit schematic of three fixed-frequency transmon qubits ($q_{ A}$, $q_{ B}$, $q_{ C}$) with nearest-neighbor capacitive couplings. The microwave-activated transition is driven by an external microwave field applied to $q_{ B}$, characterized by amplitude $\Omega_d$ and frequency $\omega_d$ and phase $\phi_{d}$. (b) Energy-level diagram of the coupled system with a maximum total excitation number below four, where $\left|{n_{ A} n_{ B} n_{ C}}\right\rangle$ denotes the collective state with $n_{ A,B,C}$ excitations in each qubit. Within the computational subspace spanned by $\{\left|{000}\right\rangle, \left|{001}\right\rangle, \left|{010}\right\rangle, \left|{011}\right\rangle, \left|{100}\right\rangle, \left|{101}\right\rangle, \left|{110}\right\rangle, \left|{111}\right\rangle\}$. The microwave-activated transition mediates resonant state exchange between $\left|{001}\right\rangle\leftrightarrow\left|{110}\right\rangle$ (blue branch) and $\left|{100}\right\rangle\leftrightarrow\left|{011}\right\rangle$ (red branch), as depicted by the corresponding blue and red double arrows in the energy-level diagram.
• Figure 2: Pulse shape profile and microwave-activated state-transfer dynamics. (a) Numerically simulated pulse envelope (blue trace) featuring a flat-top profile with Gaussian edges ($\sigma=10$ ns, total edge duration $4\sigma$). (b) microwave-activated resonant state exchange between $\left|{001}\right\rangle$ and $\left|{110}\right\rangle$, achieving $>99.9\%$ population transfer fidelity from the initial $\left|{001}\right\rangle$ state. Notably, all non-target computational states remain unpopulated ($<0.1\%$), highlighting intrinsic state selectivity of the protocol. (c) Population of the $\left|{110}\right\rangle$ state for the microwave-activated transition as a function of drive frequency $\omega_d$ and duration time, obtained by a numerical simulation of the transmon Hamiltonian (Eq. \ref{['eq:eq003']}) when the system is initialized in $\left|{001}\right\rangle$ state. (d) Extracted oscillation frequencies from (c) align with the theoretical value of approximately $1.25$ MHz. (e) Broadband spectroscopy. The spectrum reveals spectral features, including a primary blue brand resonance (blue) , a red branch sideband (red) , sidebands around $\omega_{ B}$ and distant features.
• Figure 3: Phase correction protocol and truth table. (a) Schematic of the phase correction protocol, integrating the microwave-activated unitary $\widetilde{U}(\Omega_d, \omega_d, \phi_{d})$ and the correction unitary $U_{\text{phase}}$. In $\widetilde{U}(\Omega_d, \omega_d, \phi_{d})$, $\phi_{d}$ represents the dynamically optimized pulse phase designed to suppress transient phase errors. The correction unitary $U_{\text{phase}}$, applied after the microwave-activated operation, consists of two CPhase terms ($\theta_{ AB}$, $\theta_{ BC}$) for qubit pairs $AB$ and $BC$, as well as three single-qubit virtual $Z$-rotations ($\varphi_{ A}, \varphi_{ B}, \varphi_{ C}$). By optimizing these six parameters ($\theta_{ AB}, \theta_{ BC}, \varphi_{ A}, \varphi_{ B}, \varphi_{ C}, \phi_{d}$), the protocol achieves high-fidelity three-qubit gates with minimized static and dynamic phase errors. (b) Truth table at $\Omega_d/2\pi \approx 90$ MHz, comparing the ideal gate unitary (left) with numerical simulation results (right). The protocol achieves an average fidelity of 99.9% and demonstrates near-perfect population transfer.
• Figure 4: Gate error versus drive amplitude and qubit detuning. (a) Average gate error $1 - F$ as a function of drive amplitude $\Omega_d$ for varying NN detunings $\Delta_{ AB}/2\pi = \{0.677, 0.777, 0.877, 0.977\}\,$GHz (blue, green, red, purple curves). Increasing $\Delta_{ AB}$ reduces errors from $\sim 10^{-2}$ to $\sim 10^{-4}$, demonstrating the advantage of large detuning. (b) Gate duration decreases with increasing drive amplitude $\omega_d$ and decreasing $\Delta_{ AB}$, reflecting the inverse proportionality between oscillation frequency $\nu$ and $\Delta_{ AB}$ (Eq. \ref{['eq:eq004']}). (c) Static ZZ couplings $\xi_{ AB}$ and $\xi_{ BC}$ increase with reduced $\omega_{ B}$, while the NNN coupling $\xi_{ AC}$ remains suppressed below $10$ kHz due to the large detuning. Dashed lines correspond to fourth-order perturbative calculations of static ZZ couplings. (d) Transition frequency $\omega_{\mathrm{s}}$ as a function of $\Omega_d$, showing systematic shifts under varying drive amplitudes and detunings. (e) Numerical results for $\nu$ (points) compared to perturbative predictions. Numerical results (points) deviate from perturbative predictions (solid line, Eq. \ref{['eq:eq004']}) at high $\Omega_d$, particularly for small detunings, highlighting the need for beyond-perturbative modeling in strongly driven regimes. (f) $|\xi_{ AC}|$ versus $\omega_{ B}$ and $\omega_{ A}$ (fixed $\omega_{ C}/2\pi = 5.507\,$GHz). Black solid-line traces highlight four selected NN detuning configurations, corresponding to $\Delta_{ AB}/2\pi = \{0.677, 0.777, 0.877, 0.977\}\,$GHz, concentrated within near-zero $\xi_{ AC}$ regions, revealing strong long-range ZZ suppression in the large-detuning regime.
• Figure 5: Gate error dependence on qubit coupling parameters. (a) Gate error (cyan curve, left axis) and gate duration (purple curve, right axis) as functions of the NN coupling strengths $g_{ AB}/2\pi = g_{ BC}/2\pi$, under a fixed microwave drive amplitude $\Omega_d = 100$ MHz. The gate error remains below $10^{-3}$ across a broad NN coupling range of 40–60 MHz, demonstrating robust performance. Gate durations exhibit an inverse proportionality to coupling strength, while gate durations inversely scale with coupling strength, requiring longer operation times for weaker couplings. (b) Static ZZ couplings ($\xi_{ AB}, \xi_{ BC}, \xi_{ AC}$) as functions of $g_{ AB}/2\pi = g_{ BC}/2\pi$, with dashed curves showing perturbative predictions. (c) Gate error (green curve) and duration (purple curve) versus NNN coupling $g_{ AC}/2\pi$ under fixed $g_{ AB}/2\pi = g_{ BC}/2\pi = 40$ MHz and $\Omega_d = 100$ MHz. Error suppression below $10^{-3}$ persists for $g_{ AC}/2\pi \leq 4$ MHz but degrades with stronger couplings. (d) Static ZZ couplings ($\xi_{ AB}, \xi_{ BC}, \xi_{ AC}$) as functions of $g_{ AC}/2\pi = g_{ BC}/2\pi$, with dashed curves showing perturbative predictions. Growth of NNN static ZZ coupling $\xi_{ AC}$ as a function of $g_{ AC}/2\pi$, correlating with the fidelity degradation observed in (c).