High-Coherence Kerr-cat qubit in 2D architecture
Ahmed Hajr Bingcheng Qing Ke Wang Gerwin Koolstra Zahra Pedramrazi Ziqi Kang Larry Chen Long B. Nguyen Christian Junger Noah Goss Irwin Huang Bibek Bhandari Nicholas E. Frattini Shruti Puri Justin Dressel Andrew Jordan David Santiago Irfan Siddiqi































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    These authors contributed equally; Correspondence should be addressed to: hajr@Berkeley.eduQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAThese authors contributed equally; Correspondence should be addressed to: hajr@Berkeley.eduQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAComputational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAComputational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USADepartment of Physics and Astronomy, University of Rochester, Rochester, NY, USAInstitute for Quantum Studies, Chapman University, Orange, CA, USAPresent address: Nord Quantique, Sherbrooke, QC J1J 2E2, Canada.Department of Applied Physics and Physics, Yale University, New Haven, CT 06520, USADepartment of Applied Physics and Physics, Yale University, New Haven, CT 06520, USAInstitute for Quantum Studies, Chapman University, Orange, CA, USADepartment of Physics and Astronomy, University of Rochester, Rochester, NY, USAInstitute for Quantum Studies, Chapman University, Orange, CA, USAComputational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USAComputational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
    
    The Kerr-cat qubit is a bosonic qubit in which multi-photon Schrödinger cat states are stabilized by applying a two-photon drive to an oscillator with a Kerr nonlinearity. The suppressed bit-flip rate with increasing cat size makes this qubit a promising candidate to implement quantum error correction codes tailored for noise-biased qubits. However, achieving strong light-matter interactions necessary for stabilizing and controlling this qubit has traditionally required strong microwave drives that heat the qubit and degrade its performance. In contrast, increasing the coupling to the drive port removes the need for strong drives at the expense of large Purcell decay. By integrating an effective band-block filter on-chip, we overcome this trade-off and realize a Kerr-cat qubit in a scalable 2D superconducting circuit with high coherence. This filter provides 30 dB of isolation at the qubit frequency with negligible attenuation at the frequencies required for stabilization and readout. 
     We experimentally demonstrate quantum non-demolition readout fidelity of 99.6% for a cat with 8 photons. Also, to have high-fidelity universal control over this qubit, we combine fast Rabi oscillations with a new demonstration of the X(π/2) gate through phase modulation of the stabilization drive. Finally, the lifetime in this architecture is examined as a function of the cat size of up to 10 photons in the oscillator achieving a bit-flip time higher than 1 ms and only a linear decrease in the phase-flip time, in good agreement with the theoretical analysis of the circuit. Our qubit shows promise as a building block for fault-tolerant quantum processors with a small footprint.
    
     High-Coherence Kerr-cat qubit in 2D architecture
        Irfan Siddiqi
        April 29, 2024
    =================================================
    
    
    
    § INTRODUCTION
    
    Superconducting quantum circuits offer a rich platform for engineering Hamiltonians that encode and manipulate quantum information faster than the decoherence rates <cit.>. Integrating the Josephson junction into those circuits enables a wide range of nonlinear terms in the Hamiltonian that can be used to implement multi-photon wave mixing processes in the microwave range <cit.>. 
    
    In particular, the family of cat qubits uses this nonlinearity to stabilize the superposition of macroscopically distinct so-called Schrödinger cat states. Depending on the details of the encoding scheme, those qubits exhibit either exponential protection or parity changes with single-photon loss, making them a valuable resource for implementing quantum error correction schemes <cit.>. One approach to stabilize those states with autonomous protection from bit-flips is by engineering two-photon addition and subtraction in the case of the dissipative cat. This approach offers exponential protections with increasing cat size but requires external ancillas to engineer the dissipation and the perform readout <cit.>. 
    
    An alternative approach to stabilize the cat states is by subjecting a Kerr nonlinear oscillator to a coherent two-photon drive <cit.>. This approach has the advantage of creating a relatively large energy gap (5-100 MHz) between the computational space and the rest of the Hilbert space, enabling fast logic operations (20-400 ns) and high-fidelity readout with no added ancillas. This qubit gained a lot of attention in terms of examining  its noise-biased behavior both theoretically <cit.> and experimentally <cit.>. Also, the Kerr-cat qubit makes a promising candidate to implement the XZZX surface code provided it can be realized in a scalable architecture. <cit.>. 
    
