Controlled-Z Gates via Floquet Coupling to Two Auxiliary States in Transmon Circuits

Introduction

Superconducting circuits provide a versatile platform for gate-based quantum information processing, combining engineered qubit interactions with programmable microwave and flux control [1, 2, 3]. Progress in device fabrication, coherent control, and readout has enabled experiments spanning quantum algorithms [4, 5, 6], digital quantum simulation [7, 8], and quantum error correction [9, 10, 11, 12, 13, 14]. Large-scale random-circuit sampling has further demonstrated the capabilities of programmable superconducting processors [15, 16, 17, 18, 19]. These advances rely on programmable sequences of quantum gates. Entangling operations play a central role by establishing correlations between qubits that cannot be generated by single-qubit control alone [20]. Their physical implementation connects the available circuit interactions to the logical operations required by a quantum circuit [21, 22]. Exploring different mechanisms for generating and controlling these interactions, and understanding the conditions under which they realize an entangling gate, therefore remains an important direction in superconducting quantum control.

The controlled-Z (CZ) gate, together with single-qubit rotations, forms a universal gate set [20] and is widely used in quantum algorithms and error-correction circuits [12, 23]. A common implementation in transmons exploits the coupling of $\lvert11\rangle$ to an auxiliary state outside the computational subspace, typically $\lvert02\rangle$ or $\lvert20\rangle$, to accumulate a conditional phase through adiabatic evolution or a nonadiabatic population return [23, 24, 25, 26]. Within this approach, pulse shaping suppresses unwanted transitions and endpoint leakage [22, 24, 27, 28], while tunable couplers provide control over the gate coupling and residual interactions [22, 29, 30, 31, 32] [33]. Other implementations use resonator-mediated interactions to generate conditional phases through microwave driving [34] or continuous-variable geometric evolution [35]. Parametric modulation further expands the available control by selectively activating transitions [36, 37, 38] and enables geometric gate operations through phase control [39, 40], as well as superadiabatic two-qubit gates [41]. Holonomic entanglement generation using higher transmon levels has also been demonstrated in an effective three-level manifold [42]. Together, these developments motivate exploring how selective control of noncomputational levels can be extended to the coherent participation of multiple auxiliary states in a CZ gate.

In this work, we propose a Floquet-engineered approach to CZ gates that coherently couples $\lvert11\rangle$ to both auxiliary states $\lvert02\rangle$ and $\lvert20\rangle$ in transmon circuits. Periodic longitudinal modulation activates the two auxiliary transitions while suppressing the leading exchange between $|01\rangle$ and $|10\rangle$. The resulting three-state manifold supports two complementary CZ gate protocols: a nonadiabatic population cycle through a bright auxiliary superposition and adiabatic evolution along a zero-energy dark state. Circuit-level simulations yield coherent average gate fidelities above 99.99% for both protocols. Control of the modulation phase further enables conventional and unconventional geometric bright-state CZ gates. The unconventional protocol improves tolerance to residual $ZZ$ coupling and qubit-frequency deviations. Finally, we examine the effects of neighboring

spectator qubits on the original bright- and dark-state protocols, motivated by the substantial qubit-frequency excursions required during operation. Both retain high fidelities in the configurations considered, with a residual dependence on spectator frequencies and logical states. Together, these results establish a unified framework for transmon CZ gates based on selective Floquet coupling to two auxiliary states.

Floquet coupling to two auxiliary states

Circuit model and effective description

We consider two transmon qubits, labeled $Q_1$ and $Q_2$, coupled through a controllable exchange interaction mediated by a tunable coupler, labeled $C$ [29]. The system is described by the following Hamiltonian ($\hbar=1$):

where $\hat{a}_k$ is the annihilation operator of mode $k\in\{1,2,c\}$, where the indices $1$, $2$, and $c$ correspond to $Q_1$, $Q_2$, and $C$, respectively, and $\hat{n}_k=\hat{a}_k^\dagger\hat{a}_k$. $\omega_k$ and $\alpha_k$ denote the mode frequencies and anharmonicities, respectively, while $g_{ij}$ characterizes the capacitive coupling between modes $i$ and $j$, including both rotating and counter-rotating terms. The mode frequencies can be dynamically tuned through external flux biases, $\omega_k\to\omega_k(t)$, as illustrated in Figure 1a.

By eliminating the coupler in the dispersive regime, we obtain an effective two-transmon Hamiltonian

where $\omega_j(t)$ and $\alpha_j$ now denote the renormalized transmon parameters, and $g(t)$ includes both direct and coupler-mediated exchange interactions. The relation between these effective parameters and the underlying tunable-coupler circuit is detailed in Appendix A.

