Quantum control and noise protection of a Floquet $0-\pi$ qubit

The Kapitzonium circuit (Fig. 1(b)) is identical to that of a capacitively shunted superconducting quantum interference device (SQUID). The external flux $\Phi_{\text{ext}}(t)$ threading the SQUID loop is used to emulate the effect of a time-modulated pivot point of a pendulum. The Kapitzonium Hamiltonian is derived with the established circuit quantization procedures Vool and Devoret (2017); You et al. (2019); Rajabzadeh et al. (2022); Chitta et al. (2022). The branch flux variable $\hat{\varphi}_{k}$ across each Josephson junction and the conjugate charge variable $\hat{n}_{k}$ satisfy $[\hat{\varphi}_{k},\hat{n}_{k}]=i$ for $k=1,2$. The circuit Hamiltonian is

where $E_{Jk}$ are the Josephson energies and $E_{Ck}=e^{2}/2C_{k}$ are the charging energies with $C_{k}$ being the junction capacitances.

Flux quantization sets the constraint of $\hat{\varphi}_{1}-\hat{\varphi}_{2}=\Phi_{\text{ext}}(t)$. In the symmetric case of $E_{J1}=E_{J2}$ and $E_{C1}=E_{C2}$, Eq. (1) reduces to the Kapitzonium Hamiltonian

where $\phi_{\text{ext}}(t)=\Phi_{\text{ext}}(t)/2$. The flux variable $\hat{\phi}=(\hat{\varphi}_{1}+\hat{\varphi}_{2})/2$ corresponds to the pendulum rotation angle, and $\hat{n}=\hat{n}_{1}+\hat{n}_{2}$ is the conjugate charge variable satisfying $[\hat{\phi},\hat{n}]=i$. The charging energy is $E_{C}=e^{2}/2(C_{1}+C_{2})$ and the Josephson energy is $E_{J}=E_{J1}+E_{J2}$.

To realize the analog of the Kapitza pendulum, we set the external flux to $\phi_{\text{ext}}(t)=\omega t$. The idling dynamics are described by a Floquet Hamiltonian

with a time period of $T=2\pi/\omega$. Throughout this paper, we choose Kapitzonium parameters $E_{J}/2\pi=100~{}\text{GHz}$, $\omega/2\pi=10~{}\text{GHz}$ and $E_{C}/2\pi=0.01~{}\text{GHz}$ unless otherwise specified. To understand the emergence of the degenerate “ground” states, we derive the effective Hamiltonian Eckardt (2017); Venkatraman et al. (2022) of $\hat{H}_{0}(t)$ by integrating over the fast modulation, and find

where $\tilde{E}_{J}=E_{C}E_{J}^{2}/\omega^{2}$. Here we only keep terms up to the second order in the effective Hamiltonian, while the third order term is 0 and the fourth order term scales as $E_{C}^{3}E_{J}^{2}/\omega^{4}\ll\tilde{E}_{J}$, which can be neglected Eckardt (2017); Venkatraman et al. (2022).

The static effective Hamiltonian $\hat{H}_{\text{eff}}$ possesses some of the key features of a protected qubit: disjoint wavefunctions and energy degeneracy. Since $\hat{\phi}$ is $2\pi$-periodic, $\hat{H}_{\text{eff}}$ has two near-degenerate ground states $(\left|0\right\rangle\pm\left|\pi\right\rangle)/\sqrt{2}$ in the deep transmon regime of $\tilde{E}_{J}/E_{C}=100$. Here $\left|0\right\rangle$ and $\left|\pi\right\rangle$ are localized at the two minima of the effective potential $V_{\text{eff}}(\phi)=-\tilde{E}_{J}\cos 2\phi$ (Fig. 1(c)) with exponentially small overlap.