    The protection against single-photon loss in this qubit can be intuitively understood by noting that the coherent states |±α⟩ are quasi-orthogonal eigenstates of the annihilation operator (â) with eigenvalues ±α, so the dominant process of single-photon loss only leads to a linear increase in the qubit dephasing at the benefit of exponential suppression in the relaxation rate. The first realizations of this qubit were in 3D architectures and relied  on strong microwave drives which led to notable heating and degraded the qubit lifetime <cit.>. Therefore, it is still an open question to see whether possible to realize large cat sizes with high coherence in 2D chips while maintaining high-fidelity operations. This uncertainty arises because the 3D environment is known to result in longer lifetimes but weaker light-matter coupling due to the larger spread of the  electromagnetic field in the extra dimension <cit.>.  
    
    Realizing a 2D circuit with strong light-matter coupling without sacrificing the lifetime is crucial to scaling up the Kerr-cat qubit.  The inherent trade-off is that coupling the drive port strongly to the qubit (i.e., large coupling capacitance)  leads to large Purcell decay. In contrast, coupling weakly at the expense of strong driving heats the qubit and degrades its performance, as shown in other strongly driven qubits <cit.>.
    
    In this work, we overcome this challenge by using an effective band-block filter that enables strong coupling off-resonance (at the frequency of the stabilization drive ω_S and the cat  readout drive ω_CQR) while suppressing the qubit Purcell decay time orders of magnitude larger than the limit imposed by materials defects. Operating at a Kerr nonlinearity of 1.2 MHz, we observe improvements in the bit-flip time with increasing  drive strength up to cat of size 10 photons. The bit-flip time exceeds 1 ms while only sacrificing the phase-flip time linearly with the cat size. Also, the large energy gap at this Kerr value enabled us to perform the highest fidelity readout and gates on Kerr-cats to date. Our work, therefore, provides a path for scaling noise-protected qubits toward fault-tolerant quantum computers.
    
    
    § ENCODING
    
    The Kerr-cat qubit has the following Hamiltonian in the rotating frame <cit.>:
    
        Ĥ_KC/ħ=-K â^†2â^2+ϵ_2â^†2 +ϵ^*_2â^2
    
      where K is the strength of the Kerr nonlinearity, ϵ_2 is the amplitude of the two-photon drive (i.e., the stabilization drive), and ħ is the reduced Planck constant. This Hamiltonian stabilizes two coherent states  |±α⟩ provided the cat size α^2=ϵ_2/K. The cat states are the orthogonal even/odd parity states formed by the superposition of the coherent states |C_α^±⟩=1/√(1± e^-2α^2)1/√(2)(|α⟩±|-α⟩ ) and span the Bloch sphere along the x-axis as shown in Fig. <ref>(a). The orthogonal states along the z-axis are approximately the coherent states (i.e., |± Z⟩=1/√(2)(|C_α^+⟩±|C_α^-⟩)≈|±α⟩). The energy pseudopotential of this Hamiltonian is a double-well with an energy gap E_gap/ħ≈4Kα^2<cit.> separating the Hilbert space from the rest of the computational space as shown in Fig. <ref>(b). 
      
    This Hamiltonian is now traditionally realized by charge pumping a SNAIL (Superconducting Nonlinear Asymmetric Inductive eLement) oscillator with a coherent drive at approximately twice the oscillator (i.e. qubit) frequency ω_S≈ 2 ω_Q<cit.>. Utilizing the third-order nonlinearity of the SNAIL φ̂^3 (where φ̂  is the superconducting phase operator), a photon from the drive is consumed to create two photons in the oscillator (a^†2) at the rate ϵ_2; this is accompanied by the complementary process of eliminating two photons from the oscillator (â^2) and creating a pump photon <cit.>. The other nonlinearity of the SNAIL is a fourth-order (φ̂^4), which in the rotating wave approximation (RWA) adds the Kerr nonlinearity -Kâ^†2â^2 to the Hamiltonian. We refer to this element in the Fock encoding as the SNAILmon (i.e., transmon with small anharmonicity provided by the SNAIL).
    
    
    § DESIGN
    
    The 2D  superconducting circuit used to realize the Kerr-cat qubit with the SNAILmon is shown in  Fig. <ref>(e-g). The design of the SNAILmon contains two large niobium capacitor paddles and is based on high-coherence transmon qubit designs <cit.>. In addition, our design contains two SNAILs consisting of aluminum Josephson junctions fabricated with a bridge-free two-angle deposition process. Concatenating multiple SNAILs has allowed large cat sizes in previous Kerr-cat qubit experiments <cit.>. The readout circuitry contains a readout resonator coupled to a Purcell filter  (green), which allows for fast readout while protecting the qubit from Purcell decay. For control, a weakly coupled port (c' in Fig. <ref>(c)) is used for on-resonance driving. For stabilization and readout, a strongly coupled port is used for efficient pumping (c_p in Fig. <ref>(c)) with an effective band-block filter to suppress the Purcell decay. 
    
