Enhancing quantum coherence of a fluxonium qubit by employing flux modulation with tunable-complex-amplitude

We propose to protect fluxonium qubits that are away from half flux quantum against environmental noises, especially 1/f flux noise, by adopting a modulated flux with tunable-complex-amplitude. Using open-system Floquet theory, we derive a Lindblad equation and extract decoherent rates for pure-dephasing, excitation and relaxation. After examining intrinsic attributes of the flux driven fluxonium qubit, we put forward an analytic manner to locate dynamical sweet spots for fast and weak driving. Continuous families of dynamical sweet spots are found in the parameter plane of relative amplitude factor and relative phase. Around a family of dynamical sweet spots or between two families of dynamical sweet spots, there exist continuous regions with long coherent times that exceed 100 μs . Taking advantage of the two noise-insensitive channels: relative amplitude factor and relative phase, a flux driven fluxonium qubit can become immune to flux noises from both the dc and ac flux amplitudes. And the optimal driving amplitudes are no longer isolated at a certain driving frequency, but become continuous. This is in sharp contrast to the usual schemes based on flux modulation with real-amplitude. As a result, there are plenty of manipulating flexibility in our flux driving scheme with tunable-complex-amplitude, which may be useful in logical operations among flux driven fluxonium qubits or other flux qubits.


Introduction
Rapid advances in superconducting quantum computation have ushered us into the noisy intermediate-scale quantum era [1]. We are fortunate enough to witness quantum computational advantages in problems of quantum random circuits sampling at superconducting platforms of Google's sycamore and Chinese Zuchongzhi as well as Bose sampling in Jiuzhang photonic quantum computer [2][3][4][5][6]. Nowadays scaling up superconducting quantum processors is becoming a main focus of much attention in the field of quantum information science [7][8][9]. As quantum decoherence of qubits will result in errors during quantum information processing, understanding and preventing decoherence have all the time been the most central mission on roads to this challenging goal [10][11][12][13].
Inspired by decoupling techniques from high-resolution nuclear magnetic resonance spectroscopy [29], diverse DD pulses sequences are proposed to overcome decoherence errors, such as Hahn spin-echo [30], Carr-Purcell-Meiboom-Gill echo [31], Uhrig DD [32], concatenated DD and periodic DD [33]. Pulses in DD have to be short and strong enough and their corresponding control procedures are also sophisticated [34]. Therefore, continuously driven fields are naturally introduced to decouple qubits with their surrounding noises [35]. Continuous driving fields can also be used to implement quantum error correction [36,37]. Another benefit of continuous driving is that it can be easily merged into logical operations in quantum information processing [38]. Continuous DD has been suggested to be beneficial to defending tunable transmon qubits from 1/f flux noise [39,40]. Continuously bichromatic flux control has been applied to reduce error rates and delay times of parametric entangling operations between transmon qubits [41]. Floquet flux modulations can offer hope to raise fidelities of some entangling gates that are accomplished away from the half flux [42]. Dynamical sweet spots that are first-order insensitive to 1/f noise from dc flux are created in both theory and experiment by using Floquet flux modulation with real-amplitude in fluxonium circuits and encoding qubits into time-dependent states [43,44].
In the Floquet flux driving scheme with real-amplitude, a fluxonium qubit is extremely sensitive to 1/f noises in ac flux amplitude and the optimal driving amplitudes are usually isolated for a fixed diving frequency [44], which greatly limits the selections of flux driving parameters and thus renders its experimental implementation challenging. Hence, we propose a Floquet flux driving scheme with tunable-complex-amplitude to lift this kind of restriction, and study quantum coherence in the flux driven fluxonium qubit based on open-system Floquet theory [45,46]. We put forward an analytic approach to locate dynamical sweet spots for fast and weak flux driving. In our flux modulating scheme, a flux driven fluxonium qubit become insensitive to 1/f noises in both dc and ac flux amplitudes and optimal driving amplitudes are no longer isolated but continuous for a certain driving frequency. Furthermore, continuous families of dynamical sweet spots will appeal in the parameter plane of relative amplitude factor and relative phase. Around a family of dynamical sweet spots or between two families of dynamical sweet spots, there are continuous regions with long coherent times (>100 µs). However, for fluxonium qubits with fast and high-fidelity logical gates implemented in experiment, their coherent times are less than 100 µs [47].