The effective Hamiltonian provides a useful approximate picture of the potential and the “ground” states of the Kapitzonium – note that the states $(\left|0\right\rangle\pm\left|\pi\right\rangle)/\sqrt{2}$ are only the ground states of the effective Hamiltonian. To better understand the states, gates, and noise, we will need to move beyond the effective description and consider the Floquet eigenstates of the qubit. The Floquet eigenstates $\left|\Psi_{\alpha}(t)\right\rangle$ of the idling Kapitzonium Hamiltonian $\hat{H}_{0}(t)$ satisfy

and have the form of

Here $\left|\Phi_{\alpha}(t)\right\rangle=\left|\Phi_{\alpha}(t+T)\right\rangle$ are the periodic Floquet modes and $\varepsilon_{\alpha}$ are the Floquet eigenenergies.

In Fig. 2(a), we plot the even and odd superpositions of the Floquet eigenstates defined as

for $k=0,1,2$. The phases of $\left|\Psi_{\alpha}(t=0)\right\rangle$ are chosen such that $\left|\psi_{2k}\right\rangle$ ($\left|\psi_{2k+1}\right\rangle$) are mostly localized around $\phi=0$ ($\phi=\pi$). In Fig. 2(b), we compare the exact Floquet eigenenergies $\varepsilon_{\alpha}$ with the spectrum of $\hat{H}_{\text{eff}}$ and find a close match. Here the index $\alpha$ are sorted based on the overlaps between $\left|\Psi_{\alpha}(t=0)\right\rangle$ and the $n$th eigenstates of $\hat{H}_{\text{eff}}$. The disjoint wavefunctions of $\{\left|\psi_{2k}\right\rangle,\left|\psi_{2k+1}\right\rangle\}$ and the near degeneracy $\varepsilon_{2k}\approx\varepsilon_{2k+1}$ for $k=0,1,2$ confirm the effective double well potential $V_{\text{eff}}(\phi)$. Finally, Floquet eigenstates are not stationary within one period. For example, the time evolution of $\left|\psi_{0}\right\rangle$ and $\left|\psi_{1}\right\rangle$ from $0$ to $T$ shows varying $\phi$ space wavefunctions (Fig. 2(c)).

A single qubit state $(c_{0},c_{1})^{T}$ can be encoded in Kapitzonium as

where $c_{0}$ and $c_{1}$ are invariant under $\hat{H}_{0}(t)$. For $(\varepsilon_{1}-\varepsilon_{0})/2\pi\approx 4.7$ kHz, $1/(\varepsilon_{1}-\varepsilon_{0})$ is much longer than the time scale of any relevant Kapitzonium operations. We thus assume $\varepsilon_{0}=\varepsilon_{1}$ and only consider the system at $t=nT$ to simplify our discussions. In this case, the qubit basis states $\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{1}\right\rangle$ become static, and are related to $\left|0\right\rangle$ and $\left|\pi\right\rangle$ by a Hadamard transformation

The encoded qubit state can be manipulated by engineering the flux drive $\phi_{\text{ext}}(t)$. The resulting effective potentials (Fig. 2(d)) provide the intuition for the Kapitzonium gates. Here we focus on the gate Hamiltonians and details on the required flux drive are discussed in Sec. IV.

X rotation. The X gate Hamiltonian $\hat{H}_{x}(t)=\hat{H}_{0}(t)+\alpha_{x}\cos\hat{\phi}$ generates the rotation along X axis. The effective potential now becomes an asymmetric double well $V_{\text{eff}}^{(x)}(\phi)=-\tilde{E}_{J}\cos 2\phi+\alpha_{x}\cos\phi$, which lifts the degeneracy between $\left|0\right\rangle$ and $\left|\pi\right\rangle$. Therefore, in the $\{\left|0\right\rangle,\left|\pi\right\rangle\}$ basis $\hat{H}_{x}(t)\approx\alpha_{x}\hat{\sigma}_{z}$ and in the $\{\left|\Psi_{0}\right\rangle,\left|\Psi_{1}\right\rangle\}$ basis, $\hat{H}_{x}(t)\approx\alpha_{x}\hat{\sigma}_{x}$, leading to Rabi oscillation between $\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{1}\right\rangle$.