    The nonlinearities of our circuit can be tuned with an external flux piercing each SNAIL. In this work, we fix the external flux to realize a Kerr non-linearity of K/2π=1.2 MHz, a value between the first realization of the Kerr-cat (6.7 MHz) <cit.> and the smaller value (0.3 MHz) in <cit.>. At this operating point, we observe an improvement in the bit-flip time with increasing cat size while maintaining the ability to  perform fast logic gates and coherent mapping between the Fock qubit and the cat qubit if needed (see Fig. <ref> in Appendix <ref>). 
    
    
    The band-block filter we use to couple strongly to the stabilization and readout drives is shown in Fig. <ref>(e, g). This filter consists of multiple stubs, each leading to destructive interference of the microwave signal around the qubit frequency. This is achieved by adjusting the length of the meandering section (i.e., the quarter wave resonators) and simulating the scattering parameters in Ansys-HFSS <cit.>. Ideally, this has high isolation at the qubit frequency (ω_Q/2π=5.9 GHz) with zero attenuation at the frequency of the stabilization drive (ω_S/2π=11.8 GHz) and the frequency of the cat quadrature readout (ω_CQR/2π=1.2 GHz). In Fig. <ref>, we present an independent measurement of the transmission coefficient S_21 of the filter in a dedicated chip (for more details about the setup see Appendix <ref>). In Fig. <ref>(a), we show the behavior of normalized scattering parameter S̅_21 for the full range of interest. The 30 dB of isolation at the qubit frequency  corresponds to a three-order of magnitude improvement in the lifetime, pushing the Purcell limit through the strongly coupled port to 12 ms, which is much higher than the measured single photon decay time of the Fock qubit of  T_1= 38.5μs. Material defects in the interfaces have been shown to limit the lifetime of 2D architectures at this range. <cit.>. 
    
    In Fig. <ref>(b), we zoom in around the qubit frequency to examine the range of 20 dB and 30 dB of isolation. The large bandwidth of this filter makes it suitable for multi-qubit chips with strong qubit-qubit coupling while maintaining a negligible Purcell effect and efficient pumping for all the qubits. This filter can also be incorporated in any circuit QED experiment in which a strong off-resonance drive is required to implement the desired interaction <cit.>. The idea of integrating on-chip filters to enable strong coupling with mitigated Purcell effects was first introduced in the context of fast readout <cit.>, and the use of band block filters to enable strong coupling off-resonance was also explored more recently <cit.>. 
    
    
    
    § QUBIT CONTROL AND READOUT
    
    To create the Kerr-cat qubit, we ramp up the stabilization drive to transform the SNAILmon Hamiltonian to the Kerr-cat qubit Hamiltonian in the rotating frame (see Appendix <ref> for derivation). In this section, we focus on characterizing the readout and the single qubit gates for a cat of size α^2=4 (see Appendix <ref> for more details). Starting with the cat quadrature readout in Fig. <ref>(a), we can measure the qubit along the Z-axis by sending a microwave pulse to the qubit at frequency ω_CQR/2π=(ω_R-ω_S/2)/2π= 1.2 GHz which implements a beam-splitter interaction between the qubit and readout mode Ĥ_CQR≈ϵ_CQRαẐ(b̂+b̂^†) which is derived in Appendix A.2<cit.>. The output signal is a coherent state with one of two opposite phases according to the state of the qubit, as shown in Fig. <ref>(a). For a cat of size 4 and 4 μs readout time, we measure a QNDness  𝒬=(p(+α|+α )+p(-α|-α ))/2 of 98.3 % where p(+α|+α ) is the probability that the second measurement gives |+α⟩ consistent with the first measurement result. This high QND readout enables us to use the measurement to initialize the qubit in the Z basis. For a cat of size 8, the QNDness of the readout increases to 99.6%, as shown in Fig. <ref>. Those values are equal to the state-of-the-art across all superconducting qubits <cit.>. 
        
    
    Universal single-qubit control typically requires driving on resonance to do rotations along two orthogonal axes, depending on the phase of the drive <cit.>. However, this protocol in the cat encoding leads to an enhancement factor 2α in the speed of the Z(θ) rotations and an exponential suppression of the Y(θ) rotations <cit.>(For more details, check Appendix <ref>). To overcome this challenge, it is sufficient to use the Z(θ) rotations with X(π/2) gates to have universal control. The first proposal to do an X(π/2) gate was demonstrated in <cit.> by abruptly turning off the stabilization drive for a period π/2K=216 ns. However, this protocol removes protection the stabilization drive adds to the qubit, and it tends to have worse performance for large cats <cit.>. 
    