To use both auxiliary states for a CZ gate, we seek to activate $|11\rangle\leftrightarrow|20\rangle$ and $|11\rangle\leftrightarrow|02\rangle$ simultaneously while suppressing $|01\rangle\leftrightarrow|10\rangle$ exchange. Because all three transitions arise from the same exchange term in Eq. (2), controlling $g(t)$ alone cannot provide this selectivity. Motivated by the Floquet engineering of transmon interactions demonstrated by Wan et al. [43], we introduce longitudinal modulation of the qubit-frequency difference, $\omega_1(t)-\omega_2(t)=\Delta+A\cos\mu t$, where $\Delta$ is the mean detuning, $A$ is the modulation amplitude, and $\mu$ is the angular frequency [44, 45]. The opposite frequencycuntegn excursions of the two qubits modulate the level energies without directly driving transitions between transmon levels. In the rotating frame, this longitudinal drive instead gives the exchange interaction a periodic phase, $g\exp[i\varepsilon\sin(\mu t)]$, with $\varepsilon=A/\mu$. Its Fourier components provide sidebands weighted by $J_m(\varepsilon)$, following the frequency-modulation mechanism used to generate first-order sidebands in circuit QED [46, 47]. Choosing $\mu$ near the magnitudes of the anharmonicities brings the first sidebands of both auxiliary transitions near resonance Figure 1(b), while logical exchange is governed by the zeroth-order component. This difference in sideband order allows the desired and unwanted channels to be controlled through distinct Bessel weights.

Retaining the zeroth-order logical exchange and the first auxiliary sidebands gives the leading-order Floquet Hamiltonian

where $\varepsilon=A/\mu$, $J_m$ is the Bessel function of the first kind, $U=\mu+(\alpha_1+\alpha_2)/2$, and $\delta=\Delta-(\alpha_2-\alpha_1)/2$. The rotating-frame transformations and averaging conditions are provided in Appendix B. At the first Bessel zero, $\varepsilon_0\simeq2.40483$, $J_0(\varepsilon_0)=0$ and $J_1(\varepsilon_0)\simeq0.51915$, suppressing the leading logical exchange while retaining both auxiliary couplings Figure 1(c).

Hamiltonian in the bright–dark basis

At $\varepsilon=\varepsilon_0$, the two auxiliary states in Eq. (3) can be combined into a bright state $|B\rangle$ and a dark state $|D\rangle$ [48],

The opposite signs of the two auxiliary couplings make their amplitudes add for $|B\rangle$ and cancel for $|D\rangle$, so only the bright combination couples directly to $|11\rangle$. With $G=2gJ_1(\varepsilon_0)$, the Hamiltonian in the ordered basis $\{|11\rangle,|B\rangle,|D\rangle\}$ is

Equation (5) provides a common description of the two CZ gate mechanisms. At $U=\delta=0$, $|D\rangle$ is decoupled and the dynamics reduce to a resonant $|11\rangle\leftrightarrow|B\rangle$ transition, enabling a nonadiabatic population-return cycle. A finite $\delta$ instead couples the bright and dark auxiliary combinations. For $U=0$, the resulting Hamiltonian supports a zero-energy eigenstate formed from $|11\rangle$ and $|D\rangle$; adiabatically varying $G$ and $\delta$ provides the second CZ route. We develop the pulse conditions and gate dynamics for these two protocols below.

CZ protocol by nonadiabatic bright-state evolution

Resonant evolution and conditional phase

We first consider the resonant bright-state protocol by setting $U=\delta=0$ in Eq. (5), corresponding to the resonance conditions $\mu=-(\alpha_1+\alpha_2)/2$ and $\Delta=(\alpha_2-\alpha_1)/2$. The dark state decouples from the resonantly coupled $\{|11\rangle,|B\rangle\}$ subspace, while the dark state $|D\rangle$ remains decoupled. Starting from $|11\rangle$, the time evolution is given by

The condition $\Theta(T)=\pi$ completes a coherent population-return cycle, yielding $|11\rangle\rightarrow-|11\rangle$ with no residual auxiliary-state occupation in the ideal model. For a constant coupling, the ideal return time is $T=\pi/|G|$, with $G=2gJ_1(\varepsilon_0)$. At the leading-order Bessel-zero operating point, exchange between $|10\rangle$ and $|01\rangle$ is suppressed. Together with the phase of $\pi$ acquired by $|11\rangle$ during the bright-state cycle, this yields a CZ gate up to local phases, which can be compensated by virtual-$Z$ rotations [22, 49].

Numerical implementation and results

To assess the bright-state protocol beyond the effective two-state approximation, we simulate the complete gate evolution under the full Hamiltonian in Eq. (1), including the finite-duration ramps between the idle and working points. Each transmon and the coupler is truncated to its four lowest levels, yielding a Hilbert-space dimension of $4^3=64$. The controls are parameterized by the mode-frequency trajectories, which can be mapped to flux pulses through the device-specific flux–frequency relations. We use a parameter set with idle frequencies $\omega_{1,\mathrm{idle}}/2\pi=4.55\ \mathrm{GHz}$, $\omega_{2,\mathrm{idle}}/2\pi=4.45\ \mathrm{GHz}$, and $\omega_{c,\mathrm{idle}}/2\pi=7.65\ \mathrm{GHz}$. The anharmonicities are $\alpha_1/2\pi=-250\ \mathrm{MHz}$, $\alpha_2/2\pi=-240\ \mathrm{MHz}$, and $\alpha_c/2\pi=-250\ \mathrm{MHz}$. The coupling strengths are $g_{12}/2\pi=4\ \mathrm{MHz}$ and $g_{1c}/2\pi=g_{2c}/2\pi=100\ \mathrm{MHz}$.