Z rotation. The depth $\tilde{E}_{J}$ of the effective double well potential can be controlled dynamically with the flux driving frequency $\omega$. Increasing $\omega$ reduces $\tilde{E}_{J}/E_{C}=(E_{J}/\omega)^{2}$ and induces stronger coupling between $\left|0\right\rangle$ and $\left|\pi\right\rangle$. We choose $\omega_{z}/2\pi=20$ GHz for the Z gate, which lifts the degeneracy between $\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{1}\right\rangle$ with a splitting $(\varepsilon_{1}-\varepsilon_{0})/2\pi\approx 1.8$ MHz. This implements a phase gate for $\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{1}\right\rangle$, i.e., the rotation along Z axis.

Two qubit XX rotation. We couple two Kapitzonium with a Josephson junction to realize the XX gate Hamiltonian $\hat{H}_{xx}(t)=\hat{H}_{0}(t)+\alpha_{xx}\cos(\hat{\phi}_{1}-\hat{\phi}_{2})$. Replacing the Josephson junction with a SQUID makes the coupling tunable. The joint effective potential is $V_{\text{eff}}^{(xx)}(\phi_{1},\phi_{2})=-\tilde{E}_{J}\cos 2\phi_{1}-\tilde{E}_{J}\cos 2\phi_{2}+\alpha_{xx}\cos(\phi_{1}-\phi_{2})$, which lifts the degeneracy between $\{\left|00\right\rangle,\left|\pi\pi\right\rangle\}$ and $\{\left|0\pi\right\rangle,\left|\pi 0\right\rangle\}$. In the $\{\left|0\right\rangle,\left|\pi\right\rangle\}$ basis $\hat{H}_{xx}(t)\approx\alpha_{xx}\hat{\sigma}_{z}\otimes\hat{\sigma}_{z}$, and in the $\{\left|\Psi_{0}\right\rangle,\left|\Psi_{1}\right\rangle\}$ basis $\hat{H}_{xx}(t)\approx\alpha_{xx}\hat{\sigma}_{x}\otimes\hat{\sigma}_{x}$, generating the XX rotation.

In an open quantum system, a coupling between the system and bath allows the bath to induce transitions between different eigenstates of the system. The form of the coupling and the temperature of the bath both go into determining the transitions and their rates. For simplicity, we only consider a zero-temperature bath in this paper. For time-independent systems, any transition from lower to higher energy will require energy to be absorbed from the bath. Because of this, static systems coupled to zero-temperature baths eventually decay to their ground states.

Considering the level structure in Fig. 2(a), we would naively expect the zero-temperature bath to induce only the downward transition $\left|\Psi_{2}\right\rangle\rightarrow\left|\Psi_{0}\right\rangle$, with the opposing transition $\left|\Psi_{0}\right\rangle\rightarrow\left|\Psi_{2}\right\rangle$ being suppressed as that would require absorption of energy from the bath. This intuition is incorrect because the Kapitzonium is a Floquet system, which is constantly exchanging energy with the Floquet drive. As a result, the bath induced transitions happen in both directions between the Floquet eigenstates: both $\left|\Psi_{2}\right\rangle\rightarrow\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{0}\right\rangle\rightarrow\left|\Psi_{2}\right\rangle$ transitions are allowed in a Kapitzonium interacting with a zero-temperature bath. Therefore in contrast to the static $0-\pi$ qubit Brooks et al. (2013), the Floquet $\left|0\right\rangle$ and $\left|\pi\right\rangle$ can still decay out of the qubit subspace through what looks like a heating process.

Consider a generic Floquet system $\hat{H}_{0}(t)=\hat{H}_{0}(t+T)$ coupled to some bath degrees of freedom $\hat{B}(t)$ via the system operator $\hat{O}$. The system-bath Hamiltonian in the rotating frame of the bath is

where

Here $\omega_{k}\geq 0$ is the frequency of the $k$th bath mode and $g_{k}$ is the coupling between the $k$th mode and the system.

The emission spectrum of the Floquet system can be calculated in the interaction picture of $\hat{H}_{0}(t)$, which we call the Floquet frame. More specifically, the unitary $\hat{U}_{0}(t)$ generated by $\hat{H}_{0}(t)$ is given by

where $\left|\Psi_{\alpha}(t)\right\rangle$ are the Floquet eigenstates of $\hat{H}_{0}(t)$, and $\hat{U}_{0}(t,t_{0})$ satisfies the Schrödinger equation

Now we could perform the unitary transformation $\hat{U}_{0}(t,t_{0})$ to enter the Floquet frame, where the system-bath Hamiltonian becomes

and

with $O_{\alpha\beta}(t)=\left\langle\Psi_{\alpha}(t)\middle|\hat{O}\middle|\Psi_{\beta}(t)\right\rangle$. We set $t_{0}=0$ without loss of generality.

