Nonadiabatic geometric quantum computation with cat qubits via invariant-based reverse engineering

We propose a protocol to realize nonadiabatic geometric quantum computation of small-amplitude Schr\"odinger cat qubits via invariant-based reverse engineering. We consider a system with a two-photon driven Kerr nonlinearity, which provides a pair of dressed even and odd coherent states, i.e., Schr\"odinger cat states for fault-tolerant quantum computations. An additional coherent field is applied to linearly drive a cavity mode, to induce oscillations between dressed cat states. By designing this linear drive with invariant-based reverse engineering, nonadiabatic geometric quantum computation with cat qubits can be implemented. The performance of the protocol is estimated by taking into account the influence of systematic errors, additive white Gaussian noise, and decoherence including photon loss and dephasing. Numerical results demonstrate that our protocol is robust against these negative factors. Therefore, this protocol may provide a feasible method for nonadiabatic geometric quantum computation in bosonic systems.


I. INTRODUCTION
Quantum coherence and quantum entanglement are arguably the most fascinating properties of quantum mechanics [1][2][3][4]. These are the main resources for quantum information processing [1,5] and quantum technologies of second generation [6]. Their recent applications include: demonstrations of quantum advantage using superconducting programmable processors [7] or boson sampling with squeezed states of photons [4], a quantum communication network over 4,600 km [8], and quantum-enhanced gravitational-wave detectors using squeezed vacuum [9][10][11][12]. As a very important subfield of quantum information processing, quantum computation has shown a potentially great power in solving many specific problems [13,14]. In a practical implementation of quantum computation, quantum algorithms are usually (but not always, e.g., in quantum annealing) designed as a sequence of quantum gates. Therefore, high-fidelity quantum gates are essential elements of quantum computation. Unfortunately, experimental imperfections, including operational errors, parameter fluctuations, and environment-induced decoherence, may affect the desired dynamics, limiting the fidelities of quantum gates. The problem how to overcome these experimental imperfections has to be solved * The first two authors contributed equally. † jsong@hit.edu.cn ‡ xia-208@163.com for the constructions of practical quantum computers.
Because geometric phases are determined by the global geometric properties of the evolution paths, geometric quantum computation [15,16] has shown robustness against local parameter fluctuations over a cyclic evolution [16][17][18]. As an extension of geometric quantum computation, holonomic quantum computation [19][20][21] based on non-Abelian geometric phases can be used to construct a universal set of single-qubit gates and several two-qubit entangling gates. Early implementations of geometric quantum computation involve adiabatic evolutions to suppress transitions between different eigenvectors of the Hamiltonian. This makes the evolution slow, and decoherence may destroy these geometric gates [22][23][24][25][26].
Recently, based on cat codes, various protocols [47-51, 59, 76] for preparing, stabilizing, and manipulating cat qubits have been put forward. Moreover, quantum computation [50] and adiabatic geometric quantum control [66] of cat qubits have also been considered. However, so far, only a few protocols [66,77] have been proposed to implement geometric computation using cat qubits. Because of the difficulty to arbitrarily manipulate a bosonic mode, it is still a challenge to realize NGQC using bosonic cat qubits, which are both robust and fault tolerant.
In this manuscript, we propose to use cat qubits to implement NGQC via invariant-based reverse engineering. To construct cat qubits, a two-photon driven Kerr nonlinearity is used to restrict the evolution of cavity modes to a subspace spanned by a pair of cat states. We apply an additional coherent field to linearly drive a cavity mode in order to induce oscillations between dressed cat states. With the control fields designed by invariant-based reverse engineering, the system can have a cycling evolution, which acquire only pure geometric phases. Hence, NGQC with cat qubits can be implemented.
An amplitude-amplification method, using light squeezing [78,79], is applied to increase the distinguishability of different cat states, so that it can be easier to detect input and output states in practice. Moreover, two-qubit quantum gates of cat qubits are also considered by using couplings between two cavity modes. Controlled two-qubit geometric quantum gates can be implemented almost perfectly.
Finally, the performance of the protocol in the presence of systematic errors, additive white Gaussian noise (AWGN), 1/f noise, and decoherence (including photon loss and dephasing) are investigated via numerical simulations. Our results indicate that the protocol is robust against these negative factors.
The article is organized as follows. In Sec. II, we briefly introduce the basic theory for invariant-based NGQC. In Sec. III, we describe how to implement single-and twoqubit NGQC with cat qubits. In Sec. IV, we consider experimental imperfections and estimate the performance of the protocol via numerical simulations. In Sec. V, we introduce an amplification method based on quadrature squeezing for the amplitudes of cat states, so that the detection of input and output states can be performed easily. In Sec. VI, we discuss a possible implementation of our protocol using a superconducting quantum parametron. Finally, our conclusions are given in Sec. VII. Appendix A includes a derivation of a dynamic invariant and the choice of parameters for eliminating dynamical phases.