The bare-qubit detuning $\Delta_q(t)/2\pi$ is ramped from $100\ \mathrm{MHz}$ at idle to $5\ \mathrm{MHz}$ at the working point and back using error-function ramps [50], and we set the parameter $\sigma_q$ as the characteristic time scale. During the interaction window, the two qubit frequencies are sinusoidally modulated under a smooth envelope. Meanwhile, the coupler frequency follows a compressed-Slepian trajectory from idle to its working point and back, with edge shaping inspired by Ref. [24]. These frequency waveforms are shown in Figure 2(a), the details are given in Appendix § 2.

To examine the bright-state dynamics in the full physical system, we focus on the evolution of the qubit $\{|11\rangle,|B\rangle\}$ subspace introduced in the effective description, as illustrated by the Bloch-sphere representation in Figure 2(b). The trajectory illustrates a nearly complete population-transfer-and-return cycle between $|11\rangle$ and the Floquet bright state $|B\rangle$, consistent with the resonant two-state dynamics. The Bloch trajectory is plotted in a rotating frame, with residual micromotion retained. The frame transformations and projection conventions are specified in Appendix B. Figure 2(c) shows the evolution of the dressed-state populations. With the system initialized in the idle dressed state $|\widetilde{11}\rangle$, predominantly corresponding to the two-qubit configuration $|11\rangle$, population is transferred into both auxiliary states $|\widetilde{20}\rangle$ and $|\widetilde{02}\rangle$ before returning to the initial state. The simultaneous participation of the two auxiliary channels supports the bright-state mechanism predicted by the effective Floquet Hamiltonian. Together, these results demonstrate that the bright-state population-return dynamics remain dominant in the full QCQ evolution.

We next verify that this mechanism realizes a CZ gate and evaluate its performance on the entire computational subspace. Specifically, we quantify the agreement between the numerical operation $U$ and the ideal CZ operation $U_{\mathrm{CZ}}$ using the average gate fidelity[51]:

where $d=2^n$ for an $n$-qubit system. By optimizing the gate parameters, we evaluate the leakage and swap errors, defined as $\epsilon_{\mathrm{leak}}=1-P_{11}$ and $\epsilon_{\mathrm{swap}}=1-P_{ij}$ ($ij=10,01$), respectively, where $P_{ij}$ denotes the return probability of the corresponding initial dressed state. As shown in Figure 2(d), the large transient population transfer from $|\widetilde{11}\rangle$ is followed by a nearly complete return, while the single-excitation states maintain high return probabilities. Beyond the low leakage and swap error, Figure 2(e) shows that the conditional phase approaches $\pi$ at the end of the gate. These results demonstrate a CZ gate with a duration of 72 ns and an average fidelity exceeding 99.99%.

Geometric control of the bright-state CZ gate

The resonant protocol above realizes a CZ gate through a population-return cycle with a fixed coupling phase. To explore whether additional control of the evolution path can improve robustness, we now allow the phase of the longitudinal modulation to vary, extending the same bright-state interaction to geometric CZ protocols [39, 40, 52]. Replacing $\cos\mu t$ by $\cos[\mu t+\varphi(t)]$ preserves the relative sign of the two auxiliary couplings while changing their common phase. At $U=\delta=0$, the leading-order Hamiltonian becomes

where $\Omega_B(t)=4g(t)\mathcal{J}_1(\epsilon_0)$. Thus, $\varphi(t)$ controls the rotation axis in the $\{|11\rangle,|B\rangle\}$ subspace. For comparison, the single-channel protocol replaces $|B\rangle$ by $|02\rangle$ and uses $\Omega_{02}(t)=2\sqrt{2}g(t)\mathcal{J}_1(\epsilon_0)$. Both couplings retain the Bessel renormalization.

A conventional geometric gate uses two resonant segments, each with pulse area $\int\Omega(t)\,\mathrm{d}t=\pi$. For equal coupling phases, the state completes the closed path $S_1\rightarrow S_2$ in Figure 3(a,b), returning to $|11\rangle$ with geometric phase $\pi$ and zero dynamical phase. We denote the resulting gates by $U_{02}^{\mathrm{G}}(\pi)$ and $U_B^{\mathrm{G}}(\pi)$. The unconventional protocol instead follows the three-segment path

$P_1\rightarrow P_2\rightarrow P_3\rightarrow P_1$ in Figure 3(c,d) [53]. Here the dynamical and geometric phases satisfy $\gamma_d=\eta\gamma_g$, following the unconventional geometric phase relation [54], so that the total phase $\gamma=(1+\eta)\gamma_g=\pi$ yields $U_{02}^{\mathrm{UG}}(\pi)$ or $U_B^{\mathrm{UG}}(\pi)$. The rotating-frame construction and segment conditions are given in Appendix D 1.

We compare these four protocols in the six-dimensional effective Hilbert space at the same single-channel coupling $\Omega_0=\Omega_{02}$, retaining either one or both auxiliary transitions. For the unconventional gates, we use $\chi=0.43\pi$ and $\gamma=\pi$ following Ref. [53]. Residual $ZZ$ coupling is modeled by

Figure 3(e) shows that the bright-state gates are more robust against this perturbation than their single-channel counterparts, and that unconventional control further improves the fidelity for both configurations.