Since the Floquet modes $\left|\Phi_{\alpha}(t)\right\rangle$ are periodic in time, we Fourier expand $O_{\alpha\beta}(t)$:

where $O_{\alpha\beta n}$ together with $\{\varepsilon_{\alpha}\}$ gives the emission spectrum. Since all $\hat{b}_{k}$ modes are in vacuum, transition $\alpha\rightarrow\beta$ from $\left|\Psi_{\alpha}(t_{0})\right\rangle$ to $\left|\Psi_{\beta}(t_{0})\right\rangle$ is only possible if $\varepsilon_{\alpha}-\varepsilon_{\beta}+n\omega>0$, with the transition rate determined by $|O_{\alpha\beta n}|^{2}$ and the bath spectral density at frequency $\varepsilon_{\alpha}-\varepsilon_{\beta}+n\omega$. Furthermore, transition $\alpha\rightarrow\beta$ could emit photons at multiple frequencies, and both $\alpha\rightarrow\beta$ and $\beta\rightarrow\alpha$ could occur, which is different from the relaxation of static systems.

In Fig. 3(a), we plot $|O_{\alpha\beta n}|^{2}$ for various transitions of the Kapitzonium under charge noise with $\hat{O}=\hat{n}$. The dominant heating processes $0\rightarrow 2$ (red cross) and $1\rightarrow 3$ (red dot) occur at near degenerate frequency around $\omega_{02}/2\pi\approx\omega_{13}/2\pi\approx 9.5~{}\text{GHz}$. The reverse cooling processes $2\rightarrow 0$ (blue cross) and $3\rightarrow 1$ (blue dot) are also allowed at a different frequency around $\omega_{20}/2\pi\approx\omega_{31}/2\pi\approx 10.5~{}\text{GHz}$.

Assuming the bath spectral density is flat, we trace out the bath degree of freedoms and derive a master equation for the Kapitzonium. In the Floquet frame, the master equation is $\dot{\hat{\rho}}=\kappa_{h}D[\hat{O}_{-}(t)](\hat{\rho})$ (see Appendix A), where $D[\hat{A}](\hat{\rho})=\hat{A}\hat{\rho}\hat{A}^{\dagger}-\frac{1}{2}\{\hat{A}^{\dagger}\hat{A},\hat{\rho}\}$. Here $\kappa_{h}$ is the heating rate due to intrinsic loss and

where

Without the Floquet driving, $\kappa_{h}$ is approximately the amplitude damping rate of a transmon corresponding to its $1/T_{1}$.

The frequency dependence of the cavity emission suggests that we can enhance a specific transition rate by increasing the bath spectral density at the transition frequency Murch et al. (2012); Putterman et al. (2022). More concretely, we propose to capacitively couple the Kapitzonium to a bandpass filter around 10.5 GHz with 800 MHz bandwidth (Fig. 3(a) grey shade region). The bandpass filter enhances the cooling processes without causing extra heating, which preserves the qubit basis states $\{\left|\Psi_{0}\right\rangle,\left|\Psi_{1}\right\rangle\}$. Furthermore, the environment cannot distinguish whether the emitted photon comes from the $2\rightarrow 0$ or $3\rightarrow 1$ transition since $\omega_{20}\approx\omega_{31}$. Therefore the phase coherence between $\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{1}\right\rangle$ is also preserved by the cooling processes, enabling fully autonomous protection of the qubit subspace.

The bandpass filter can be modeled as a chain of linearly coupled harmonic ocsillators $\hat{a}_{1},...,\hat{a}_{N}$ Putterman et al. (2022). The first filter mode $\hat{a}_{1}$ also couples capacitively to the Kapitzonium via the interaction $g\hat{n}\left(\hat{a}_{1}+\hat{a}_{1}^{\dagger}\right)$. The full Hamiltonian is (Fig. 3(b))

where $\omega_{f}$ is the center frequency of the filter and $J$ is the coupling rate between two adjacent filter modes. The last filter mode $\hat{a}_{N}$ decays into a zero temperature bath at rate $\kappa_{f}$, which is described by the Lindblad dissipator $\kappa_{f}D[\hat{a}_{N}]$.