II. NONADIABATIC GEOMETRIC QUANTUM COMPUTATION BASED ON A DYNAMIC INVARIANT
For details, we first recall the Lewis-Riesenfeld invariant theory [80]. Assuming that a physical system is described by a Hamiltonian H(t), a Hermitian operator I(t) satisfies the following equation ( = 1) For a non-degenerate eigenvector |φ l (t) of To realize NGQC, one can select a set of timedependent vectors, {|φ l (t) }, spanning a computational subspace S. According to Ref. [81], {|φ l (t) } should satisfy the three conditions: (i) the cyclic evolution condition |φ l (0) = |φ l (T ) with T being the total operation time; (ii) the von Neumann equatioṅ with Ξ l (t) = |φ l (t) φ l (t)|; and (iii) annihilation of the dynamical phase When satisfying the three conditions, the evolution in subspace S can be described by with a pure geometric phase Ref. [82] has shown that, a non-degenerate eigenvector |φ l (t) of an invariant obeys the von Neumann equation given in Eq. (3). Therefore, when the parameters of I(t) are designed with the cycling boundary conditions and the dynamical part of the Lewis-Riesenfeld phase is eliminated, the three conditions are satisfied. In this case, one can implement NGQC with a dynamic invariant I(t).

III. NONADIABATIC GEOMETRIC QUANTUM COMPUTATION OF CAT QUBITS
A. Arbitrary single-qubit gates

Hamiltonian and evolution operator of a single resonator
We consider a resonant single-mode two-photon (i.e., quadrature) squeezing drive applied to a Kerr-nonlinear resonator. In the frame rotating at the resonator frequency ω 1 , the system is described by the Hamiltonian [49,51] where K is the Kerr nonlinearity, a 1 (a † 1 ) is the annihilation (creation) operator of the resonator (cavity) mode, 2 is the strength of the two-photon drive assumed here to be real, and ξ is its phase. The coherent states | ± α 1 (where α = 2 /Ke iξ is the complex amplitude) are two degenerate eigenstates of H cat . Therefore, the even (|C + 1 ) and odd (|C − 1 ) coherent states, often referred to as Schrödinger cat states, which are defined as includes a control Hamiltonian defined as [51] H c (t) = χ(t)a † 1 a 1 + (t)a † 1 + * (t)a 1 , with χ(t) and (t) being the detuning and strength of a single-photon drive, respectively. When the energy gap E gap between the cat states |C ± 1 and their nearest eigenstate of H cat is much larger than χ(t) and (t), the system can be restricted to the subspace S spanned by |C ± 1 . We can accordingly use the cat states to define the Pauli matrices as in terms of the raising (σ + ) and lowering (σ − ) qubit operators, Then, the Hamiltonian of the system can be simplified to H c (t) = Ω(t) · σ, where, Ω(t) = [Ω x (t), Ω y (t), Ω z (t)] is a set of driving amplitudes to be determined. We find that I(t) = ζ(t) · σ is a dynamic invariant, where ζ(t) = [ζ x (t), ζ y (t), ζ z (t)] is the three-dimensional timedependent vector satisfying the conditions˙ ζ(t) = 2 Ω(t)× ζ(t) and | ζ(t)| = const (see Appendix A for details). The three components [ζ x (t), ζ y (t), ζ z (t)] of ζ(t) denote the projections of the dynamic invariant I(t) along the directions (σ x , σ y , σ z ) in the SU(2) algebra.
By introducing two time-dependent dimensionless parameters η and µ, we can parametrize ζ(t) as (sin η sin µ, cos η sin µ, cos µ) and the eigenvectors of the dynamic invariant I(t) can be derived as We can design the parameters (t) and χ(t) as with effective driving amplitudes (see Appendix A for details) , Both dynamical phases, acquired by the eigenvectors shown in Eq. (13), vanish due to φ ± (t)|H c (t)|φ ± (t) = 0, while the geometric phases acquired by |φ ± (t) , defined in Eq. (13), are According to Eq. (5), the evolution of the system in the subspace S, after a cycling evolution with period T , is calculated as where θ = T 0η sin 2 (µ/2) dt is the final geometric phase of the cycling evolution, and µ 0 (η 0 ) is the initial value of µ (η). The evolution operator U s (T, 0) represents a rotation on the Bloch sphere that can generate arbitrary single-qubit gates [83,84]. For a cycling evolution, the parameters can be interpolated by trigonometric functions as where Λ is an auxiliary parameter to be determined according to the requirements of different gates.  Table I.

