Abstract
The tradeoff between robustness and tunability is a central challenge in the pursuit of quantum simulation and faulttolerant quantum computation. In particular, quantum architectures are often designed to achieve high coherence at the expense of tunability. Many current qubit designs have fixed energy levels and consequently limited types of controllable interactions. Here by adiabatically transforming fixedfrequency superconducting circuits into modifiable Floquet qubits, we demonstrate an XXZ Heisenberg interaction with fully adjustable anisotropy. This interaction model can act as the primitive for an expressive set of quantum operations, but is also the basis for quantum simulations of spin systems. To illustrate the robustness and versatility of our Floquet protocol, we tailor the Heisenberg Hamiltonian and implement twoqubit iSWAP, CZ and SWAP gates with good estimated fidelities. In addition, we implement a Heisenberg interaction between higher energy levels and employ it to construct a threequbit CCZ gate, also with a competitive fidelity. Our protocol applies to multiple fixedfrequency highcoherence platforms, providing a collection of interactions for highperformance quantum information processing. It also establishes the potential of the Floquet framework as a tool for exploring quantum electrodynamics and optimal control.
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Main
The capability to coherently choreograph interactions between qubits is the foundation for the recent advances in quantum technologies. A quintessential example is the manipulation of the quantum Heisenberg model for the simulation of manybody quantum spin systems^{1,2,3,4}, which has led to the recent discoveries of intriguing physical phenomena such as discrete time crystal^{5}, phantom spinhelix states^{6} and formation of photon bound states^{7}. The Heisenberg interactions are also the primitives for expressive multiqubit gates^{8} which play important roles in quantum algorithms^{9} and quantum error correction^{10,11}. Therefore, endowing quantum architectures with such archetypal interactions considerably extends their capabilities and performance.
The required tunability in solidstate quantum devices entails additional decoherence channels, demanding design overhead and increased operational complexity. For example, in the domain of superconducting circuits, the performance in fluxtunable devices is typically limited by unavoidable 1/f noise arising from the surrounding environment. Meanwhile, fixedfrequency platforms such as singlejunction transmon^{12,13} and fluxonium^{14} biased at the halfinteger flux quantum^{15} have the best coherence times to date, but their native interactions are limited to the crossresonance^{16,17} and longitudinal couplings^{18,19}. Parametric longitudinal^{20,21,22} and transverse interactions^{23} can also be accomplished by introducing additional tunable couplers, but the performance could be undermined by the couplers’ coherence and spurious couplings.
In this Article, we present a reliable and hardwareefficient protocol to synthesize Floquet qubits^{24,25,26} from statically coupled singlejunction transmon qubits using timeperiodic microwave drives, showing that the adiabatic mapping procedure can be hastened by exploiting a shortcutstoadiabaticity (STA) technique^{27,28,29}. Then, we implement an XXZ Heisenberg interaction between these Floquet qubits, described by the Hamiltonian
and demonstrate that the transverse spinexchange and longitudinal spin–spin interaction terms can be adjusted independently by tailoring the drive parameters.
To validate the robustness and practicality of the protocol, we characterize twoqubit iSWAP, CZ and SWAP gates which correspond to different anisotropy Δ = J_{ZZ}/J_{XY} values, achieving estimated fidelities of 99.32(3)%, 99.72(2)% and 98.93(5)%, respectively. In addition, we show that the Floquetengineered interactions can be broadly applied to other levels in the system. Specifically, we explore the swapping between the qutrit states \(\left\vert 11\right\rangle\) and \(\left\vert 02\right\rangle\), then employ it to implement a threequbit controlledcontrolledZ (CCZ) gate which is locally equivalent to the Toffoli gate^{30}, achieving an estimated fidelity of 96.18(5)%. Our work exemplifies the operational principles of Floquet qubits and illustrates their broad potential, thus opening promising pathways for future developments of the Floquet framework in enhancing the capabilities of fixedfrequency solidstate quantum platforms.
Synthesizing Floquet qubits
Figure 1a depicts the superconducting device used as a testbed in the experiment. It consists of singlejunction transmon qubits^{12} that are pairwise coupled via mutual coplanar stripline resonators^{31,32}. Details of the quantum device and experimental setup are presented in Supplementary Notes 1 and 2. Although the frequencies of the qubits are fixed after fabrication, Floquet engineering has recently emerged as a powerful tool that allows the sculpting of effective Hamiltonians that are otherwise unavailable^{33}, thus promising an additional dimension to tune the system. Here, we synthesize Floquet qubits using detuned periodic microwave drives and tailor them to implement the Hamiltonian given by equation (1).