The qubit subspace is autonomously protected when the engineered cooling rate is sufficiently larger than the intrinsic heating rate $\kappa_{h}$ of Kapitzonium. Following Ref. Putterman et al. (2022), we choose $N=3$, $\kappa_{f}=2J$ and $g=\kappa_{f}/5$, such that the filter bandwidth is $2\kappa_{f}$ and the filter modes are only weakly excited. The cooling rate is about $\kappa_{c}=4g^{2}/\kappa_{f}=4\kappa_{f}/25$ after adiabatically eliminating all filter modes Reiter and Sørensen (2012); Chamberland et al. (2022). For $\kappa_{f}/2\pi=400$ MHz we have $\kappa_{c}/2\pi=64$ MHz. In Fig. 3(d), we plot the average fidelity of Kapitzonium idling for 50 ns with (blue cross) and without (blue dot) the filter for different values of $1/\kappa_{h}$. The cooling enhanced by the filter has reduced the idling infidelity by more than 2 orders of magnitude.

The filter that protects idling may not protect the Kapitzonium gates since the emission spectrum could be different during the gates. The filter performance is determined by two quantities of the emission spectrum. One quantity is the frequency spacing $\mathcal{B}\equiv|(\omega_{20}+\omega_{31})-(\omega_{02}+\omega_{13})|/2$ between the dominant heating and cooling transitions. $\mathcal{B}$ limits the filter bandwidth $\kappa_{f}$ and thus the maximal cooling rate $\kappa_{c}$. The other quantity is the degeneracy of the two heating (cooling) transitions measured by $\Delta\equiv|\omega_{02}-\omega_{13}|=|\omega_{20}-\omega_{31}|$. In the degenerate regime where $\Delta\ll\kappa_{c}$, such as idling with $\Delta/2\pi\approx 0.2$ MHz, the qubit subspace is protected (Fig. 3(c)). In the non-degenerate regime where $\Delta\gtrsim\kappa_{c}$, the environment could distinguish which cooling transition the emitted photon comes from and dephase the qubit (Fig. 3(c)). Therefore the qubit basis states $\left|\Psi_{0}\right\rangle$ and $\left|\Psi_{1}\right\rangle$ are protected but not their coherent superpositions.

For X gate with relatively small $\alpha_{x}/2\pi\approx 5.2$ MHz, the emission spectrum is similar to the idling emission spectrum. Therefore the idling filter also protects the X gate, with $\Delta/2\pi\approx 0.8$ MHz well within the degenerate regime.

For Z gate with $\omega/2\pi=20$ GHz, we have $\mathcal{B}/2\pi\approx 468$ MHz and $\Delta/2\pi\approx 27$ MHz. A different filter is required with center frequency at about 20 GHz. Z gate is unprotected, since the largest possible cooling rate $4\mathcal{B}/25\approx 75$ MHz is comparable with $\Delta$.

The full protection of the X gate can be verified numerically, and the gate infidelity is reduced by about 2 orders of magnitude with the filter (Fig. 3(d) orange diamond and dot). The partial protection of the qubit basis states during the Z gate is verified in Appendix B. In principle, the XX gate should also be fully protected for $\alpha_{xx}$ on the order of a few MHz, since its $\Delta$ and $\mathcal{B}$ are similar to the X gate. However, we didn’t simulate the XX gate due to the high computational cost.

Kapitzonium measurements can be performed in the unprotected regime. By measuring the frequency of the emitted photon, we learn about which transition it comes from and randomly project the system to either $\left|\Psi_{0}\right\rangle$ or $\left|\Psi_{1}\right\rangle$ (Fig. 4(a)). We could apply the X gate but with a much larger $\alpha_{x}$ to implement the X measurement (Fig. 4(b)). Larger $\alpha_{x}$ lifts the degeneracy between $\omega_{20}$ and $\omega_{31}$, and $\{\left|0\right\rangle,\left|\pi\right\rangle\}$ are the measurement basis. On the other hand, the unprotected Z gate dephases the qubit and naturally perform the Z measurement with measurement basis $\{\left|\Psi_{0}\right\rangle,\left|\Psi_{1}\right\rangle\}$ (Fig. 4(b)).