This paper is organized as follows. In section 2, we present a detailed derivation of the effective Hamiltonian of the flux driven fluxonium qubit, its quasi-energies and eigenstates. In section 3, we introduce open-system Floquet theory to derive decoherent rates for pure-dephasing, excitation and relaxation induced by 1/f flux noise and dielectric loss. In section 4, we demonstrate advantages of the flux modulation with tunable-complex-amplitude and discuss families of dynamical sweet spots, noise-insensitive channels. We conclude the paper in section 5. Technical details can be found in appendices.

Fluxonium qubit with modulated external flux
We consider a fluxonium circuit with a superconducting loop biased by a modulated external magnetic flux Φ ext (t) (see figure 1). Its Hamiltonian can be described as [48] where E C , E L and E J represent capacitive, inductive and Josephson energies respectively, n = Q/(−2e) is the Cooper-pair number operator, ϕ = 2πΦ/Φ 0 stands for the reduced flux operator. Here Φ 0 = h/(2e) is the flux quantum with Planck constant h = 2πℏ and an elementary charge e. Charge operator Q and flux operator Φ obey commutation relation: The reduced external-magnetic flux ϕ ext (t) = 2πΦ ext (t)/Φ 0 = ϕ dc + ϕ ac (t) including a dc part ϕ dc and an ac modulating part = ϕ 0 1 + sin(2α) cos(θ) cos(w d t +θ) with relative amplitude factor α ∈ (0, π), relative phase θ ∈ (0, π) and tunable-complex-function F(α, θ) = 1 + sin(2α) cos(θ)e −iθ . Definition of the effective phaseθ can be founded in equation (C.7). When the real modulating amplitude ϕ 0 = 0, the external flux ϕ ext becomes static and effective Hamiltonian with transition energy ∆ = E 1 − E 0 of a static qubit at ϕ dc = π, B = 2E Lφ01 (ϕ dc − π) related with the dc bias, ac driving amplitude A(t) = E Lφ01 ϕ ac (t) andφ 01 = |⟨ψ 0 |ϕ|ψ 1 ⟩|. So for a fluxonium qubit without ac flux modulating, its transition energy is Ω = √ ∆ 2 + B 2 . Due to periodicity in the time-dependent Hamiltonian: H q (t) = H q (t + T) with T = 2π/w d , we can acquire its eigenvalues and eigenstates by using Floquet theory [49][50][51]. Usually, we need to numerically diagonalize a Floquet Hamiltonian H F with matrix elementary blocks (see equation (C.5)) with real driving amplitude A = E Lφ01 ϕ 0 . In addition, K max should be large enough so that the numerical eigenvalues are equidistant and satisfy ϵ j ± − ϵ j−1 ± = ℏw d guaranteed by time translation invariance of H q (t) (see figure 2(b)). More details can be found in C. In numerical calculations, we set that K max = 50. So for the driven fluxonium qubit, its computation basis vectors can be selected as with Floquet modes in which we have ignored the symbol 'j = 0' for simplicity, Floquet modes |ũ ± (t)⟩ are closely related with the eigenstates |U j=0 ± ⟩. State evolutions of this driven fluxonium qubit are governed by the unitary operator Its transition energy can be defined asΩ whose value at certain driving amplitude and frequency is plotted in figure 2(a) and can change gently no matter when it moves along the θ or α axis except for α = 0, π/2, π and θ = π/2. Figure 2(b) displays its Floquet quasi-energy spectra varying with relative phase θ at relative amplitude factor α = 0.25π.