    To improve on this implementation, we demonstrate a new X(π/2) gate by phase modulating the stabilization drive according to  ϵ_2 →ϵ_2 e^-itδ_d(t). This modulation modifies the double well potential and allows the coherent states to tunnel. With the appropriate unitary transformation, the phase modulation of the drive adds the following term to the Kerr-cat Hamiltonian:
    
        H_d(t)=-1/2(δ_d(t)+tδ̇_d(t)) â^†â
    
    Fig. <ref>(b) shows the expectation value of Pauli Z after implementing two X(π/2) pulses as a function of the maximum detuning δ_0 in the phase modulation waveform and the gate time T_g. This gate implementation is resource-efficient since it does not require additional tones and does not disturb the qubit parameters compared to a fast DC flux. The coherent tunneling between the states inside the two energy wells using a controlled time-dependent detuning to perform logic gates was first examined theoretically  <cit.>, and the static detuning was shown to improve the lifetime 
    experimentally <cit.>. The simulation of this gate according to Eq. <ref> is shown in Fig. <ref> (more details in Appendix <ref>). From the chevron plot, we chose T_g=320 ns and a detuning  δ_0/K=-8.2 for maximum population transfer. 
    
    Next, we calibrate the phase of the drive to implement Z(θ) rotations by driving on resonance. The added term to the Kerr-cat Hamiltonian has the form ≈ 2α |Ω_ z|cos(θ_ z)Ẑ-2αΩ_ zsin(θ_ z) e^-2|α|^2Ŷ as long as the drive amplitude Ω_ z is small compared to the energy gap E_gap/ħ<cit.>. Fig<ref>(e) shows the protocol to calibrate the phase of the drive (Arg(Ω_ z )=θ_ z) by leveraging the cat quadrature readout and the X(π/2) gate introduced earlier. The second X(π/2) gate enables us to measure Rabi oscillations in ⟨Y|$⟩ as we sweep the phase of the Rabi driveθ_ zwith maximum contrast|⟨Y||⟩≈ 0.9. When using the traditionalX(π/2)gate where the stabilization drive is turned off <cit.>, the maximum contrast drops to|⟨Y||⟩≈ 0.5. We can see in Fig. <ref>(c) that driving in phase leads to the maximum Rabi rate while driving aroundθ_ z= ±π/2leads to suppressed rates.
    
    For a cat of size 4, we choose a gate time of120ns to implement theZ(π/2)gate. With theX(π/2)and theZ(θ)calibrated, we have a universal set of single-qubit gates. Using process tomography, we measure gates fidelitiesℱ_X_π/2=90.8%andℱ_Z_π/2=91.7%. For the identity, we measureℱ_I=92.7%, which implies that state preparation errors are the limiting factor. The Pauli transfer matrices for those processes with the details of the analysis are provided in Appendix <ref>.
    
    
    § CAT QUBIT LIFETIME
    
    We start by measuring the lifetime of the cat qubit as a function of the cat sizeα^2=ϵ_2/Kby varying the strength of the two-photon driveϵ_2as shown in Fig. <ref>. Every sequence begins with a readout signal to initialize the qubit in the Z-basis. To measure the lifetime of the cat statesT_ C(i.e., phase-flip time), we use the KerrX(π/2)gate introduced in <cit.>, since it does not depend on the cat size in the calibration. For every cat size, we vary the delay timeΔ tto observe the exponential decay of the coherence and perform anotherX(π/2)since the cat readout projects the qubit to the Z basis. The measured lifetime in Fig. <ref>(a)  follows the trend in grey, which is the linear trade-off for large cats described byT_ C=T_1 / 2⟨n̅⟩with⟨n̅⟩ =|α|^2 (1+e^-4|α|^2)/(1-e^-4|α|^2)is the time-averaged number of photons in the qubit <cit.>. 
    
    The linear increase in the phase-flip rate of the cat states should come at the benefit of exponential protection against the single photon loss in the bit-flip rate. In Fig. <ref>(b), we examine the lifetime of the coherent statesT_α(i.e. bit-flip time) as a function of the cat size. The staircase shape we observe implies that thermal excitations limit the lifetime. A bit-flip error can happen through tunneling to the higher energy states, which become asymptotically degenerate at the points where the lifetime increases sharply with the cat size. This trend was first observed in <cit.>. The staircase behavior is captured by heating effects in the master equation likeκ_ω_d/2 n_ω_d/2𝒟{â^†}while the exact location of the plateaus in this model depends on the spectrum of the dephasing noiseκ_ϕ(ω)<cit.> (see Appendix <ref> for more details). Our data up to cats of size 8 can be understood in terms of the single and multi-photon heating and dephasing of the SNAIL with thermal populationn_ω_d/2=2%. This value is the lowest compared to other Kerr-cat experiments or similar strongly driven qubits<cit.>. However, this is still higher than the upper bound of0.3%value we measure from the residual population of the|n=1⟩state in the Fock basis <cit.>. The details of the master equation model are presented in Appendix <ref>. 
    