The mapping is described by Floquet formalism as follows. The Hamiltonian of a twolevel spinhalf system subjected to a periodic driving field with amplitude A, frequency ω_{d} and phase φ is given as
where ℏω_{q} is the energy gap of the twolevel system, and \({\hat{\sigma }}_{{{{\rm{z}}}}}\) and \({\hat{\sigma }}_{{{{\rm{x}}}}}\) represent the Pauli operators. There exists no static eigenenergies and eigenstates of the system as solutions of the timedependent Schrödinger equation \({\rm{i}}\hslash {\partial }_{t}\left\vert \psi (t)\right\rangle ={\hat{{{{\mathcal{H}}}}}}_{{{{\rm{q}}}}}(t)\left\vert \psi (t)\right\rangle\). However, due to the periodicity of \({\hat{{{{\mathcal{H}}}}}}_{{{{\rm{q}}}}}(t)\), the Schrödinger equation can be modified into the Floquet equation^{34,35} \(\left({\hat{{{{\mathcal{H}}}}}}_{{{{\rm{q}}}}}(t){\rm{i}}\hslash {\partial }_{t}\right){\left\vert {u}_{n}(t)\right\rangle }_{{{{\rm{F}}}}}=\hslash {\varepsilon }_{n}{\left\vert {u}_{n}(t)\right\rangle }_{{{{\rm{F}}}}}\), and static quasienergies ℏε_{n} can be found for timeperiodic Floquet states \({\left\vert {u}_{n}(t)\right\rangle }_{{{{\rm{F}}}}}={\left\vert {u}_{n}(t+2\uppi /{\omega }_{\rm{d}})\right\rangle }_{{{{\rm{F}}}}}\). Here, the Floquet states are denoted with subscript F to distinguish them from the bare states in the lab frame. The Floquet and bare states are interconvertible following the relation
Interestingly, \({\rm{e}}^{{\rm{i}}k{\omega }_{{{{\rm{d}}}}}t}{\left\vert {u}_{n}(t)\right\rangle }_{{{{\rm{F}}}}}\) with integer k also satisfies the Floquet equation and has quasienergy ℏ(ε_{n} + kω_{d}), resulting in an infinite transition spectrum^{35} \({\varepsilon }_{1}{\varepsilon }_{0}=k{\omega }_{{{{\rm{d}}}}}\pm \sqrt{{A}^{2}+{({\omega }_{{{{\rm{d}}}}}{\omega }_{{{{\rm{q}}}}})}^{2}}\), where the plus(minus) sign corresponds to red(blue)detuned drive. In addition, the drive phase φ acts as a time translation operator on the Hamiltonian in equation (2), \({\left\vert {u}_{n}(t)\right\rangle }_{{{{\rm{F}}}}}\to {\left\vert {u}_{n}(t+\varphi /{\omega }_{{{{\rm{d}}}}})\right\rangle }_{{{{\rm{F}}}}}\). These show how the Floquet states and their quasienergies depend on the drive parameters A, φ and ω_{d}, which we can use to tailor the driven systems (see Supplementary Notes 3 and 4 for detailed Floquet formalism).
To prepare a Floquet qubit with the desired properties, we have to continuously map the undriven qubit to the Floquet basis, as shown in Fig. 1b. If the transformation is performed abruptly, finite tunnelling exists between the Floquet basis states, and the process becomes nonadiabatic. According to the Adiabatic Theorem^{35}, the tunnelling rate is proportional to dA/dt, that is, the target Floquet qubit corresponding to a larger drive amplitude must be transformed using a longer ramp time.
We experimentally explore this by irradiating qubit Q_{1} initialized in \(\left\vert 0\right\rangle\) using a cosineramp pulse with different pulse durations τ_{g} and ramp times τ_{r}. The drive amplitude is set to be 100 MHz in terms of onresonant Rabi frequency, and the drive frequency is reddetuned by 40 MHz from Q_{1}’s transition frequency. As the qubit should remain in its instantaneous eigenstate under the adiabatic process, nonadiabatic effects manifest as finite excited state populations \({P}_{\left\vert 1\right\rangle }\) after the pulse, which oscillates with respect to the pulse duration due to the dynamical phase accumulation of the Floquet qubit (Fig. 1c). Evidently, shorter ramp times correspond to more severe nonadiabatic effects. In addition, the result in Fig. 1d confirms that a larger drive amplitude requires a longer ramp time to satisfy the adiabatic condition.