For qubit-frequency deviations, the comparison depends on the control protocol and detuning direction. Along $\delta_2=0$ [Figure 3(f)], the conventional bright-state gate is slightly less robust than the single-channel gate: differential detuning couples the bright and dark states and opens an additional leakage channel. Unconventional control suppresses the final dark-state population and improves the fidelity, with the bright-state protocol outperforming its single-channel counterpart in the scans shown in Figure 3(f–h). The conventional gates’ two-dimensional comparison and the residual dark-state populations are discussed in Appendix D 2. These results identify phase control as a means of improving bright-state CZ robustness within the effective model. The static-error scans considered here do not characterize the full response to time-dependent flux noise. That response also depends on the modulation operating point and noise spectrum, as established in studies of ac sweet spots and parametric-gate noise sensitivity [55, 56, 57].

CZ PROTOCOL BY ADIABATIC DARK-STATE EVOLUTION

Dark-state path and conditional phase

In contrast to the nonadiabatic bright-state protocol, which relies on a resonant cycle, the same three-state manifold spanned by $|11\rangle$, $|20\rangle$, and $|02\rangle$ can also be configured as an effective $\Lambda$ system in the $|11\rangle$, $|B\rangle$, $|D\rangle$ basis, enabling STIRAP-like adiabatic passage analogous to that in atomic and trapped-ion systems [58, 59]. Building on this analogy and inspired by studies of STIRAP geometric control [60] and doublon-mediated geometric gates in optical lattices [61], We now develop an adiabatic dark-state protocol for realizing a CZ gate, exploiting the detuning $\delta(t)$ and the coupling $G(t)$ to control the instantaneous dark eigenstate. Setting $U=0$, the effective Hamiltonian in $\{|11\rangle, |B\rangle, |D\rangle\}$ reads:

the Hamiltonian admits a zero-energy dark eigenstate for arbitrary $G(t)$ and $\delta(t)$, provided they do not vanish simultaneously:

Here the subscript ad distinguishes the instantaneous dark eigenstate from the auxiliary dark state $|D\rangle$. This eigenstate has no $|B\rangle$ component due to destructive interference between the two coupling pathways. Introducing the mixing angle $\theta(t)$ through $\tan\theta(t)=G(t)/\delta(t)$, with a continuous choice of branch, the dark state can be written as $|D_{\mathrm{ad}}(t)\rangle=\cos\theta(t)|11\rangle-\sin\theta(t)|D\rangle$. By varying $\theta(t)$ from $0$ to $\pi$ and satisfying the adiabatic condition $|\dot{\theta}(t)|\ll\Omega(t)$, where $\Omega(t)=\sqrt{G^2(t)+\delta^2(t)}$ is the instantaneous energy gap, the system follows the dark eigenstate and evolves from $|11\rangle$ to $-|11\rangle$. In the ideal effective frame, the dark-state eigenenergy remains zero throughout the evolution, so no dynamical phase is accumulated. The sign change upon completing the adiabatic path gives rise to a geometric phase of $\pi$. Provided that the $|01\rangle\leftrightarrow|10\rangle$ exchange remains suppressed during the Floquet modulation, the geometric phase acquired by $|11\rangle$ produces the desired conditional phase of $\pi$ in the effective model. After compensating the remaining single-qubit phases by local $Z$ rotations, the resulting operation is equivalent to $U_{\mathrm{CZ}}=\operatorname{diag}(1,1,1,-1)$. To assess the feasibility of this protocol beyond the ideal effective description, we next numerically examine the dynamics and conditional phase in the full physical model.

Numerical implementation and results

Following the approach adopted for the bright-state protocol in Sec. III B, we implement the adiabatic dark-state protocol with the same circuit parameters. The corresponding waveforms are shown in Figure 4(a). During the interaction window, the detuning control $\delta(t)$ follows a cosine profile whose argument varies smoothly from $0$ to $\pi$ according to a beta cumulative distribution function, taking the detuning between equal and opposite endpoint values. The auxiliary coupling $G(t)$ is controlled through the coupler-frequency waveform, which is determined using the calibrated relation between the effective qubit exchange coupling and the coupler frequency. Further details are given in Appendix § 2 2.

The population dynamics shown in Figure 4(b) can be understood from the dark-state composition in Eq. (11). In the ideal description, the dark state is dominated by $|11\rangle$

when $|G(t)| \ll|\delta(t)|$, whereas near $\delta(t)=0$ its weight shifts to $|D\rangle=(|20\rangle+|02\rangle)/\sqrt{2}$. The comparable populations of the two auxiliary states near the middle of the numerical pulse are consistent with this structure, with their combined occupation reaching approximately 0.974. As the detuning changes sign and the coupling is reduced, the ideal dark-state path returns to $-|11\rangle$. The numerical evolution correspondingly shows a near-complete recovery of $|11\rangle$ with a small residual error, while keeping the final swap errors below $10^{-4}$ [Figure 4(c)]. This behavior is consistent with the Floquet modulation suppressing logical exchange while retaining the auxiliary couplings needed for the gate. The accompanying accumulation of a conditional phase close to $\pi$ [Figure 4(d)] completes the intended CZ action and the resulting gate achieves an average fidelity of 99.9941% with a total duration of 277 ns.