    Next, we focus on cats of size 4, which offer the most gain inT_αwithout sacrificingT_ Cmuch. It is also possible to red-tune the stabilization drive to boost the lifetime of the coherent states by adding the harmonic termΔâ^†âwithΔ=ω_Q-ω_S/2to the Kerr-cat Hamiltonian in Eq.<ref><cit.>. Fig. <ref>(c) shows the improvement inT_αforϵ_2/K=4as we sweep the detuning. The lifetime peaks around even multiples of K due to the Hamiltonian picking extra degeneracies in the excited states <cit.>. However, in the case of the detuned cat, the ground state of the Fock qubit|n=0⟩does not map to the even cat state of the detuned cat qubit, so instead, we keep the stabilization drive for a timeτ_th≈ T_1necessary to put the detuned cat in a thermal state, then we use the cat quadrature readout to initialize the qubit.  Those results demonstrate the potential of the Kerr-cat qubit to benefit from the suppressed bit-flip rate with only a linear increase in the phase-flip rate while maintaining the ability to perform high-fidelity operations.
    
    
    § CONCLUSION 
    
    Our results demonstrate a high-coherence Kerr-cat qubit in a scalable 2D architecture with high-fidelity single-qubit operations and minimal heating. This demonstration is enabled by integrating a wide band-block filter for efficient energy delivery, enabling strong light-matter coupling with negligible Purcell effect. We report improvement in the bit-flip time up to cats of sizeα^2=10with lifetimes higher than 1 ms in good agreement with our theoretical modeling of the circuit, including multi-photon processes. Our qubit was based on a SNAILmon oscillator with a single photon decay timeT_1=38.5 μ s.  
    
    Operating at a Kerr nonlinearity of 1.2 MHz, we get an energy gap4K α^2=9.6-48 MHz for cats of size 2-10, which enables fast operation compared to the decay rates. For those device parameters, we demonstrate quantum non-demolition readout with QNDness of 98.3% for a cat of size 4 and 99.6% for a cat of size 8. We also introduce a resource-efficient way to implement theX(π/2)gate with phase modulation of the stabilization drive. Combined with continuous and fastZ(θ)rotations, this is sufficient to have universal control over the Kerr-cat qubit. For a cat of size 4, we implement single qubitZ(π/2)in 120 ns and theX(π/2)in 320 ns with gate fidelities of91.7%and90.8%respectively, limited by state preparation since for the identity, we measure process fidelity of92.7%using process tomography. A detailed analysis of those gates in terms of noise bias and the removal of spam errors is the subject of future work. 
    
    With the goal of implementing quantum error correction codes with many physical Kerr-cat qubits the problem of finding an efficient pumping scheme that does not heat the cryogenic environment is of most importance. Our solution relied on integrating an effective band-block filter to remove the trade-off between strong capacitive coupling to the qubit and large Purcell decay to enable strong light-matter interactions. With 30 dB of isolation at the qubit frequency, large bandwidth, and negligible attenuation at the pump frequencies, this design has the potential to realize multi-qubit processors.
    
    In conclusion, we introduced a novel scheme for efficient power delivery in 2D architecture to realize the Kerr-cat qubit with high coherence, fast logic gates, and state of art QND readout.
    This work extends the reach of the Kerr-cat qubit to strongly coupled multi-qubit 2D processors, which can be used to examine the fidelity of the noise-bias-preserving two-qubit CZ and CX gates <cit.> and quantum error correction schemes tailored for noise-biased qubits  with the Kerr cats <cit.>. Future single-qubit experiments can focus on increasing the Kerr nonlinearity of the qubit without jeopardizing the lifetime to enable faster single-qubit gates with improved fidelity and examine the root causes of the saturation in the lifetime with controlled dissipation.  
    
    
    
    § ACKNOWLEDGMENTS
    
     We thank Chuan-Hong Liu, Akel Hashim, and Ravi Naik for valuable discussions. This work was supported by the U. S. Army Research Office under grant W911NF-22-1-0258 
    
    
    
    
    
    § REALIZING THE KERR-CAT HAMILTONIAN FROM THE SNAIL
    
    
     §.§ The Kerr-cat Hamiltonian
    
    The Kerr-cat HamiltonianĤ_KCcan be realized by adding a stabilization driveĤ_Sat twice the SNAILmon frequency to the SNAIL Hamiltonian (Ĥ_a) and moving to a rotating frame as follows (ħ=1) <cit.>:
    
        Ĥ_0=Ĥ_a+Ĥ_S,
    
        Ĥ_a=ω_a,0â^†â+g_3 (â^†+â)^3+g_4 (â^†+â)^4,
    
        Ĥ_S=2 Re(e^iω_st)(ϵ_ s,0â^†+ϵ^*_ s,0â).
    