The measurement rate is proportional to the occupation of the excited states $\left|\Psi_{2}\right\rangle$ and $\left|\Psi_{3}\right\rangle$. We could increase the measurement rate by capacitively driving $\left|\Psi_{0}\right\rangle\leftrightarrow\left|\Psi_{2}\right\rangle$ and $\left|\Psi_{1}\right\rangle\leftrightarrow\left|\Psi_{3}\right\rangle$ at the heating transition frequency (Fig. 4(a)). The charge operator in the Floquet frame is

where we only include $\left|\Psi_{0\sim 3}\right\rangle$ terms from Eq. (15) and the dominant Fourier coefficients $n_{02}\approx n_{20}\approx n_{13}\approx n_{31}\approx 1$. We apply a charge drive $2\Omega(\cos(\omega_{d1}t)+\cos(\omega_{d2}t))\hat{n}(t)$ to the Kapitzonium where $\omega_{d1}=\omega_{02}$, $\omega_{d2}=\omega_{13}$ and $\Omega\ll\Delta,\kappa_{c}$. This leads to the coupling $\Omega(n_{02}\left|\Psi_{0}\right\rangle\left\langle\Psi_{2}\right|+n_{13}\left|\Psi_{1}\right\rangle\left\langle\Psi_{3}\right|+\text{h.c.})$ after the rotating wave approximation (RWA). Furthermore, $\{\left|\Psi_{2}\right\rangle,\left|\Psi_{3}\right\rangle\}$ are only weakly excited since $\Omega\ll\kappa_{c}$, which decay back to $\{\left|\Psi_{0}\right\rangle,\left|\Psi_{1}\right\rangle\}$ after the measurement with the charge drive turned off.

We simulate the Kapitzonium measurements with quantum trajectory methods Wiseman and Milburn (2009) (see Appendix C). Starting from an initial state of $(\left|\Psi_{0}\right\rangle+\left|\Psi_{1}\right\rangle)/\sqrt{2}$, we monitor the output field from the filter with a heterodyne measurement for $2~{}\mu\text{s}$, and then calculate the power spectrum $S(\omega)$ for each measurement record $S(t)$. Depending on whether the system is projected into $\left|\Psi_{0}\right\rangle$ or $\left|\Psi_{1}\right\rangle$, $S(\omega)$ show a peak at $\omega_{20}$ or $\omega_{31}$ (Fig. 4(a)). We therefore define the signal as $S=S_{31}-S_{20}$ where $S_{ij}$ is the integrated power within a narrow frequency window around $\omega_{ij}$. $S<0$ or $S>0$ represents a measurement result of 0 or 1. The measurement fidelity is about 99.4%, estimated from a total of 3000 trajectories (Fig. 4(c)). We plot a single trajectory in Fig. 4(d) showing the measurement result $S(t)$, the occupation $p_{i}$ of $\left|\Psi_{i}\right\rangle$ during the measurement for $i=0,1$, and the spectrum $S(\omega)$.

Ideally, the SQUID in Fig. 1(b) should be symmetric to engineer the Kapitzonium Hamiltonian Eq. (2). However in actual experiments, there always exists some amount of disorder which requires a more general treatment of the circuit.

The circuit Hamiltonian in presence of parameter disorder is You et al. (2019); Riwar and DiVincenzo (2022)

where $C_{\Sigma}=C_{1}+C_{2}$ is the total junction capacitance and $E_{C}=e^{2}/2C_{\Sigma}$. To symmetrize the flux allocation inside the two cosines, we move to another reference frame by performing the unitary transformation

where the Hamiltonian becomes

We will drop the unwanted term $\propto\dot{\Phi}_{\text{ext}}(t)\hat{n}$ for now, since in principle it can be compensated for with the gate voltage.