Spectator effects on bright- and dark-state CZ gates

The CZ-gate implementations in Secs. III and IV involve qubit-frequency modulation with amplitudes of order $10^2$ MHz. This frequency-tuning requirement presents a practical limitation in multiqubit circuits, where the active qubits may approach transitions of neighboring qubits or parasitic two-level systems and experience unwanted interactions. To assess spectator-induced errors in this setting, we examine the two calibrated gates in a four-qubit square lattice with tunable couplers [50], as illustrated in Figure 5(a).

The CZ gate acts on the active qubits $Q_1,Q_2$, while the spectator qubits $S_1,S_2$ and the additional couplers $C_2$–$C_4$ remain at their idle points. Each protocol retains its previously calibrated active parameters and control waveforms. For each spectator-frequency configuration, we evaluate all four spectator logical states. The endpoint leakage $L$ denotes the population outside the computational subspace of $Q_1,Q_2$, averaged over the four active logical inputs.

When both spectators are at 3.0 GHz, the gate error and leakage show little variation with the spectator state [Figure 5(b,Figure 5c)]. The 72 ns bright-state gate maintains an average fidelity $F > 99.95\%$, while the 277 ns dark-state gate achieves $F > 99.9\%$. For both protocols, the leakage remains below $5 \times10^{-4}$ in every spectator state. Thus, the gate performance is only weakly dependent on the neighboring logical states in this configuration.

Raising the spectator frequencies to $(f_{S1}, f_{S2}) = (3.8, 3.9)$ GHz produces a more pronounced state dependence, particularly for the dark-state gate. The largest gate error and leakage occur for $\lvert11\rangle_S$ in both protocols. Nevertheless, the bright-state gate maintains $F > 99.95\%$ and $L < 5 \times10^{-4}$ for all spectator states, while the dark-state gate retains $F > 99.8\%$ and $L < 10^{-3}$. With the active control waveforms unchanged, this variation demonstrates that residual spectator interactions depend on the surrounding frequency configuration. The frequencies and logical states of neighboring qubits therefore remain relevant to gate calibration in multiqubit implementations.

Conclusion

We have presented a Floquet-engineered approach to CZ gates that uses both auxiliary states $\lvert20\rangle$ and $\lvert02\rangle$ in transmons with negative and unequal anharmonicities. Longitudinal modulation selectively activates their couplings to $\lvert11\rangle$ while suppressing the leading single-excitation exchange. The resulting bright–dark basis provides a common description of two complementary gate mechanisms: a resonant population-return cycle through the bright state and adiabatic evolution along a zero-energy dark eigenstate. Circuit-model simulations yield coherent average gate fidelities above 99.99% after local-$Z$ corrections, with durations of 72 ns and 277 ns for the bright- and dark-state protocols, respectively.

Control of the modulation phase also enables conventional and unconventional geometric implementations of the bright-state CZ gate. Effective-model comparisons show that the unconventional protocol improves tolerance to residual $ZZ$ coupling and suppresses detuning-induced population remaining in the dark state. The response to qubit-frequency errors nevertheless depends on the control path and detuning direction, so robustness is not guaranteed by the participation of both auxiliary transitions alone. These results identify the evolution path as an additional control resource within the same Floquet-engineered manifold.

With the active gate waveforms fixed, simulations including neighboring spectator qubits show that gate performance depends on both spectator frequencies and logical states. This dependence connects the isolated-pair protocols to the frequency-allocation and calibration requirements of multiqubit circuits. Together, our results establish coherent control of two auxiliary transitions as a framework for realizing CZ gates through distinct population-transfer and geometric evolution paths.

Acknowledgments

We appreciate the support from the Synergetic Extreme Condition User Facility (SECUF) in Huairou District, Beijing. This work was supported by the National Natural Science Foundation of China (Grants No. U25A6009, No. 92265207, No. 92565301, No. 92365301, No. T2121001, No. T2322030, No. 12122504, No. 12274142, No. 12475017, and No. 12604773), the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0301800), the Beijing Nova Program (Grant No. 20220484121), the Ministry of Science and Technology project (Grant No. 2025YFE0217600), and the China Postdoctoral Science Foundation (Grants No. GZC20252227, 2025M783453, and 2026T190913).

Circuit model and dispersive reduction

Here we detail the dispersive reduction connecting Eqs. (1) and (2). For convenience, we define $\Delta_{jc}=\omega_j-\omega_c$ and $\Sigma_{jc}=\omega_j+\omega_c$. In the dispersive regime $|g_{jc}|,|\alpha_\mu|\ll|\Delta_{jc}|$, we assume that the coupler remains in its ground state and apply the Schrieffer–Wolff transformation $\hat{H}=e^{\hat{S}}\hat{H}_{\mathrm{QCQ}}e^{-\hat{S}}$ with [29]

To second order in the qubit–coupler couplings, the effective Hamiltonian takes the static form of Eq. (2) after the rapidly rotating pair terms are neglected. The effective exchange coupling is

The local dispersive frequency shifts are absorbed into $\omega_j$, following the notation in the main text, while corrections to $\alpha_j$ are neglected at this order. The $1/\Sigma_{jc}$ terms account for virtual counter-rotating processes.