    The operatorâis the bosonic annihilation operator of excitations in the SNAILmon oscillator. The oscillator frequency isω_ a,0, and third, and fourth-order nonlinearities areg_3andg_4, respectively. Now, to eliminate the time dependence, we apply two consecutive transformations. First, we move to a displaced Fock basis characterized by-ξ_sand then to a rotating frame characterized byω_ rÛ= Û_rotÛ_dis=e^iω_rtâ^†â· e^ξ_sâ^†-ξ^*_sâ
    
    This leads to the effective HamiltonianĤ_KC=ÛĤ_0Û^†+iÛ̇Û^†. The second term inĤ_KCeliminates the harmonic part inÛĤ_0Û^†ifω_r=ω_a,0. More precisely: 
     
        iÛ̇Û^†=-ω_râ^†â+ie^iω_rt â^†â(ξ̇_sâ^†-ξ̇^*_sâ)e^-iω_rt â^†â
    
     
    The impact of sandwichingĤ_0is summarized by first displacinga(a^†) byξ_s(ξ_s^*)  and then adding a time-dependent  factore^-i ω_rt (e^i ω_rt) , which is crucial when we apply the rotating wave approximation (RWA) as follows:
    
        Û_dis â Û^†_dis=â-ξ_s
    
        Û_dis â^† Û^†_dis=â^†-ξ^*_s
    
        Û_rot â Û^†_rot=â e^-i ω_rt
    
        Û_rot â^† Û^†_rot=â^† e^i ω_rt
    
    Now we can expandĤ_KCto get:
    
        Ĥ_KC   =-ω_râ^†â +iξ̇_sâ^† e^i ω_rt-iξ̇^*_sâ e^-i ω_rt
           +ω_a,0 (â^† e^i ω_rt-ξ^*_s) (â e^-i ω_rt-ξ_s) 
           +2 Re(e^iω_st)[ϵ_ s,0 (â^† e^i ω_rt-ξ^*_s)+ϵ^*_ s,0(â e^-i ω_rt-ξ_s)]
        
           +g_3 (â e^-i ω_rt-ξ_s+h.c.)^3 
           +g_4 (â e^-i ω_rt-ξ_s+h.c.)^4
    
    
    Now, to determineξ_s, we collect the terms withâ^†  e^i ω_ rt:
    
    -ξ_sω_a,0+iξ̇_s+2 Re(e^iω_st)ϵ_s=0
    
        ξ_s=- (ϵ_ s,0/ω_s-ω_a,0e^-iω_st-ϵ_ s,0/ω_s+ω_a,0e^iω_st )
    
    Considering that bothξ_sand its complex conjugate will appear inside the nonlinear terms we writeξ_s+ξ^*_s=ξ_s,eff  e^-iω_st+h.c.ξ_s,eff =- (ϵ_ s,0/ω_s-ω_a,0-ϵ_ s,0/ω_s+ω_a,0 ).
    
    This means the first three lines in (A10) cancel up to some time-varying terms, which average to zero very quickly. Now, to eliminate the time dependence, we chooseω_rso that the term containingâ^† 2,â^ 2(which comes from the third-order non-linearity) does not oscillate with time. This condition can be satisfied ifω_r=ω_s/2:
    
    
    3g_3  (â^† e^i ω_rt)^2  ξ_s,eff· e^-iω_st=3g_3 â^† 2ξ_s,eff· e^-i(ω_s-2ω_r)t.
     
    When this condition is satisfied, we can simplifyĤ_KCby neglecting every oscillating term to get:
     
        Ĥ_ KC= Δ_a,râ^†â-Kâ^† 2â^2+ϵ_2 â^† 2+ϵ^*_2 â^ 2-4K â^†â |ξ_s,eff|^2
    
    The first term is a harmonic term weighted by the detuning (Δ_a,r=ω_a,0-ω_ r). The second term is the Kerr non-linearity (K=-6g_4). The third term is the two-photon drive (ϵ_2=3g_3ξ_s,eff), and the last term is the Stark shift induced by the drive. The coherent states|±α⟩are eigenstates of this Hamiltonian withα^2=ϵ_2/K<cit.>. The cat states are the orthogonal even- and odd-parity  states formed by the superposition of the coherent states and form two, degenerate ground states of this Hamiltonian :
    
        |C_α^±⟩=1/√(1± e^-2α^2)1/√(2)(|α⟩±|-α⟩ ).
    