Interestingly, we find that using an STA technique known as derivative removal by adiabatic gate (DRAG)^{28,29} helps reduce nonadiabatic effects substantially. As shown in Fig. 1e, the excited state leakage \({P}_{\vert 1 \rangle }^{\mathrm{max}}\) corresponding to a shortramp pulse can be suppressed by adding a quadrature component to the pulse with amplitude A_{Q} = λ_{DRAG} × dA(t)/dt. In this case, τ_{r} can be reduced from 60 ns to 30 ns by employing a DRAG coefficient λ_{DRAG} = −0.7. This suggests that advanced optimal control techniques can be explored to further accelerate the mapping procedure. Importantly, our results below show that once adiabaticity is satisfied, the protocol is robust against adverse effects from the strong drives, including microwave crosstalk, calibration fluctuations and leakage to higher levels.
Tailoring Heisenberg interactions
Having established the general conditions for adiabatic mapping between undriven qubit states and Floquet states, we next generate the microwave pulses that establish the XXZ Heisenberg interaction in equation (1) between the Floquet qubits. The interaction Hamiltonian describing the coupling between Q_{1} and Q_{2} in Fig. 1a is \({\hat{{{{\mathcal{H}}}}}}_{{{{\rm{int}}}}}/\hslash =J{\hat{\sigma }}_{{{{\rm{x}}}}}^{(1)}{\hat{\sigma }}_{{{{\rm{x}}}}}^{(2)}\), where J is the static coupling strength, the superscripts are qubit indices, and the Pauli operators are defined in the undriven basis. Using the relation given by equation (3), this interaction can be described by a Floquet Hamiltonian:
where \({\varepsilon }_{ab}^{(k)}\equiv {\varepsilon }_{b}^{(k)}{\varepsilon }_{a}^{(k)}\), \({c}_{ab}^{(k)}(t)=\left\langle {\psi }_{a}^{(k)}(t) {\hat{\sigma }}_{{{{\rm{x}}}}}^{(k)} {\psi }_{b}^{(k)}(t)\right\rangle\), \({\hat{f}}_{ab}^{\,(k)}(t)={\left\vert {u}_{a}^{(k)}(t)\right\rangle }_{{{{\rm{F}}}}} {\left\langle {u}_{b}^{(k)}(t)\right\vert }_{{{{\rm{F}}}}}\) for qubit Q_{k} and a, b, c, d ∈ {0, 1} for two qubits. The fast oscillation dynamics can be neglected by invoking the rotating wave approximation, leaving only the terms that follow energy conservation law, \({\varepsilon }_{ab}^{(1)}+{\varepsilon }_{cd}^{(2)}=0\) for abcd ∈ {0110, 1001, 0000, 0011, 1100, 1111}. Inspecting the reduced Hamiltonian then gives us insight on the types of interactions present between the qubits.
On one hand, the terms satisfying \({\varepsilon }_{01}^{(1)}={\varepsilon }_{01}^{(2)}\) correspond to the transverse XY spinexchange interaction in equation (1) with \({J}_{{{{\rm{XY}}}}}=\)\(J{\langle {c}_{01}^{(1)}{c}_{10}^{(2)}\rangle }_{t}=J{\langle {c}_{10}^{(1)}{c}_{01}^{(2)}\rangle }_{t}\), where 〈…〉_{t} denotes the timeaverage value. This process follows the conventional wisdom that an XY exchangetype interaction between two coupled spins occurs when they are brought into resonance with each other. On the other hand, the rest of the reduced Hamiltonian produces the longitudinal ZZ spin–spin coupling in equation (1), \({J}_{{{{\rm{ZZ}}}}}=J{\langle {c}_{11}^{(1)}{c}_{11}^{(2)}+}\)\({{c}_{00}^{(1)}{c}_{00}^{(2)}{c}_{00}^{(1)}{c}_{11}^{(2)}{c}_{11}^{(1)}{c}_{00}^{(2)}\rangle }_{t}\). Consequently, we can program the transverse and longitudinal interactions independently by tailoring the quasienergies with periodic microwave drives.