Quantum coherence of a fluxonium qubit under flux modulation with tunable-complex-amplitude
For a realistic qubit, its quantum coherence is inevitably weakened by various noises, which are usually viewed as a bath in thermal equilibrium ρ B . The composite system under consideration is composed of a flux driven fluxonium qubit and a bath. Its Hamiltonian can be expressed as with in which H B is single-particle Hamiltonian of the bath containing operator β, H int represents coupling between the fluxonium qubit and bath, which causes decoherence in a qubit. It is reasonable to assume that this coupling H int is weak enough that it will not affect thermal equilibrium state ρ B of the bath. Quantum coherence of a driven fluxonium qubit can be analyzed by utilizing open-system Floquet theory [45,46]. In interaction representation, the time evolution operator U(t) can be defined as and interaction Hamiltonian becomes with β(t) = U † B (t)βU B (t) and σ z (t) = U † q (t)σ z U q (t). Combining equation (8), the time dependent quasi-spin operator σ z (t) can be written out in a new form with redefined quasi-spin operatorτ ν =τ ν (0) (ν = z, ±) and On account of periodicity of the Floquet modes |ũ ± (t)⟩, We can have Fourier expansions and anti-Fourier expansions which are bound up with |ũ To inspect quantum coherence of the driven fluxonium qubit, we can investigate evolutions of its density matrixρ q (t) with the aid of Redfield equation [52]. Followed by rotating-wave approximation in frequency domain (see more details in appendix D), we find that evolutions ofρ q (t) satisfy a Lindblad equation with Lamb shift HamiltonianH LS defined in equation (D.13), damping superoperator The coefficientsK ν are related with decoherent rates and have following concrete forms in which a z = 4, a ± = 1, ν | 2 with f k,ν (w) = t π sinc (w + w k,ν )t , w k,± = ±Ω/ℏ − kw d , w k,z = −kw d and sinc(x) = sin(x)/x. S(w) is the noise spectral density. When the interested time t ≫ 1/w k,ν , f k,ν (w) serves as a delta function δ(w + w k,ν ). χ ν (w, t) functions as a frequency filter like that in dynamic decoupling and multi-photon transitions in the Floquet driving play roles of pulse sequences in DD [53][54][55]. External noises with w = −w k,ν will lead to decoherence. If k = 0, then w = −w 0,ν and coefficientsK ν reduces to that of a static qubit with |g τ ]| 2 (see equations (B.10) and (B.11)). Multi-photon transitions with sidebands w = −w k,ν (k ̸ = 0) make it possible to protect quantum coherence by means of Floquet modulations. For a fluxonium qubit, there exist two main noises: 1/f flux noise and dielectric loss, whose noise spectral densities are given by [56] containing Boltzmann constant k B and temperature T . Here we assume the loss tangent tan(δ C ) = 1.1 × 10 −6 , flux amplitude δ f = 1.8 × 10 −6 and temperature T = 15 mK [56]. Total noise spectral density under consideration is S(w) = S f (w) + S d (w). It is reasonable to make approximation in long time evolution limit: t ≫ 1/w 0 , when there is no divergence in S(w) around w = w 0 . We adopt steps displayed in appendix B and derive decoherent rates for pure-dephasing, excitation and relaxatioñ in which we consider an infrared cutoff w ir = 1 Hz that is much smaller than the inverse of characteristic time t m . The factor | ln(w ir t m )| can be found to be close to 4 [57,58]. Similar to that of a static fluxonium qubit, the decoherent rate of pure-dephasingγ z can be proven to connect tightly with differential of transition energyΩ to dc flux bias ϕ dc or B by exploiting perturbation method. A small change δB will result in a deviation in the transition energy, to first order, the transition energy shift reads which is equivalent to At an extreme point, ∂ BΩ = 0, thus we have that g z = 0, around which the dynamical sweet spots emerge [39,40,43,44]. Decoherence rate of pure-dephasingγ z will reach a minimal value at a dynamical sweet spot. As complex relationship exists between pure-dephasing rateγ z and Fourier coefficients g [k] z (see equations (21) and (22)), g z = 0 usually can not sufficiently guarantee a minimal pure-dephasing rate, nevertheless, it is really helpful for physically judging a dynamical sweet spot [44].

Families of dynamical sweet spots
For a fluxonium qubit with static flux, when the flux diverges from its static sweet spot, its coherent time of pure-dephasing will decrease exponentially (see figure B1 in appendix B). To protect quantum coherence in this case, a Floquet flux modulating scheme with real amplitude was proposed and the quantum coherence can be enhanced when operating at its dynamical sweet spots [44]. However, this kind of flux modulation can not reduce the sensitivity of a fluxonium qubit to 1/f flux noises stemming from the ac flux amplitude (see the blue line in figure 3(a) and table 1). So, we advise to employ a flux modulation with tunable-complex-amplitude to improve its resistance to noises in ac flux amplitude by introducing another two adjust parameters: a relative amplitude factor α and a relative phase θ. To acquire more intuitive awareness, we plot coherent times varying along with ac flux amplitude ϕ 0 in case of dc flux ϕ dc = 1.03π and modulating frequency w d = 0.98Ω/ℏ in figure 3. From this figure, we can realize that: when α = 0, relative phase θ is trivial and our flux modulation will reduce to that with real amplitude. The pure-dephasing coherent timeT ϕ is maximum at an isolated dynamical sweet spot ϕ 0 ≈ 0.01941π and will decrease exponentially if it is far away from this point (see the blue line in figure 3(a) and table 1). While, when α = π/4, the pure-dephasing coherent time can be largely lengthened through adjusting relative phase θ and a continuous range of sweet spots appear (see the red line in figure 3(a)). In the continuous range, coherent time of polarization almost remain fixed at 718 µs (see the red line in figure 3(b)). In a word, the flux modulating scheme with tunable-complex-amplitude can not only make a fluxonium qubit less sensitive to flux noises, but also gives us more choices in selection of optimal driving parameters so that we can simultaneously have long coherent times of pure-dephasing and polarization.