Examples of single-qubit-gate implementations
We now discuss how to use the evolution operator U s (T, 0), given in Eq. (17), to realize: where 1 is the identity operator acting on the cat qubit.
To determine the initial values of µ 0 and η 0 , we can exploit the evolution operator U s (T, 0) shown in Eq. (17). Because the system evolves through a cycling evolution [|φ ± (T ) = |φ ± (0) ], the evolution operator only relies on the acquired geometric phase θ and the initial values of µ 0 and η 0 . For example, to implement the NOT gate, we should make the diagonal elements of the evolution operator U s (T, 0) in Eq. (17) to become zeros. There-  fore, we set θ = µ 0 = π/2. In addition, the off-diagonal elements of U s (T, 0) should be equal to 1. Thus, we select η 0 = π/2. Dropping a global phase π/2, U s (T, 0) becomes the NOT gate Moreover, to implement the Hadamard gate, the matrix for U s (T, 0) should be equal to Comparing U H and U s (T, 0) in Eq. (17), we find that µ 0 = π/4 and η 0 = θ = π/2. In this case, up to a global phase π/2, U s (T, 0) becomes the Hadamard gate. Finally, for the π-phase gate, the off-diagonal elements of U s (T, 0) should vanish. Therefore, we set µ 0 = 0. In this case, U s (T, 0) is independent of the value of η 0 , so that we can choose η 0 = 0 for simplicity. Omitting the global phase θ, U s (T, 0) becomes Then, the π-phase gate can be realized with θ = π/2. For the sake of clarity, the corresponding parameters to realize the three gates for θ = π/2 are listed in Table I. According to the parameters given in Table I, on the Bloch sphere in Fig. 1, we plot the trajectories of the eigenvectors |φ ± (t) , i.e., where e k is the unit direction vector along the k-axis. As shown in each panel of Fig. 1, both vectors |φ ± (t) evolve along their cycling paths individually, and the geometric phases acquired by them are equal to half of the solid angles of the areas surrounded by the corresponding paths.
In addition, the solid angles of the paths of r ± (t) have opposite signs, because the paths are on the upper and lower half spheres, respectively. These numerical results are in agreement with the theoretical results in Eq. (16). The average fidelity of the gates over all possible initial states in the subspace S can be calculated by [85,86] with M = P c U † G U 1 P c , while P c and D are the projector and dimension of the computational subspace, respectively. The subscript "G" denotes the desired gate, e.g., G = NOT when one wants to implement the NOT gate. Figure 2(a) shows the infidelity (1 −F NOT ) versus the amplitude α and the Kerr nonlinearity K for the NOT gate, as an example.