We validate this principle as follows. First, we achieve a pure transverse XY spinexchange interaction corresponding to an XX Heisenberg model where the anisotropy is zero, Δ = J_{ZZ}/J_{XY} = 0. Given that Q_{1}’s frequency is lower than that of Q_{2}, their quasienergy differences \({\varepsilon }_{01}^{(k)}\) can be brought into resonance if Q_{1}(Q_{2}) is driven with red(blue)detuned microwaves (Supplementary Note 5). After preparing the qubits in \(\left\vert 10\right\rangle\), we apply two such pulses (p_{1} and p_{2} in Fig. 2a) with the same duration τ_{g} and amplitude A_{XY} at a detuning frequency of 40 MHz.
We observe a coherent population transfer to state \(\left\vert 01\right\rangle\) that forms a chevron pattern as a function of τ_{g} and A_{XY}, signifying a transverse coupling between the qubits (Fig. 2b). Notably, although the interaction occurs between the Floquet qubits in the dressed frame, the adiabatic connection ascertains the exchange between the bare qubit states after the reverse mapping, which bears resemblance to the latching mechanism in classical electronics. Indeed, at the optimal drive amplitude A_{XY}/2π = 65.2 MHz (Fig. 2c), \(\left\vert 10\right\rangle\) and \(\left\vert 01\right\rangle\) exhibit coherent oscillations at a rate of 3.2 MHz, which is limited by the static coupling constant J (Supplementary Note 5). The lack of fast oscillatory behaviour is a clear indication of the high mapping fidelity.
Next, we induce a pure longitudinal ZZ spin–spin coupling corresponding to an Ising interaction between the Floquet qubits. This can be accomplished by irradiating microwave drives p_{1} on Q_{1} and p_{3} on Q_{2} (Fig. 2a) at a frequency 40 MHz reddetuned from Q_{1}. The amplitude of p_{1} is fixed at A_{XY}/2π = 65.2 MHz, while p_{3} has parameterized amplitude A_{ZZ} and phase φ. For weak driving, A_{XY,ZZ} ≪ ω_{q,d}, the ZZ rate is given as \({J}_{{{{\rm{ZZ}}}}}\approx 2J{A}_{{{{\rm{XY}}}}}{A}_{{{{\rm{ZZ}}}}}\cos (\varphi )/\sqrt{({A}_{{{{\rm{XY}}}}}^{2}+{\delta }_{1}^{2})({A}_{{{{\rm{ZZ}}}}}^{2}+{\delta }_{2}^{2})}\), where δ_{k} is the detuning from Q_{k}’s frequency (Supplementary Note 5). Importantly, while the transverse coupling rate J_{XY} shown above is limited by the static coupling strength J, the longitudinal coupling rate J_{ZZ} can be tuned by two knobs, namely the drives’ amplitudes and phase difference. We characterize the interaction by first initializing the two qubits in the superposition state \((\left\vert 0\right\rangle +\left\vert 1\right\rangle )\otimes (\left\vert 0\right\rangle +\left\vert 1\right\rangle )/2\), applying the pulses as specified and then extracting the entangling phase \({\Phi}_{{{{\rm{ZZ}}}}}({\tau }_{{{{\rm{g}}}}})=\int\nolimits_{0}^{{\tau }_{{{{\rm{g}}}}}}{J}_{{{{\rm{ZZ}}}}}(t)\,{\rm{d}}t\) using tomographic reconstruction assisted by numerical optimization. As shown in Fig. 2d, this phase depends on p_{3}’s amplitude A_{ZZ} and phase φ, consistent with our description.
Leveraging the independent controls of the transverse and longitudinal interactions, we now tailor the interplay between them to adjust the anisotropy of the XXZ Heisenberg interaction model. To this end, we apply p_{1} and p_{2} pulses with their amplitude A_{XY} and duration τ_{g} tuned to induce a full \(\left\vert 10\right\rangle \leftrightarrow \left\vert 01\right\rangle\) swap (Fig. 2c). Pulse p_{3} is then jointly applied, albeit with parameterized amplitude A_{ZZ} and phase φ. Incorporating the swap condition into the tomography analysis, we extract the longitudinal entangling phase Φ_{ZZ} which depends on p_{3}’s parameters as shown in Fig. 2e. This demonstrates the versatility available in programming the anisotropy of the model given by equation (1). This demonstration only includes A_{ZZ} ≲ 20 MHz to alleviate the effect from p_{3} on the swap condition, which stems from the large microwave crosstalk in the experimental device (Supplementary Note 2).