External fluxes tune the Josephson energies and hence the mode frequencies, $\omega_\mu\to\omega_\mu(t)$ [2, 62]. For approximately constant $g_{ij}$, the instantaneous exchange becomes

where $\Delta_{jc}(t)=\omega_j(t)-\omega_c(t)$ and $\Sigma_{jc}(t)=\omega_j(t)+\omega_c(t)$ are evaluated from the circuit frequencies in Eq. (1). The corresponding local frequency shifts are absorbed into $\omega_j(t)$ in Eq. (2). This instantaneous approximation requires dispersive separation throughout the pulse and negligible time-derivative corrections, with $|\dot{\Delta}_{jc}(t)|\ll|\Delta_{jc}(t)|^2$ for constant $g_{jc}$ [63].

Effective Floquet Hamiltonian and Bloch-vector representation

Derivation of the effective Hamiltonian

The exchange interaction in Eq. (2) drives both logical exchange and the auxiliary transitions, so varying its strength alone does not select the desired channels. For negative transmon anharmonicities, the bare auxiliary resonance conditions, $\Delta+\alpha_1=0$ and $-\Delta+\alpha_2=0$, cannot be satisfied simultaneously [50]. Longitudinal modulation addresses both constraints: its periodic phase on the exchange interaction generates Floquet sidebands, allowing the auxiliary transitions to be selected independently of the leading logical exchange.

In the working region, take the effective exchange $g$ in Eq. (2) to be locally constant and write the longitudinal modulation as

Here $\bar{\omega}$ is the mean effective transmon frequency. With $\hat{N}_q=\hat{n}_1+\hat{n}_2$ and $\hat{D}_q=\hat{n}_1-\hat{n}_2$, the common-frequency and longitudinal-drive rotations combine into

For $|\psi_1\rangle=\hat{V}_1^\dagger|\psi_q\rangle$, the transformed Hamiltonian is

The derivative term cancels the common-frequency and longitudinal-modulation terms. The Jacobi–Anger expansion gives

We next introduce the auxiliary-state rotation

With transformed Hamiltonian,

where the last term shifts both auxiliary-state energies by $\mu$. Averaging over one modulation period, $\langle F\rangle_\mu=(\mu/2\pi)\int_0^{2\pi/\mu}F(t)\,\mathrm{d}t$, yields

The auxiliary matrix elements select $m=1$ in Eq. (18), while their Hermitian conjugates select $m=-1$. Their opposite signs follow from $\mathcal{J}_1(-\varepsilon)=-\mathcal{J}_1(\varepsilon)$ in this frame convention.

Together with the diagonal energies $U\pm\delta$, these period-averaged terms give Eq. (3). The simultaneous-resonance condition $U=\delta=0$ requires

At $\varepsilon=\varepsilon_0$, the leading logical exchange vanishes. In the two-excitation manifold, transforming to the bright–dark basis $\{|11\rangle,|B\rangle,|D\rangle\}$ gives Eq. (5), with $G=2g\mathcal{J}_1(\varepsilon_0)$. This leading-order averaging requires the exchange, residual auxiliary detunings, and envelope variation rates to be small compared with $\mu$, with the discarded harmonics off resonance.

The opposite signs of the auxiliary matrix elements make their contributions add for $|B\rangle= (|20\rangle- |02\rangle)/\sqrt{2}$ and cancel for $|D\rangle= (|20\rangle+ |02\rangle)/\sqrt{2}$. At $\delta= 0$, $|D\rangle$ is decoupled; setting $U = 0$ also makes $|11\rangle\leftrightarrow|B\rangle$ resonant. A finite $\delta$ mixes the bright and dark auxiliary combinations, producing the chain $|11\rangle\leftrightarrow|B\rangle\leftrightarrow|D\rangle$. For $U = 0$, destructive interference yields a zero-energy eigenstate formed from $|11\rangle$ and $|D\rangle$, with no bright-state component. Varying $G$ and $\delta$ adiabatically therefore enables dark-state passage analogous to STIRAP [64, 65, 66, 67, 68].

Dressed-state basis and projection frame

The full-QCQ state $|\psi_{\mathrm{lab}}(t)\rangle$ is initialized in the idle dressed state $|11\rangle$. The reference states satisfy $\hat{H}_{\mathrm{idle}}|jk\rangle= E_{jk}^{\mathrm{idle}}|jk\rangle$, with $jk \in\{11, 20, 02\}$ and the coupler label suppressed. Figure 2(c) shows the laboratory-frame populations $P_{jk} = |\langle jk|\psi_{\mathrm{lab}}\rangle|^2$. In Figure 2(b), the label $|B\rangle$ denotes the fixed ideal antisymmetric projection state

To specify the projection frame, write the longitudinal drive in Eq. (28) as $\mathcal{A}w(t)\cos\chi(t)\hat{D}_{q}$, where $\hat{D}_{q} = \hat{n}_{1} - \hat{n}_{2}$ and $\chi(t) = \nu(t-r_q)$. Here $\nu$ is an angular frequency and $\mathcal{A}$ is the single-qubit amplitude, with $\mathcal{A} = 2A$ in Eq. (15). The full-pulse primitive and its canonical form are

For any plateau time $t_p$, choose $C = \eta(t_p) - (\mathcal{A}/\nu)\sin\chi(t_p)$. This single constant gives $\eta_{\mathrm{can}} = (\mathcal{A}/\nu)\sin\chi$ on the plateau while retaining the envelope integral on both edges.