To elaborate advantages of flux modulation with tunable-complex-amplitude over that with real amplitude [44], we also need to prove that a fluxonium circuit driven by an external flux with tunable-complex-amplitude is less sensitive to noises coming from fluctuations of the relative amplitude factor α and relative phase θ. We will study its quantum decoherence in cases of fast and slow flux modulations and find out an analytical method to identify the optimal modulating parameters as well as continuous families of dynamical sweet spots in the parameter plane of relative amplitude factor α and relative phase θ.

fast flux modulation
In our flux driving scheme with tunable-complex-amplitude, Floquet eigen-spectra of a driven fluxonium qubit can be controlled by external modulating parameters, such as driving frequency w d , dc flux bias ϕ dc (or B), ac flux amplitude ϕ 0 (or A), relative amplitude factor α and phase θ, which construct a five dimensional hyper-plane-space. It is highly complicated and nearly impossible to forecast dynamical sweet spots in this 5D space. To a certain extent, it is bewildering and hard to address the optimal operating points in advance for concrete modulating parameters. However, for realistic implementation, it would be helpful to strengthen such theoretical prediction capability, and we manage to make it more predictable by making use of the intrinsic attributes in this flux driven fluxonium qubit with Floquet Hamiltonian H F (α, θ) (see equation (E.3)). When the driving frequency is close to transition frequency, i.e. w d ∼ Ω/ℏ, Floquet Hamiltonian H F (α, θ) can be analytically diagonalized with perturbation theory elucidated in appendix E, from which we are aware of that in weak driving (2Ã ≪ ℏw d ) the transition energyΩ approximates and its partial differential to B is Here will reduce to be the transition energy Ω of a static fluxonium qubit at A = 0 (see equation (B.2)),Ã = A 1 + sin(2α) cos(θ), J 0/1 (x) are the first kind of Bessel functions. So sweet spots should meet that B = 0 (static sweet spots) or This equation manifests that driving frequency w d should be less than the transition frequency Ω/ℏ of a static fluxonium qubit to reach a dynamic sweet spot. Therefore we are equipped with the ability to predict locations of dynamical sweet spots in fast and weak driving.  For our flux modulation with tunable-complex-amplitude, its effective driving amplitude is A = A 1 + sin(2α) cos(θ). It can turn into the flux modulation with real-amplitude in following two cases: (a) When θ = π/2, |F| = 1 andÃ → A. Relative amplitude factor α becomes a global phase that can be eliminated through a unitary transformation. So at this special case, changing α does not affect physical characteristics; (b) when α = 0, π/2, π, |F| = 1 andÃ → A. Relative phase θ now also does not affect the physics. In a flux driving scheme with real-amplitude [44], a dynamical sweet spot may satisfy equation (26) with driving amplitudeÃ replaced by A. So for a certain driving frequency w d ∼ Ω, equation (26) only has a solution A = A 0 with 2A 0 ≪ ℏw d , which is in the range of weak driving (see figure 4(a)). However, additional modulating parameters α and θ in our flux driving with tunable-complex-amplitude can make choices of driving amplitude A more flexible because we can always makeÃ = A 0 to be valid if the driving amplitude Thus, compared with flux modulation with real-amplitude, our flux modulating scheme can offer continuous rather than isolated optimal driving amplitudes at a certain driving frequency and larger flexibility in selection of optimal driving frequency w d and driving amplitude A (see region between the two lines y = O(Ã, w d )/Ω| min/max in figure 4(a)).