We find that the average fidelityF NOT decreases sharply when the amplitude α increases. This effect can be understood because the control parameters χ(t) and (t) increase with exp(|α| 2 ) according to Eq. (14). Consequently, the ratio between the energy gap and parameters of H c (t) reduces, and the leakage to other eigenstates of H cat becomes significant. To manipulate the cat qubit with a larger α, one may increase the Kerr-nonlinearity K and the strength 2 of the squeezing drive, but we should notice that K and 2 both have their upper limits in experiments [51]. We can also consider a longer interaction time T to reduce values of the parameters in H add (t), but such a long-time evolution may increase the influence of decoherence. For a realistic value of the Kerr nonlinearity K = 2π × 12.5 MHz [51] (E gap = 161 MHz), the parameters χ(t) and (t) are shown in Fig. 2(b), when the total interaction time T = 1 µs and the amplitude of coherent states is |α| = 0.5. With these parameters, we obtain F NOT = 0.9997, indicating that the NOT gate can be implemented almost perfectly. To show the performance of different types of quantum gates, we plot the populations of different output states with different input states in the implementation of the NOT, Hadamard, and the π-phase gates in Figs. 3(a), 3(b), and 3(c), respectively. As shown, the populations of the output states are all very close to the ideal values of the theoretical results, and the leakage to unwanted levels is negligible.
For example, we calculate the final fidelities F H (T ) of the Hadamard gate with different input states. For the input state |C + , we obtain Moreover, for the input state |C + , the gate fidelity is For the input state |C − , we obtain Accordingly, the gate fidelity in this case is Therefore, the fidelities of the Hadamard gate for the two input states |C ± are both nearly unity.
In addition, we also plot the average fidelities of the NOT, Hadamard and π-phase gates in Fig. 3(d). The average fidelities of the three gates areF NOT = 0.9997, F H = 0.9999, andF π = 0.9998, respectively. The results show that the three gates are implemented with very high fidelities.

Hamiltonian and evolution operator of two coupled resonators
Now we consider a generalized scheme when two cavity modes are driven by two Kerr-nonlinear resonators, as described by the Hamiltonian The product coherent states |±α 1 ⊗|±α 2 of the modes a 1 and a 2 are the four degenerate eigenstates of H cat2 . Therefore, the product cat states {|C ± 1 ⊗ |C ± 2 } span a four-dimensional computational subspace S 2 useful for implementing two-qubit gates.   Additionally, we consider a control Hamiltonian [47,50,87] where χ 12 (t) is the cross-Kerr parameter, λ(t) is the strength of a resonant longitudinal interaction between the modes a 1 and a 2 ; χ 1/2 (t) are the detunings, and (t) is the strength of an additional coherent driving of the mode a 2 . We assume that the parameters in H c2 (t) should be much smaller than the energy gap E gap2 of the eigenstates of H cat2 to limit the evolution to the subspace S 2 .
To realize geometric controlled θ-rotation gates, we choose the parameters in Eq. (25) as follows The evolution operator of the system reads where 1 2 is the identity operator acting on the cat qubit 2 and U s (T, 0) is the single-qubit operation acting on the cat qubit 2 defined by Eq. (17).