Benchmarking Heisenberg interactions
The programmable Heisenberg interaction endows quantum processors with an extensive quantum gate set and the capability to simulate manybody spinhalf systems. Here we benchmark our Floquetengineered Heisenberg interactions by characterizing a suite of representative twoqubit gates: the iSWAP, CZ and SWAP gates resulting from the XX Heisenberg model \({\hat{{{{\mathcal{H}}}}}}_{{{{\rm{XX}}}}}/{\hbar}={J}_{{{{\rm{XY}}}}}({\hat{\sigma }}_{{{{\rm{x}}}}}{\hat{\sigma }}_{{{{\rm{x}}}}}+{\hat{\sigma }}_{{{{\rm{y}}}}}{\hat{\sigma }}_{{{{\rm{y}}}}})\), Ising model \({\hat{{{{\mathcal{H}}}}}}_{{{{\rm{ZZ}}}}}/{\hbar}={J}_{{{{\rm{ZZ}}}}}{\hat{\sigma }}_{{{{\rm{z}}}}}{\hat{\sigma }}_{{{{\rm{z}}}}}\) and XXX Heisenberg model \({\hat{{{{\mathcal{H}}}}}}_{{{{\rm{XXX}}}}}/{\hbar}=J\left(\right.{\hat{\sigma}}_{{{{\rm{x}}}}}{\hat{\sigma }}_{{{{\rm{x}}}}}+{\hat{\sigma }}_{{{{\rm{y}}}}}{\hat{\sigma }}_{{{{\rm{y}}}}}+{\hat{\sigma }}_{{{{\rm{z}}}}}{\hat{\sigma }}_{{{{\rm{z}}}}}\left.\right)\), respectively. Accordingly, an iSWAP unitary arises naturally from a pure transverse XY interaction with the pulse duration τ_{g} corresponding to a full swap in Fig. 2c, implemented by applying p_{1} and p_{2}. In practice, there is a dynamical ZZ coupling originating from microwave crosstalk, which can be tracked and compensated by simultaneously applying p_{3} with appropriate amplitude and phase. Likewise, a CZ gate is realized when p_{1} and p_{3} are calibrated to bring up an entangling phase Φ_{ZZ}/2π = 0.25. Finally, we tailor all three pulses to sculpt an isotropic XXX Heisenberg interaction that leads to a SWAP gate at the correct gate time τ_{g}, at which both iSWAP and CZ conditions are satisfied. Our calibration steps are detailed in Supplementary Notes 6–8.
To quantify the gates’ performances without state preparation and measurement errors, we employ cycle benchmarking (CB)^{36}, which tailors all errors into stochastic Pauli channels via Pauli twirling and results in tight bounds on the estimated fidelity (Supplementary Note 9). Besides the dressed cycles that include the implemented gates, we also measure the reference cycle and extract its errors to estimate the relevant gate fidelities. Figure 3a shows the Pauli fidelity distribution histograms of both the reference and dressed cycles corresponding to the intended twoqubit gates. Comparing the dressed cycle data to the reference cycle result allows us to estimate the average gate fidelities of the implemented iSWAP, CZ and SWAP gates to be 99.32(3)%, 99.72(2)% and 98.93(5)%, respectively. We note that these gates are expandable to a continuous fSim gate set^{8,37}, which can be integrated into arbitrary quantum circuits compatible with fixedfrequency qubits by using more advanced circuit compilation tools^{38} or efficient physical Zgates.