In order to fully illustrate advantages of flux driving with tunable-complex-amplitude, we first select a dc flux ϕ dc (for example, ϕ dc = 1.03π) that is around the static sweet spot ϕ dc = π, and then set driving frequency w d ∼ Ω and appropriate driving amplitude A that meet the constraint of equation (26). Next, we calculate quantum coherent times of pure-dephasing and depolarization:T ϕ = 1/γ z andT 1 = 1/(γ + +γ − ) using equation (21) for various θ and α. As shown in figure 4(b), the entire parameter space of α − θ can be divided into four parts according to plus or minus sign of (sin(2α), cos(θ)), which can be marked as (+, +), (−, +), (−, −), (+, −). For flux driving frequency w d = 0.98Ω/ℏ, we compute pure-dephasing and depolarization coherent times (T ϕ andT 1 ) in plane of α − θ in cases of driving amplitude A = 0.17Ω, 0.5Ω (see the two red dots in figure 4(a)). When |F| = 1, its corresponding optimal driving amplitude is A 0 ≈ 0.239Ω. Form figure 5, we can learn that in plane of α − θ, changes of depolarization coherent timeT 1 are relatively gentle ranging from 493 µs to 718 µs. While pure-dephasing coherent timeT ϕ grows to be maxima at a (inverse) U-shaped pattern determined by function |F| (see figure 4(b)), which we named as a family of dynamical sweet spots. It rapidly decreases when tuning relevant parameters away from this curve. Thus decoherence of depolarization is comparatively stable in our flux modulation and more attentions should be focused on decoherence of pure-dephasing. In figure 5(a), A = 0.17Ω < A 0 , |F| must be larger  To deeply understand the fundamental relation between pure-dephasing and differential of transition energy to dc flux ∂ BΩ in the flux driven fluxonium qubit, we plot behaviors of ∂ BΩ varying along with relative phase factor θ in figure 6(a). From it three essential points can be extracted: (a) Zero points of ∂ BΩ turn out to be the dynamical sweet spots. (b) In regions where the pure-dephasing coherent timeT ϕ >100 µs, it can be guaranteed that |∂ BΩ | < 0.01. Hence the continuous regions with relatively long coherent time (>100 µs) can be determined by ensuring a region has a small differential |∂ BΩ | < 0.01, which gives us another way to determine appropriate parameter regimes for the relative amplitude factor α and relative phase θ. Existence of continuous regions with long coherent times indicates that a flux driven fluxonium qubit can be less sensitive to noises induced by fluctuations of α and θ than to that by fluctuations of the ac flux amplitude ϕ 0 (or A). We can succeed to protect a flux driven fluxonium qubit from affecting by flux noises.

slow flux modulation
When flux driving frequency becomes much smaller than the transition frequency Ω/ℏ, we can no longer use equation (26) to locate dynamical sweet spots. To explain advantage of flux driving with tunable-complex-amplitude at small driving frequencies and large driving amplitudes, we need to calculate zero-points of ∂ BΩ at |F| = 1 and find its corresponding dynamical sweet spots by numerical difference method.
In figure 6(b), we take driving frequency w d = 0.74Ω/ℏ as an example and plot ∂ BΩ vs driving amplitude A. From it, we know that five isolate dynamical sweet spots A 1 − A 5 exist for flux driving with real-amplitude. In the case of flux driving with tunable-complex-amplitude, dynamical sweet spots can be found at driving amplitudes that satisfyÃ = A i (i = 1, 2, . . . , 5). So the optimal driving amplitudes are in range [A 1 / √ 2, +∞) and dynamical sweet spots are continuous but not isolated. We also calculate pure-dephasing coherent timẽ T ϕ at four driving amplitudes (see red dots in figure 6(b)), which are plotted in figure 7. Pure-dephasing coherent timeT ϕ turns out to be maximal at families of dynamical sweet spots whose locations in plane of α − θ can be determined by equationÃ = A i (i = 1, 2, . . . , 5). With increasing driving amplitude A, more and more families of dynamical sweet spots will appear, whose numbers and distributions depend on the relative magnitude between driving amplitude A and dynamical sweet spots A i at |F| = 1. For example, when , only a family of dynamical sweet spots appears in region of (+, +) or (−, −) (see figure 7(a)); for there are a family of dynamical sweet spots in region of (−, +) or (+, −) and another family of dynamical sweet spots in region of (+, +) or (−, −) (see figure 7 2Ω < A 4 , three families of dynamical sweet spots appear in region of (−, +) or (+, −) and two families of dynamical sweet spots come out in region of (+, +) or (−, −) (see figure 7(c)); at last, when A = 3.2Ω > A 5 , five families of dynamical sweet spots arise in region of (−, +) or (+, −) (see figure 7(d)).