Example of a two-qubit entangling gate
As an example of the application of a two-qubit entangling gate, we show the implementation of a modified controlled-NOT (CNOT) gate defined by the operator corresponding to the parameters η 0 = π/2, µ 0 = π/2, and θ = π/2 for U 1 . The following discussion is based on the parameters α 1 = α 2 = 0.5, K = 2π × 12.5 MHz (E gap = 161 MHz), and T = 1 µs. The average fidelitȳ F CNOT (t) of the implementation of this CNOT gate over all possible initial states in the computational subspace S 2 is defined by [85,86] F CNOT (t) = 1 where given via the projector P c2 and the dimension D 2 = 4 of the computational subspace S 2 . We plot the time variation ofF CNOT (t) in Fig. 4(a), and obtainF CNOT (T ) = 0.9997. Consequently, the modified CNOT gate can be realized almost perfectly.
Populations of different output states with different input states in the implementation of our CNOT-like gate are plotted in Fig. 4(b). As seen, the system does not evolve when the cavity mode 1 is in the cat state |C + 1 . However, if the cavity mode 1 is in the cat state |C − 1 , a nearly perfect population inversion occurs to the cavity mode 2. The result of Fig. 4(b) also indicates that the CNOT gate is successfully implemented with an extremely small leakage to unwanted levels.

IV. DISCUSSIONS ON EXPERIMENTAL IMPERFECTIONS
Here, we estimate the performance of the protocol in the presence of different experimental imperfections. We consider our implementation of a Hadamard gate as an example. First, due to an imperfect calibration of the instruments, there may exist systematic errors in the control parameters. The control parameter under the influence of systematic errors can be written as where Ω e k (t) is a faulty control parameter and δ k is the corresponding error coefficient.
We plot the average fidelityF H of the Hadamard gate versus the systematic error coefficient δ k in Fig. 5(a). It is seen that, when δ x ∈ [−10%, 10%] (δ y ∈ [−10%, 10%]), the average fidelityF H remains higher than 0.9969 (0.9973). In addition, the influence onF H caused by systematic errors of Ω z (t), is larger than those of Ω x (t) and Ω y (t), but we can still obtainF H ≥ 0.9768 when δ z ∈ [−10%, 10%]. Therefore, the implementation of the Hadamard gate is robust against systematic errors.
Apart from systematic errors, due to random noise, there are also fluctuations of parameters that may disturb the evolution of the system. Additive white Gaussian noise (AWGN) is a good model to investigate random processes [82,[88][89][90]. Therefore, we add AWGN to the control parameter as Here, AWGN[Ω k (t), R SN ] is a function that generates AWGN for the original signal Ω k (t) with a signal-tonoise ratio R SN . As AWGN is generated randomly in each single simulation, we perform the numerical simulation averaged over 50 samples to estimate its average effect. Then, 1 −F H in each single simulation is plotted in Fig. 5  Apart from AWGN, the 1/f noise in coherent drives is also a noise limiting the performance of quantum devices. When considering this noise, the control parameter be-comesΩ with noise f −1 [Ω k (t),R SN ] being a function generating the 1/f noise for the original signal Ω k (t) with a signalto-noise ratioR SN . Here, we also perform numerical simulations averaged over 50 samples to estimate the average effect of the 1/f noise withR SN = 10. The results are shown in Fig. 5(c), where we find 1 −F H ∈ [4.884 × 10 −5 , 5.004 × 10 −5 ] averaged over the fifty simulations. Therefore, our implementation of the Hadamard gate is also insensitive to 1/f noise.
As the system cannot be completely isolated from the environment in experiments, the interactions between the system and the environment may result in decoherence. We consider two types of decoherence factors, i.e., a single-photon loss and dephasing. The evolution of the system is described by the Lindblad master equation [47] where κ (κ φ ) is the single-photon-loss (dephasing) rate and the Lindblad superoperator L acting on an arbi- In the presence of decoherence, the evolution is no longer unitary. For the convenience of our discussion, we take the evolution with initial state |C + 1 as an example and analyze the fidelities of the Hadamard gate as We plot the infidelity 1 − F H versus the single photon loss rate κ and the dephasing rate κ φ in Fig. 6(a) in the range [0,0.05 MHz] [48]. The results show that the influence of single-photon loss is stronger than dephasing. When κ, κ φ ≤ 0.05 MHz, 1 − F H (T ) is lower than 0.0201. Therefore, the protocol is robust against single-photon loss and dephasing. In addition, the populations of the output states corresponding to different input states in the implementation of the Hadamard gate with decoherence rates κ = κ φ = 0.05 MHz are plotted in Fig. 6(b). Compared with the results shown in Fig. 3(a), in the presence of decoherence, there exist more faulty populations of the output states. This is because the single-photon loss continuously causes quantum jumps between the cat states |C ± 1 [49]. The total populations in the subspace S with the input states |C ± 1 are both higher than 0.995, showing that the leakage to the unwanted levels outside the subspace S is still very small in the presence of decoherence.