    
     §.§ Cat Quadrature Readout 
    
    
    The implementation of the cat quadrature readout is similar  to the implementation of the two-photon drive because it relies on the three-wave mixing in the SNAIL. To derive the Hamiltonian for Cat quadrature readout, we first consider the combined Hamiltonian of the SNAIL and the readout resonatorĤ_bwith a coupling coefficientg_cmuch smaller than their detuningΔ=ω_a,0-ω_b,0:
    
    
    
        Ĥ_0=Ĥ_a+Ĥ_s+Ĥ_b+Ĥ_c,
    
        Ĥ_b=ω_b,0b̂^†b̂,
    
        Ĥ_c=g_c (â^†+â)(b̂^†+b̂)≈ g_c (â^†b̂+b̂^†â).
    
    Now, we can cast this Hamiltonian in a dressed form where the bosonic modes are decoupled  by using the following unitary <cit.>:
    
        Û_̂ĉ=e^λ(â^†b̂-b̂^†â)
    
    The impact on the bosonic operatorsâandb̂is: 
    
    
    
        Û_c â Û^†_c= cos(λ)â- sin(λ)b̂≈â-g_c/Δb̂
    
        Û_c b̂ Û^†_c=  cos(λ) b̂+ sin(λ)â≈b̂+g_c/Δâ
    
    ExpandingĤ_a+Ĥ_b+Ĥ_cwe get:
    
        ω_a,0( cos(λ)â^† - sin(λ)b̂^†) ( cos(λ)â- sin(λ)b̂)
           +ω_b,0( cos(λ)b̂^†+ sin(λ)â^†) ( cos(λ)b̂+ sin(λ)â)
           +g_c( cos(λ)â^†- sin(λ)b̂^†)( cos(λ)b̂+ sin(λ)â)+h.c.
           +g_3 ( cos(λ)â- sin(λ)b̂+h.c.)^3 
           +g_4 ( cos(λ)â- sin(λ)b̂+h.c.)^4
    
    Choosingλsuch that the coupling between the modes (i.e., the coefficient ofâ^†b̂) vanishes gives:
    
        λ=1/2 Arctan(2g_c/Δ)≈g_c/Δ.
    
    The renormalized frequencies areω_a=1/2(ω_a,0+ω_b,0+√(Δ^2+4g_c^2))≈ω_a,0+g_c^2/Δandω_b=1/2(ω_b,0+ω_a,0-√(Δ^2+4g_c^2))≈ω_b,0-g_c^2/Δ. Now, the two modes are decoupled, but the modeω_bappears in the nonlinear terms of modeω_a. SoĤ_a+Ĥ_b+Ĥ_ctakes the effective form:
    
    
        ω_aâ^†â+ ω_bb̂^†b̂+g_3 ( â+g_c/Δb̂+h.c.)^3+g_4 (â+g_c/Δb̂+h.c.)^4
    
    The third term is the key to implementing the cat quadrature readout. Adding another drive of the formĤ_CQR=2 Re(e^iω_CQRt)(ϵ_CQR,0â^†+ϵ^*_CQR,0â)can implement the desired interaction and following the same treatment as in part A1 we get:
    
        +g_3 (âe^-i ω_rt+g_c/Δb̂e^-i ω_bt-ξ_CQR,eff  e^-iω_CQRt-ξ_s,eff  e^-iω_st
           +h.c.) ^3 
           +g_4 (âe^-i ω_rt+g_c/Δb̂e^-i ω_bt-ξ_CQR,eff  e^-iω_CQRt-ξ_s,eff  e^-iω_st
           +h.c.)^4
    
    So  fromg_3, we can also stabilize the following term in the rotating frame provided thatω_=ω_b-ω_ r6g_3 g_c/Δξ^*_CQR,eff â^†b̂  e^i(ω_CQR-ω_b+ω_r)t+h.c.
    
    
    Fromg_4, there will be two additional terms corresponding to the stark shift from the CQR drive  and another term corresponding to the cross Kerr between the readout and the SNAILmon:
     
        24 g_4(â^†â+g_c^2/Δ^2b̂^†b̂) |ξ_CQR,eff|^2 +24 g_4g_c^2/Δ^2â^†âb̂^†b̂
    
    
    § MASTER EQUATION CALCULATIONS
    
    
    In order to simulate the lifetime of the coherent states, we consider the system to be linearly coupled to a macroscopic environment composed of a bath of linear resonators with continuous modes. The system environment coupling Hamiltonian in the rotating frame takes the following form
    