Define the longitudinal rotation $\hat{W}(t) = e^{i\eta_{\mathrm{can}}(t)\hat{D}_{q}}$ and the fixed dressed auxiliary projector $\hat{P}_{a} = |\widetilde{20}\rangle\langle\widetilde{20}| + |\widetilde{02}\rangle\langle\widetilde{02}|$. The auxiliary rotation $\hat{Q}(t) = e^{-i\chi(t)\hat{P}_{a}}$ subtracts the prescribed carrier phase $\chi(t)$ from both auxiliary states, yielding a clearer Bloch trajectory. The full state used for projection is

Its Hamiltonian, including both frame-derivative terms, is

Here $\dot{\eta}_{\mathrm{can}} = \mathcal{A}w\cos\chi$ cancels the explicit longitudinal drive. The projector $\hat{P}_{a}$ uses idle dressed states, unlike $\hat{P}_{a}$ in Eq. (19). These rotations are fixed by the waveform and reference basis and are applied to the full state before projection.

Projected Bloch coordinates

Let $a = \langle\widetilde{11}|\psi_{P}\rangle$ and $b = \langle\widetilde{B}_{-}|\psi_{P}\rangle$, with retained weight $N = |a|^{2} + |b|^{2}$. In the ordered basis $\{|\widetilde{11}\rangle, |\widetilde{B}_{-}\rangle\}$, the unnormalized Bloch coordinates are

If $a_{\mathrm{dr}}$ and $b_{\mathrm{dr}}$ denote the projections before $\hat{Q}$, then $a = a_{\mathrm{dr}}$ and $b = e^{-i\chi}b_{\mathrm{dr}}$. Thus the auxiliary rotation changes the azimuth by the known carrier phase while preserving $Z$ and $N$. For this projected pure state, $\sqrt{X^{2} + Y^{2} + Z^{2}} = N$, the radius records the weight retained in the chosen two-state projection. As shown in Figure 2(b), this resulting trajectory is consistent with the resonant dynamics predicted by the effective bright-state model.

Control waveforms for the CZ controls

Bright-state protocol

This appendix specifies the control waveforms used to implement the bright-state and adiabatic dark-state CZ protocols. For each protocol, we describe the control trajectories, pulse envelopes, and timing conventions used in the numerical simulations. The bright-state CZ gate in Sec. IIIB, is illustrated in Figure 2(a). The complete pulse takes the system from the idle configuration to the working region and back, combining a qubit-detuning sweep, opposite-sign longitudinal modulation of the two qubits, and a shaped coupler trajectory. For a total pulse duration $T$, the three mode frequencies are parameterized as

Here $\overline{\omega}_{q}$ is the fixed mean qubit frequency, $\Delta_{q}(t)$ is the bare-qubit detuning before adding the modulation, and $\mathcal{A}$ and $\nu$ are the single-qubit modulation amplitude and angular frequency, respectively.

The detuning $\Delta_{q}(t)/(2\pi)$ is ramped from $100\,\mathrm{MHz}$ at idle to $5\,\mathrm{MHz}$ at the working point and back using error-function ramps [50], with characteristic time scale $\sigma_q$. The modulation envelope $w(t)$ is nonzero only within $r_q < t < T-r_q$, where $r_q = 4\sqrt{2}\sigma_q$. Its turn-on and turn-off edges are constructed from the quintic smoother-step function $S_{5}(x) = 10x^{3} - 15x^{4} + 6x^{5}$. The time $r_q$ also defines the phase origin of the modulation.

The coupler frequency is controlled independently through the normalized trajectory $s_{c}(t)$, which joins the idle and working frequencies with symmetric edges and a central plateau. We use a compressed-Slepian parameterization with finite-cosine edges inspired by Ref. [24]:

The parameter $r$ sets the duration $rT/2$ of each coupler edge, while $c_{2}$ and $c_{3}$ determine its shape, with $c_{1}=1$.

Adiabatic dark-state protocol

We assume the interaction onset as $t=0$ and denote the interaction duration by $T_{\mathrm{a}}$. To control $\delta(t)$, we sweep the slow bare-qubit detuning about $\Delta_{0}=(\alpha_{2}-\alpha_{1})/2$, choosing $\Delta_{\mathrm{q}}(t)=\Delta_{0}+r\cos\vartheta(s)$, where $s=t/T_{\mathrm{a}}\in[0,1]$. The sweep angle is parameterized as

Here, $I_{\mathrm{s}}(\beta,\beta)$ is the regularized incomplete beta function, and $\beta>1$ adjusts the sweep rate while ensuring smooth endpoints. As $\vartheta$ increases from $0$ to $\pi$, the detuning offset changes from $+r$ to $-r$.

Equation (A2) shows how the exchange coupling can be controlled through the coupler frequency. For numerical waveform construction, we calibrate the effective coupling $g_{\mathrm{cal}}(\omega_{\mathrm{c}},\Delta_{\mathrm{q}})$ from the static Hamiltonian and determine $\omega_{\mathrm{c}}(t)$ by solving

where $g_{\mathrm{idle}}(\Delta_{\mathrm{q}})=g_{\mathrm{cal}}(\omega_{\mathrm{c,idle}},\Delta_{\mathrm{q}})$ denotes the small residual coupling at the idle point for the current slow detuning. The constant $g_{\mathrm{exc}}>0$ sets the maximum exchange excursion, while $p$ controls its shape. At both endpoints, the excursion vanishes, ensuring that the coupler is at the idle frequency. Near the middle of the interaction window, the coupling strength reaches its maximum, with the coupler at its lowest frequency and the detuning offset passing through zero.