Our analysis attributes the limitations of these results primarily to decoherence mechanisms (Supplementary Notes 10–12). Intriguingly, the Floquet qubits appear to exhibit coherence times deviating from those of the bare qubits, as shown in Fig. 3b. The measurements are performed using nominal energy relaxation and echo dephasing procedures on the bare Q_{1}, but with the addition of a microwave pulse applied 40 MHz reddetuned from its \(\left\vert 0\right\rangle \leftrightarrow \left\vert 1\right\rangle\) transition during idle periods. The results are postselected to yield the populations of the desired states, and the experiment is repeated over twenty iterations to eliminate any potential outlier. While the dynamics remain the same at small drive amplitudes, T_{1} tends to increase while \({T}_{2}^{E}\) tends to decrease at strong driving before nonadiabaticity sets in. Interestingly, we also discover a heating mechanism that enlarges the excited state population in the bare qubit at the end of the 355μslong \(\left\vert 0\right\rangle \to \left\vert 1\right\rangle\) measurement sequence, with \({P}_{\left\vert 1\right\rangle }^{{{{\rm{final}}}}}\) increasing with the driving amplitude (Fig. 3b, inset). We include additional details in Supplementary Note 13 and hope that future investigations can find efficient approaches to mitigate these effects, reminiscent of the recent progress in driven ultracoldatom systems^{39}.
Floquet qutrit and threequbit gate
So far, the fundamental and universal importance of spin physics motivates our discussion to portray the implemented Floquet qubits as ideal spinhalfs. Nevertheless, many solidstate systems, including the transmon, naturally include multiple relevant energy levels. Making use of them expands the Hilbert space, allowing more information to be encoded, which leads to hardwareefficient execution of quantum algorithms^{40,41} and hastens the development of faulttolerant computation^{42,43}. We now show that the presented protocol can be tailored for multilevel systems, thereby paving pathways for quantum information processing using Floquet qudits.
Specifically, we leverage the techniques described so far to induce a transverse qutrit–qutrit interaction between the states \(\left\vert 11\right\rangle\) and \(\left\vert 02\right\rangle\). Although the crossKerr coupling has been explored^{44}, such an energyexchange interaction is still absent in fixedfrequency qutrits. While this is a useful ternary gate itself, we presently show that integrating it into a sequence involving multiple qubits allows the implementation of a threebody CCZ gate^{30}, which plays an important role in quantum applications such as factorization^{45,46} and quantum error correction^{47,48}. Notably, this scheme can be extended to implement an nqubit gate^{49}. To this end, we add to the experiment Q_{3}, which is coupled to the right side of Q_{2} in Fig. 1a and use Q_{2} and Q_{3} as control qubits (subscripted c), while Q_{1} is designated as the target (subscripted t).
Figure 4a depicts the energy diagram of Q_{2} and Q_{3}. Our approach to accomplish the interaction primarily involves applying a microwave pulse to Q_{3} at a frequency reddetuned from its \(\left\vert 1\right\rangle \leftrightarrow \left\vert 2\right\rangle\) transition to create a Floquet qutrit such that the control Floquet states \({\left\vert 11\right\rangle }_{{{{\rm{c}}}}}\) and \({\left\vert 02\right\rangle }_{{{{\rm{c}}}}}\) become degenerate. After initializing the control qubits in \({\left\vert 11\right\rangle }_{{{{\rm{c}}}}}\), we apply such a pulse with ramp time τ_{r} = 170 ns, DRAG coefficient λ_{DRAG} = −0.6 and varying amplitude A_{XY} and duration τ_{g}, which are tailored to ensure adiabaticity at a reddetuning of 22 MHz. The transverse interaction between \({\left\vert 11\right\rangle }_{{{{\rm{c}}}}}\) and \({\left\vert 02\right\rangle }_{{{{\rm{c}}}}}\) then manifests into an asymmetric chevron pattern with respect to A_{XY} and τ_{g} in Fig. 4b. Interestingly, we observe that the optimal swap condition occurs at a stronger amplitude relative to the symmetry point.
A CCZ unitary can be implemented using the sequence given in Fig. 4c (Supplementary Note 14). The final CPhase gate on the control qubits is tuned to bring the effective operation on them to be \({\hat{\sigma }}_{{{{\rm{I}}}}}\otimes {\hat{\sigma }}_{{{{\rm{I}}}}}\) at the end of the sequence. After calibrating the individual gates, we verify the entanglement between the three qubits by extracting the Zphase of the target qubit (Q_{1}) for different control states and observe a phase shift of approximately π for \({\left\vert 10\right\rangle }_{{{{\rm{c}}}}}\) (Fig. 4d), which evinces the CCZ effect. The sequence can be further sandwiched between singlequbit rotations on Q_{1} to construct a Toffoli gate. The process can be straightforwardly validated by measuring the truth table^{30,49}, from which we extracted a fidelity of \({{{{\mathcal{F}}}}}_{{{{\rm{tt}}}}}={{{\rm{Tr}}}}({{{{\mathcal{U}}}}}_{\exp }{{{{\mathcal{U}}}}}_{{{{\rm{ideal}}}}}^{{\dagger} })/8=96.20(6) \%\) (Fig. 4e). Finally, we employ CB to benchmark the CCZ gate (Fig. 4f), achieving a fidelity of 96.18(5)%, with the main error resulting from decoherence (Supplementary Note 11).