        Ĥ_ SB(t) = ∑_kV_k(âe^-iω_ dt/2 + â^† e^iω_ dt/2 )(b̂_k(t)+b̂_k^†(t)),
    
    whereV_kgives the coupling strength andb̂_k(b̂_k^†)are the annihilation (creation) operators of an excitation of energyħω_kin the environment. Following Ref. Jaya_SNAIL_theory, we obtain following master equation up to O(φ_ zps^2),
    
        dρ̂_ S/dt = i/ħ[Ĥ_ KC,ρ_ S]+L̂_ RWAρ̂_ S + L̂_ NRWAρ̂_ S+ L̂_ϕρ̂_ S,
    
    where 
    
        L̂_ RWA= κ_ω_d/2[n_ω_ d/2𝒟{â^†}+(1+n_ω_ d/2)𝒟{â}]
    
    gives the contributions satisfying the rotating wave approximation. It includes single photon dissipation effects.κ(ω)is the spectral density of the environment defined throughκ (ω) = 2π∑_k |V_k|^2 δ(ħω-ħω_k). The magnitude of spectral density depends strongly on the transition energy but is independent of any other system or environment parameters. The thermal population in the environment is given by the Bose-Einstein distribution function,n(ω) = (e^ħω/k_ B T_ω-1)^-1, where T_ωis the temperature of the environment. Similarly,
    
        L̂_ NRWA= κ_ω_ d[n_ω_ d𝒟{8g_3/3ω_ dâ^†^2-(592g_3/9ω_ d^2-16g_4/g_3ω_ d)â^†âϵ_2^*}
        
        +(1+n_ω_ d)𝒟{8g_3/3ω_ dâ^2-(592g_3/9ω_ d^2-16g_4/g_3ω_ d)â^†âϵ_2}]
    
    is the beyond-rotating wave approximation contributions. It includes two-photon dissipation effects and drive-induced dephasing, which enters the master equation at O(φ_ zps^2)withφ_ zpsbeing the value of the zero-point fluctuation in the phase operator of the SNAILmon <cit.>. Finally,
    
        L̂_ϕ= κ_ϕ𝒟{â^†â}
    
    takes into account any additional drive-independent dephasing processes. 
    
    In our model, we consider single-photon effects, multi-photon heating and cooling effects, dephasing as well as non-Markovian detuning noise. The non-Markovian noise is taken to be a Gaussian random fluctuation of the detuning with a mean of0.03Kand a standard deviation ofK/500. The lifetime is averaged over 100 trials of the non-Markovian noise. We estimateg_3/2π = 15  MHzand the decay rateκ_ω_ d/2= 1/T_1≈ 26kHz from the experiment. We fit the experimental data with T_ω_ d/2, T_ω_d,κ_ω_ d, andκ_ϕ. For the fit shown in Fig. <ref>(d), we usedκ_ϕ=100Hz,κ_ω_d=7MHz, T_ω_d/2=73.5mK, and T_ω_d=515mK. We find an excellent agreement between the experimental data and theoretical plot up to the second plateau in the lifetime. The second plateau can be explained by changing the environment temperature (i.e., changing the value ofn_ thto be5%)<cit.>. More statistics on the lifetime time of the Fock qubit and the coherent states are shown in Fig. <ref>. Those variations in the lifetime are typical in superconducting qubits <cit.>.
    
    
    
    § UNIVERSAL SINGLE-QUBIT GATES FOR THE KERR CAT QUBIT
    
    A universal gate set for the Kerr cat qubit has been reported in the literature <cit.> based on continuousZ(θ)rotations and a discreteX(π/2)rotation on the Bloch sphere. However, the discreteX(π/2)relies on turning off the stabilization drive, which protects this qubit and the performance of this gate degrades very quickly with increasing cat size|α|^2<cit.>. Therefore, it is necessary to realize an alternative approach with higher fidelity. Here, we describe the implementation of gates we introduced in the main text.
    
    
     §.§ Continuous Z(θ) rotation based on the quantum Zeno effect
    
    
    Continuous Z(θ) rotations are realized by applying a microwave drive at the frequencyω_r = ω_s/2, which activates a single-photon transition between the cat states according to:
    
        Ĥ = Ω_ z/2â^† + Ω_ z^*/2â
    
    whereΩ_ z/2is the strength of the microwave drive. The strong stabilization drive of the Kerr cat qubit continuously projects its dynamics into the cat space due to the quantum Zeno effects. Therefore, if the single-photon transition rate is much smaller than the energy gap, i.e.Ω_ z≪ E_gap=4ϵ_2, one can project the bosonic mode operators into cat space to get:
    
        P̂_ CâP̂_ C=αẐ-i α e^-2 α^2Ŷ
    
        P̂_ Câ^†P̂_ C=α^* Ẑ+i α^* e^-2 α^2Ŷ
    
    WhereP̂_ C = |C_α^+⟩⟨ C_α^+|+|C_α^-⟩⟨ C_α^-|. We notice that bothâandâ^†exponentially approachαẐwith increasing cat size. WhenΩ_ zis in phase withα, the single-photon transition effectively applies continuousZ(θ)rotation according to the Hamiltonian:
    
        P̂_ CĤP̂_ C^† = 2αΩ_ zẐ