The loading and reset ramps connect the slow detuning to its idle value using the $S_{5}$ function defined in the preceding subsection, with durations $t_{\mathrm{load}}$ and $t_{\mathrm{reset}}$, respectively. The same function shapes the modulation envelope $w(t)$: it rises from zero to unity and falls back to zero through edges of duration $\tau_{w}$, with $w(t)=1$ between the edges. During loading and reset, the longitudinal modulation is off and the coupler remains at its idle frequency.

The opposite longitudinal modulations applied to the qubits are $\pm A w(t)\cos(\nu t)$, where $A$ is the single-qubit modulation amplitude and $\nu$ is the angular frequency. The corresponding auxiliary coupling is approximately

Unconventional geometric gate

Derivation of the unconventional geometric control conditions

For a linearly varying coupling phase $\varphi(t)=at+\varphi_{0}$, we use the transformation $R_{\Delta_{\mathrm{e}}}(t)=e^{-i\Delta_{\mathrm{e}}t/2}|1\rangle\langle1|+e^{i\Delta_{\mathrm{e}}t/2}|B\rangle\langle B|$. The transformed phase is $\varphi_{\mathrm{e}}(t)=(a-\Delta_{\mathrm{e}})t+\varphi_{0}$. The complete three-segment control conditions are

where $d_{\chi}=(1+\eta)\cos\chi-\eta$.

We derive the three-segment control conditions for the unconventional nonadiabatic geometric CZ gate in the effective $\{|11\rangle,|B\rangle\}$ subspace. The effective Hamiltonian is

where $\sigma_z^{(B)}=|11\rangle\langle11|-|B\rangle\langle B|$. We assume a constant Rabi frequency $\Omega_B$ within the three control segments. Starting from $|11\rangle$, the evolving state can be parameterized as

where $\chi(t)$ and $\xi(t)$ denote the polar and azimuthal angles on the Bloch sphere, respectively. Substituting Eq. (35) into the Schrödinger equation gives

We construct a closed trajectory consisting of three segments. In the first segment $P_1\to P_2$, we choose $\Delta_{\mathrm{e}}=0$ and $\varphi_{\mathrm{e}}=\xi_1+\pi/2$. Equations (D4a) and (D4b) then give $\dot{\chi}=\Omega_B$ and $\dot{\xi}=0$. Thus, the state reaches a polar angle $\chi$ at $t=\tau_1$, satisfying $\Omega_B\tau_1=\chi$.

In the second segment $P_2\to P_3$, the state evolves along a latitude with fixed polar angle $\chi$. Setting $\varphi_{\mathrm{e}}(t)=\xi(t)$ ensures $\dot{\chi}=0$, while Eq. (D4b) becomes $\dot{\xi}=-\Delta_{\mathrm{e}}-\Omega_B\cot\chi$. Writing $\dot{\xi}=-\beta$, we obtain

The total phase acquired during the cyclic evolution consists of geometric and dynamical contributions $\gamma=\gamma_g+\gamma_d$.

Since the dynamical phase vanishes along the two meridional segments, its contribution comes entirely from $P_2$:

For an unconventional geometric gate, we impose the phase relation $\gamma_d=\eta\gamma_g$. Combining Eqs. (39)–(40), we find

Consequently, the control parameters in $P_2$ satisfy

Using $\gamma=(1+\eta)\gamma_g$, the duration of the second segment is determined by

The final azimuthal angle is therefore

In the final segment $P_3 \to P_1$, we set $\Delta_e = 0$ and $\varphi_e = \xi_2 - \pi/2$. The resulting evolution satisfies $\dot{\chi} = -\Omega_B$ and $\dot{\xi} = 0$, returning the state to $\lvert11\rangle$ after $\Omega_B(\tau- \tau_2) = \chi$. With the phase reference $\xi_1 = 0$, these results reproduce the three-segment control conditions in Eq. (33). The cyclic evolution gives

Setting $\gamma= \pi$ yields the required conditional phase for the effective-frame geometric CZ gate.

Two-dimensional detuning robustness of conventional geometric gates

To complement the one-dimensional detuning analysis in the main text, Figure 6(a,Figure 6b) presents the two-dimensional fidelity landscapes of the conventional geometric gates.

Although the bright-state gate exhibits slightly lower fidelity along $\delta_2 = 0$, it outperforms the single-channel gate over most of the scanned parameter region.

In the $\{\lvert11\rangle,\lvert B\rangle,\lvert D\rangle\}$ subspace, the frequency-deviation term becomes

revealing the bright-dark coupling induced by differential detuning. Figure 6(c) shows the corresponding residual dark-state population, $P_D(T) = \lvert\langle D\lvert U(T)\rvert11\rangle\rvert^2$, which vanishes along $\delta_1 = \delta_2$ and increases with differential detuning. Its maximum reaches approximately 27.2% for the conventional gate, compared with 1.57% for the unconventional gate under the same detuning range. This comparison indicates that the unconventional protocol substantially suppresses the residual dark-state population at the end of the gate.