Around a family of dynamical sweet spots, there is a continuous region with comparatively large pure-dephasing coherent time (T ϕ >100 µs) in plane of α − θ (see figures 7(a) and (b)). Moreover, there may be a continuous region between two families of dynamical sweet spots that supports long pure-dephasing coherent time (see figures 7(c) and (d)). These phenomena are directly related with changing rates of the transition energyΩ with respect to B (see figure 8(a)), whose values are usually small (<0.01) in this kind of continuous regions. When moving away from dynamical sweet spots in plane of α − θ, decrease of coherent  timeT ϕ can sometimes be much slower than that of a static fluxonium qubit (see figures A3 and B1). This difference reveals that changing rates of transition energyΩ respect to B along with parameters θ and α are smaller than that along with dc flux ϕ dc or B, which can be comprehended from the perspective that contributions of θ and α to |∂ BΩ | have been rescaled by first kind of Bessel functions J 0/1 (x) and triangle-functions sin(x)/ cos(x) unlike that of driving frequency w d and dc flux ϕ dc (or B) (see equation (25)). We can regard θ and α as noise-insensitive channels and consider dc flux bias ϕ dc and driving frequency w d as noise sensitive channels. In principle, fluctuations in driving frequency w d of Floquet flux modulations can be diminished by frequency filters. These noise-insensitive channels on one hand can help to improve immunity of a flux driven fluxonium qubit to 1/f flux noises, on the other hand, can make our flux modulating scheme more flexible and effective. Existence of continuous regions with large pure-dephasing coherent time (T ϕ >100 µs) in controllable parameter space of α − θ can help to relax the exacting requirements between driving frequency w d and driving amplitude A in flux modulating scheme with real-amplitude [44]. Larger driving amplitude A will stimulate higher-order Fourier coefficients g [k] z as illustrated in figure 8(b) and give rise to appearance of more families of dynamical sweet spots.

Conclusion
In this paper, we have proposed a flux modulation scheme with tunable-complex-amplitude to protect a fluxonium qubit from external noises, especially the 1/f flux noise. For the driven fluxonium qubit, its computation basis vectors can be the dynamical steady states |ψ ± (t)⟩ in equation (6). By adjusting flux modulating parameters, such as driving frequency, driving amplitude, relative amplitude factor and relative phase, the driven fluxonium qubit can operate at dynamical sweet spots even its dc bias flux is not at a half flux quantum. In our flux modulation with tunable-complex-amplitude, optimal driving amplitudes are continuous rather than isolate owing to existence of additional modulating parameters: relative amplitude factor α and relative phase θ. Such multiple dimensional parameter space can host plenty of families of dynamical sweet spots. Around a family of dynamical sweet spots or between two families of dynamical sweet spots, there exist continuous regions with large pure-dephasing coherent time that exceed 100 µs. Accordingly, our flux modulation with tunable-complex-amplitude can be used to protect a fluxonium qubit from flux noises in noise sensitive channels: dc flux amplitude ϕ dc and ac flux amplitude ϕ 0 . This flux modulating scheme may help to improve fidelities of entangling logic gates that deviating from the half flux bias.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix A. Fluxonium qubit with statically external-magnetic flux
A fluxonium circuit consists of a Josephson junction with Josephson energy E J and charging energy E C shunted by a superinductor with inductive energy E L formed from an array of moderate-area (≈ 1 µm 2 ) Josephson tunnel junctions. The superconducting loop can be biased with an external-magnetic flux Φ ext making its energy spectrum tunable (see figure A1(a)). Different from other inductively shunted junction devices, parameters of a typical fluxonium circuit should satisfy E L ≪ E J and 1 ≲ E J /E C ≲ 10. Effective Hamiltonian of this fluxonium circuit is [59] with Cooper-pair number operator n = Q −2e , reduced flux operator ϕ = 2π Φ Φ0 and flux quantum Φ 0 = h 2e . h = 2πℏ is the Planck constant and e stands for an elementary charge. Charge operator Q and flux operator Φ behave as the momentum and position operators. So they obey commutation relation: [Φ, Q] = iℏ, that is, [ϕ, n] = −i. The first term in Hamiltonian H 0 plays role of kinetic energy and the last two terms constitute corresponding potential energy V(ϕ), which is non-linear due to existence of Josephson junction tunneling. Eigen-system of this fluxonium circuit can be obtained in Fock states space: {|m⟩} +∞ m=0 with creation operator b † [60]. As its Hamiltonian H 0 in equation (A.1) presents in quadratic form of charge number operator n and reduced flux operator ϕ like that of a harmonic oscillator in one dimension, we make following transformations using bosonic operators b and b † . H 0 then turns into  which can be diagonalized numerically with appropriate cutoff in m. Thus its eigen-energies E j and corresponding eigen-states |ψ j ⟩ satisfy and j = 0, 1, 2, . . .. Flux ϕ can induce transitions between eigenstates |ψ i ⟩ and |ψ j ⟩ with transition strength determined by flux matrix elements ϕ ij = ⟨ψ i |ϕ|ψ j ⟩, whose concrete form is Josephson energy E J weakens linearity of Hamiltonian H 0 . To characterize its anharmonic eigen-spectra, we define the anharmonicity η as By making a substitution ϕ → ϕ + ϕ ext in equation (A.1), we can easily realize that Hamiltonian H 0 is a periodic functional of external flux ϕ ext . Figure A1(a) shows the energy spectra of a fluxonium circuit in a period and explains the tunability in eigen-energies through manipulating external flux ϕ ext , which determines the relative position of potential wells in V(ϕ). When ϕ ext = 0, there is only a potential well and no flux tunnelings for low energy physics and V(ϕ) is symmetric around ϕ = 0 (see figure A1(b)). At ϕ ext = π, V(ϕ) turns into a symmetric double-well with symmetry axis ϕ = π and flux tunnelings occur between adjacent potential wells (see figure A1(c)). If external flux ϕ ext takes other values, symmetry of the potential energy V(ϕ) breaks down as well as parity-forbidden transition of dipole ϕ 02 (see figure A2(a)). So ϕ ext = 0 and ϕ ext = π are two specific symmetry points for a fluxonium circuit (see also figure A2). As shown in figure A2(b), anharmonicity η ≫ 1 when external flux ϕ ext ∼ π. In this range, the high energy bands E j with j > 1 are well-isolated from the lowest two energy levels E 0 and E 1 . It is quite valid to ignore all excited energy levels except for E 0 and E 1 . In this way, we can get an ideal two-level system, which forms a fluxonium qubit. We next derive effective Hamiltonian of the fluxonium qubit. Eigen-energies and corresponding eigen-states at ϕ ext = π are labeled asẼ j and |ψ j ⟩. Due to high anharmonicity η ≫ 1 near ϕ ext = π, we can safely reduce the original Hilbert space into a two-dimensional one Γ q : {|ψ 0 ⟩, |ψ 1 ⟩}. Under this approximation, Hamiltonian of a fluxonium qubit can be written as with energy gap ∆ = E 1 − E 0 , pseudo-spin operator σ z = |ψ 1 ⟩⟨ψ 1 | − |ψ 0 ⟩⟨ψ 0 | and complex number ϕ 01 = ⟨ψ 0 |ϕ|ψ 1 ⟩ =φ 01 exp(−iχ). We have used relationship |ψ 0 ⟩⟨ψ 0 | + |ψ 1 ⟩⟨ψ 1 | = 1 and ignored constant energy terms in above equation. Applying another transformation we can acquire that with D = 2E L (ϕ ext − π)φ 01 and pseudo-spin operator σ x = |ψ 0 ⟩⟨ψ 1 | + |ψ 1 ⟩⟨ψ 0 |. For a flux-type qubit, it is usual a common choice to make transformations [51]: σ z → σ x , σ x → −σ z , that is rotating along y-axis by π/4, so we obtain that whose eigenergy gap is Ω = √ ∆ 2 + D 2 (see figure A3). Usually, a fluxonium circuit would be affected by external bath with Hamiltonian H B . Interaction between them is assumed to be in which β stands for bath operators. In space Γ q , it turns to be After making a transformation like that in equation (A.8) and rotating along y-axis by π/4, interaction between a fluxonium qubit and bath becomes where we have redefined the bath operator β → −φ 01 β.

Appendix B. Quantum coherence of a static fluxonium qubit
In this section, we consider depolarization and pure-dephasing in a static fluxonium qubit caused by environmental noises such as 1/f flux noise and dielectric loss [56,61,62]. Hamiltonian of a static fluxonium qubit affected by an external bath is where H q and H B denote single-particle Hamiltonian of the fluxonium qubit and bath, H int represents the interaction between them, β f and β d in qubit-bath coupling are bath operators related with 1/f flux noise and dielectric loss. The static fluxonium qubit has following eigenenergies We now turn to interaction picture for convenience. Then coupling between the fluxonium qubit and bath should be and σ z (t) = e iHqt/ℏ σ z e −iHqt/ℏ which is assumed to be weak enough so that the bath can always stay in thermal equilibrium with density matrix ρ B satisfying [H B , ρ B ] = 0. Evolution of density matrix ρ q (t) of the static qubit can be determined by using Redfield equation [52] where we have defined self-correlation functional C(t) related with spectral density S(w) of quantum noise through a Fourier transformation [63] Substituting equation (B.5) into equation (B.6) and making rotating-wave approximation by neglecting all terms containing e iδt with δ ̸ = 0, we can