For disorder in junction energies, we define $\delta_{e}=(E_{J1}-E_{J2})/(E_{J1}+E_{J2})$. With the idling flux drive $\phi_{\text{ext}}(t)=\omega t$, the effective Hamiltonian is

Therefore disorder in junction energies reduces the effective $\tilde{E}_{J}$, and for small $\delta_{e}$ on the order of 0.05 this reduction is likely negligible.

Here we discuss the flux control for implementing the Kapitzonium gates and measurements. The flux drives $\phi_{\text{ext}}(t)=\omega t$ (idling) and $\phi_{\text{ext}}(t)=\omega_{z}t$ (Z gate and Z measurement) are not feasible experimentally, since the required bias current grows linearly with time. One natural solution is to use a triangle waveform instead

where $u(t)=\phi_{\text{ext}}(t)\pmod{4\pi}$. However, this flux choice poses inconveniences for implementing the X gate and X measurement.

Alternatively, we could set $\phi_{\text{ext}}(t)=\alpha\cos\omega t$ with

where we have applied the Jacobi–Anger expansion and $J_{n}(x)$ is the $n$-th Bessel function of the first kind. The effective Hamiltonian for Eq. (2) now becomes

Therefore we could choose different $\alpha$ such that $J_{0}(\alpha)=0$ for Kapitzonium idling, Z gate and Z measurement, and $J_{0}(\alpha)\neq 0$ for X gate and X measurement.

We first estimate the coherence properties of the Kapitzonium, while neglecting any nonidealities due to flux, quasiparticle and offset charge noise. In this idealized model, both bit-flip rate $1/T_{1}$ and phase-flip rate $1/T_{2}$ of the Kapitzonium without cooling are 0, since at any time $t$

for $\alpha,\beta\in\{0,1\}$. This is because the Kapitzonium Hamiltonian (Eq. (3)) is symmetric under the transformation of $\hat{n}\rightarrow-\hat{n}$, and both $\left|\Psi_{0}(t)\right\rangle$ and $\left|\Psi_{1}(t)\right\rangle$ have even parity under this transformation. More generally with a static offset charge, numerical results suggest that both bit-flip and phase-flip rates are exponentially small in $E_{J}/\omega$ (see Appendix D). Due to the Floquet driving, the dominant error is the leakage from the qubit subspace, e.g., from $\left|0\right\rangle(\left|\pi\right\rangle)$ to $\left|\psi_{2}\right\rangle(\left|\psi_{3}\right\rangle)$. This leakage rate $1/T_{l}$ is proportional to $\kappa_{h}$.

The filter implemented in Sec. III suppresses the leakage rate by $\kappa_{h}/\kappa_{c}$ so $1/T_{l}\propto\kappa_{h}^{2}/\kappa_{c}$. Such a cooling process induces dephasing error due to the frequency difference $\Delta$ in the emitted photons, with a rate of $1/T_{2}\propto\frac{\kappa_{h}}{2}\left(\frac{\Delta}{\kappa_{c}}\right)^{2}$. In the deep transmon regime $E_{J}\gg\omega$, we have $\Delta\propto\exp(-\frac{\pi^{2}E_{J}}{8\sqrt{2}\omega})$ and the dephasing rate is exponentially small in $E_{J}/\omega$.

In summary, the bit and phase flip rates are exponentially suppressed, while the leakage rate is reduced by the filter factor $\kappa_{h}/\kappa_{c}$. Below, we give a preliminary discussion of how other noise processes that affect superconducting circuits further reduce the coherence of the Kapitzonium.

Flux noise. The flux imposed on the loop will be invariably accompanied by additional flux noise $\delta\phi$, which typically has a $f^{-\alpha}$ spectral density and thus a small spectral weight at high frequencies. The effects of $\delta\phi$ depend on how we choose to drive the flux. For $\phi_{\text{ext}}(t)=\omega t+\delta\phi\pmod{4\pi}$, $\delta\phi$ corresponds to a shift in time which does not change the Floquet eigenenergies. Since the idling, Z gate and Z measurement can be implemented in this way (Sec. IV.2), they are robust to flux noise. However, experimental constraints in implementing a flux drive, as well as the proposed approach to X gate and X measurement, make other driving schemes such as $\phi_{\text{ext}}(t)=\alpha\cos\omega t$ (Sec. IV.2) more attractive. In this case, flux noise leads to an imposed flux of $\phi_{\text{ext}}(t)=\alpha\cos\omega t+\delta\phi$ where $\delta\phi$ has a nontrivial effect on the effective Hamiltonian.