Outlook
Our work embodies a transformative application of Floquet engineering in superconducting circuits where periodic drives are used to map static qubits to Floquet qubits with modifiable quasienergies, granting access to an unconventional tuning channel. We demonstrate the practicality and versatility of this approach by synthesizing Floquet qubits and qutrits, then realizing an XXZ Heisenberg interaction between them with fully tunable anisotropy. The robustness of the scheme against environmental noise, nonadiabaticity, leakage and calibration errors is reflected in the high gate fidelities, while overcoming the current limitations is straightforward. On one hand, the coherence times of the fixedfrequency transmon qubits in the experiment are relatively low, so we expect better performance in stateoftheart devices. On the other hand, the coupling rate is primarily limited by the static coupling constant J, which can be increased substantially in future devices. In addition, the pulse shape used in this work is quite simple, so we believe that, in the future, advanced STA techniques can be employed to design shorter gates, further reducing errors from dephasing. Our preliminary assessment (Supplementary Note 15) shows the promising extensibility of the protocol to largescale devices. We note that the experimental device is simply a testbed, and the full potential of this framework lies upon its adaptation to other synthetic fixedfrequency quantum architectures with better projected performance, such as fluxonium quantum processor with allmicrowave control^{50}.
Having illustrated the useful properties of the Floquet qubits, we envision the following avenues to further develop and propel the concept to scale up fixedfrequency platforms. The protocol presented here involves transforming back to the static qubit, so normal operations such as readout and singlequbit gates can be employed without recalibration. In future applications, a Floquet qubit can, in principle, be permanently defined by applying a continuous periodic drive, streamlining the process and unlocking opportunities for control and readout of the Floquet qubit^{25} (Supplementary Note 16). This approach also allows in situ tuning of the qubit frequencies, thus providing a practical solution for problems arising from twolevelsystem defects and spectral crowding. Alternatively, the ramp time can be reduced substantially if we operate in the diabatic regime, where the mapping is close to ideal despite finite transition between the Floquet states. We expect the potential development of optimal control within the Floquet framework to provide a reliable approach in this regime. Last but not least, the heating effect which correlates with the reduction in T_{2} is reminiscent of a similar effect in coldatom systems which has been successfully suppressed^{39}. This calls for deeper understanding of the quantum thermodynamics in driven solidstate systems and the development of new mitigation strategies.
Data availability
All analysed data are available at https://doi.org/10.6084/m9.figshare.24217011.v1. Source data are provided with this paper. All other data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank A. Morvan and W. P. Livingston for their assistance with the measurement. L.B.N. is grateful to Z. Huang, I. MondragonShem and J. Koch for valuable discussions on Floquet theory. The micrograph of the chip was obtained with support from B. Qing and K. Lee. This work was supported by the Office of Advanced Scientific Computing Research, Testbeds for Science programme, Office of Science of the US Department of Energy under Contract No. DEAC0205CH11231, the KIST research programme under grant No. 2E32241 and ARO grant No. W911NF2210258.
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L.B.N. conceived and organized the project. Y.K. initiated the measurement and applied the Floquet framework. L.B.N. and Y.K. acquired the data and analysed the results. A.H. assisted with device calibration and CB evaluation. N.G. assisted with the characterization of qutrits and the CCZ gate. B.M. assisted with crossentropy benchmarking. B.B. and D.D. assisted with Floquet theory. R.K.N. set up the microwave apparatus. J.M.K. fabricated the device. A.N.J. supervised the theoretical work. D.I.S. and I.S. oversaw the experimental effort. L.B.N. and Y.K. wrote the paper with input from A.H., N.G. and B.B.
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Nguyen, L.B., Kim, Y., Hashim, A. et al. Programmable Heisenberg interactions between Floquet qubits. Nat. Phys. 20, 240–246 (2024). https://doi.org/10.1038/s41567023023267
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DOI: https://doi.org/10.1038/s41567023023267
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