get that and coefficients K ν (t) (ν = z, ±) related with decoherence of pure-dephasing, excitation and relaxation with function sinc(x) = sin(x)/x. We consider two major noises in a fluxonium qubit: 1/f flux noise and dielectric loss. Their noise spectral density are given by [56] with noise amplitudes A d = π 2 tan(δ C )|φ 01 | 2 ℏ/E C , A f = 2πδ f E L |φ 01 |, thermal factor κ(w, T ) = | coth( ℏw 2kBT ) + 1|/2 containing Boltzmann constant k B and temperature T . Here we assume the loss tangent tan(δ C ) = 1.1 × 10 −6 , flux amplitude δ f = 1.8 × 10 −6 and temperature T = 15 mK. The total noise spectral density S(w) = S f (w) + S d (w). We next make approximation with t ≫ 1/w 0 , when there is no divergence in S(w) around w = w 0 . Decoherent rates for excitation and relaxation are There exists a divergence at w = 0 in spectral density S f (w) of 1/f flux noise and special attentions are needed in evaluation of pure-dephasing rate originating from the 1/f flux noise. Its corresponding integration can be organized in a following way in which we have regularized divergence of this integral by employing infrared cutoff w ir = 1 Hz that is much smaller than the inverse of characteristic time [57]. Then coefficients K z (t) becomes Thus off-diagonal elements of density matrix ρ q (t) decays as a Gaussian from which we can have its decay rate with characteristic time t m . And factor | ln(w ir t m )| is found to be close to 4 [57,58]. Using relationships that it can be easy to identify that decoherent rate of pure-dephasing γ z is directly proportional to |∂ D Ω| [11]. When ϕ ext = π, D = 0, so that ∂ D Ω = 0, which indicates that there is no decoherence of pure-dephasing with γ z = 0 and its coherence is limited only by depolarization. In addition, the anharmonicity η ≫ 1 at this point (see figure A2(b)). So ϕ ext = π is known as a static sweet spot [64]. From these decoherent rates (see equations (B.14) and (B.18)), coherent times (T ϕ and T 1 ) of pure-dephasing and depolarization can be defined as We plot coherent times T ϕ and T 1 at different flux bias ϕ ext . It is illustrated in figure B1 that pure-dephasing time T ϕ → +∞ at ϕ ext = π because of ∂ D Ω ∝ ∂ ϕext Ω = 0 and decreases exponentially once it does not operate at the static sweet spot. While depolarization time T ϕ increases slowly from 430 µs to 939 µs if it is far away from this point ϕ ext = π.
As the driving amplitude A(t) has a period T = 2π/w d , so that H q (t) = H q (t + T) and we can use the aforementioned method to acquire its eigenvalues and eigenstates as a special example. We just need to substitute with tunable-complex-function F(α, θ) = cos(α) + sin(α)e −iθ = 1 + sin(2α) cos(θ)e −iθ into equation (C.5), set w = w d and numerically diagonalize this sparse matrix with appropriate cutoff in the Fourier harmonic number. It needs to be stressed here thatθ is a global phase and can be eliminated by making an unitary transformation Due to existence of time translation invariance in H q (t), we will find that its eigen-spectra ϵ j ± (j = . . . , −1, 0, 1 j=0,± ⟩e −ikw d t . (C.11) We refer to neglect symbol 'j = 0' in the maintext for simplicity as presented in equations (6) and (7).

Appendix E. Analytic quasi-energies of a driven fluxonium qubit
In this section, we endeavor to derive an approximate formula for quasi-energies of a driven fluxonium qubit by applying methods introduced in [ Hereθ is a global phase and can be eliminated by applying unitary transformations (see equation (C.9)). When θ = π 2 , H [1] q = A 2 e −iα σ z and α becomes a global phase. So at this special case, changing α does not affect its physical characteristics. On the other hand, when α = π 2 , relative phase θ now turns to be a global phase, which does not affect the physics.
For general cases, Floquet Hamiltonian of the driven fluxonium qubit now becomes (E.3) When ∆ = 0, modes |ψ 1 ⟩ and |ψ 0 ⟩ are decoupled. The Floquet Hamiltonian for each mode has following eigen-values and eigen-functions with first kind of Bessel function J m (x). So we can make a transformation to change the basis vector from {|k⟩ 1/0 } to {|m⟩ +/− } and the Floquet Hamiltonian turns into (E.6) When the driving frequency is close to transition frequency, i.e. w d ∼ Ω/ℏ, off-diagonal matrices in H F are small and can be seen as a perturbation in case of weak driving (2Ã ≪ ℏw d ). We can ignore off-diagonal matrices containing Bessel function J m (x) with |m| > 1 and approximate the Floquet Hamiltonian H F using a 4 × 4 matrix By diagonalizing this smaller matrix, we can acquire following analytic quasi-energies quasi-energy gap and its partial differential to B . (E.10) Thus, it is possible to get dynamic sweet spots under following condition in fast and weak driving