Flux sensitivity at arbitrary flux bias occurs in other protected qubits subject to experimental constraints. For example it is also present in the static soft $0-\pi$ qubit Groszkowski et al. (2018); Gyenis et al. (2021b). Nonetheless, as in the case of the soft $0-\pi$ qubit Gyenis et al. (2021b), it is possible to operate at flux sweet-spot where the dephasing rate is protected to first order against flux noise. For a Kapitzonium with a noisy periodic flux drive $\phi_{\text{ext}}(t)=\alpha\cos\omega t+\delta\phi$, the effective Hamiltonian is also insensitive to first order to $\delta\phi$ around $\delta\phi=0$, and leading order terms scale as $\delta\phi^{2}$. Therefore operating at the sweet spot $\delta\phi=0$ can significantly reduce errors of X gate and X measurement.

Quasiparticle noise. Quasiparticle tunneling changes the charge parity. This may induces both energy relaxation and dephasing Lutchyn et al. (2005, 2006); Koch et al. (2007); Martinis et al. (2009); Catelani et al. (2011); Serniak et al. (2018) of the Kapitzonium. Here we note that in a highly simplified model of quasiparticle dynamics, the frequency shifts in the $\{\left|0\right\rangle,\left|\pi\right\rangle\}$ basis induced by quasiparticles are proportional to $\left\langle 0\middle|\sin\frac{\hat{\varphi}_{k}}{2}\middle|0\right\rangle\approx(-1)^{k-1}\sin\frac{\phi_{\text{ext}}(t)}{2}$, where $k=1,2$ is the index of the junction, and $\left\langle\pi\middle|\sin\frac{\hat{\varphi}_{k}}{2}\middle|\pi\right\rangle\approx\cos\frac{\phi_{\text{ext}}(t)}{2}$. Therefore for $\phi_{\text{ext}}(t)=\omega t$, $\left|0\right\rangle$ and $\left|\pi\right\rangle$ on average don’t accumulate extra phase and should be robust to quasiparticle tunneling . This analysis may not hold when $\phi_{\text{ext}}(t)=\alpha\cos\omega t$. A more accurate model of quasiparticle tunneling is also likely to be needed to better understand the effects of quasiparticles in the Kapitzonium. In particular, the modulation drive itself can cause photon-assisted effects Houzet et al. (2019). We leave such a detailed study to future work.

Offset charge noise. In addition to quasiparticle tunneling which flips the parity of gate charge, the Kapitzonium, like the charge qubit from which it is derived, suffers from the continuous variations in the gate charge offset parameter due to uncontrolled fluctuations in the background electric fields. This can be modeled by replacing $\hat{n}$ with $\hat{n}-n_{g}$ in the Kapitzonium Hamiltonian where $n_{g}$ is the random offset charge.

The Floquet eigenenergies weakly depend on $n_{g}$ during idling (Fig. 4(e), left figure). The weak dependence also holds when adding $\alpha_{x}\cos\hat{\phi}$ to the Hamiltonian. Therefore idling, X gate and X measurement are all insensitive to charge noise. On the other hand, Z gate and Z measurement are not robust to charge noise since $\varepsilon_{1}-\varepsilon_{0}$ and the emitted photon frequency $\omega_{20},\omega_{31}$ depend on $n_{g}$ (Fig. 4(e), middle and right figures). For example, at $n_{g}=0.5$ we have $\varepsilon_{1}=\varepsilon_{0}$ and $\omega_{20}=\omega_{31}$. Therefore both Z gate and Z measurement cannot be performed at $n_{g}=0.5$.

Finding ways to work around these challenges as well as other issues which we may learn about from experiments (such as photon-assisted tunneling and unforeseen effects due to the drive), will be the subject of future work.