V. AMPLIFICATION OF THE CAT-STATE AMPLITUDE BY SQUEEZING THE DRIVE SIGNAL
To increase the distinguishability of cat-state qubits, we can introduce a method to amplify the photon numbers inspired by Refs. [78,79]. When we consider a squeezing operator as S = exp[r(a †2 − a 2 )/2], the cat states {|C ± } become the amplified cat states {|C ± } with |C ± = S|C ± (i.e., squeezed cat states). Here, we omit the subscript "1" for simplicity. The squeezing operator S can be realized by two-photon (squeezing) driving for the interaction time t s = r/2 2 , i.e., switching off the Kerr interaction and the control field [H c (t)]. In addition, the inverse transformS = S † can be realized by squeezing the driving interaction Average photon number Step 1 Step 2 Step 3  3) is an antisqueezing (squeezing) process for a measurable input (output) state.
Step 2 denotes evolution of the system implementing a given gate (e.g., a Hadamard gate in the bottom panel).
for the interaction time t s = r/2 2 . In this case, the total process can be divided into three steps as shown in Fig. 7.
In step 1, we apply the transformS to the input state, which is a superposition of the squeezed cat states. In this way, the amplified cat states {|C ± } are transformed into small-amplitude cat states {|C ± }. The step 2 is the geometric gate operation, as illustrated in Sec. III. In step 3, we apply the squeezing transform S to enhance the photon number of the output state, so that the output state can be experimentally detected. The total operator acting on the amplified cat-state qubit isŨ = SU S † , which has the same matrix elements for a smallamplitude cat-state qubit, i.e., C  |Ũ |C  = C  |U |C  (,  = ±). Here, we assume r = 1.2 and α = 0.5, as an example to show the implementation of NGQC for the amplified-cat-state qubits.
We plot the time variation of the average photon number n p (t) = Tr[ρ(t)a † a] in the bottom panel of Fig. 7. As shown, the average photon number decreases during the first step, corresponding to the anti-squeezing process |C + → |C + . Then, by implementing the Hadamard gate, the cat state |C + 1 is transformed into (|C + + |C − )/ √ 2. Finally, in step 3, we amplify the output state by squeezing, i.e., The final average photon number is 6.732. The antisqueezing and squeezing processes are fast (see Fig. 7), so that decoherence in these two processes affects weakly the target state. By considering the experimentally feasible parameters: 2 = Kα 2 = 2π × 3.125 MHz, t s = r/ 2 = 30.56 ns, and κ = κ φ = 0.05 MHz, we achieve the fidelity Φ( ) F H = 0.9513 of the output state (|C + + |C − )/ √ 2 for the Hadamard gate with the initial state |C + for the amplified cat-state qubit.

VI. POSSIBLE IMPLEMENTATION USING SUPERCONDUCTING QUANTUM CIRCUITS
A. Single-qubit geometric quantum gates As shown in Fig. 8, we consider an array-type resonator composed of N superconducting quantum interference devices (SQUIDs) [91][92][93][94][95][96][97]. An ac gate voltagẽ V p = V p cos(ω p t + ϕ p ) (with amplitude V p , frequency ω p , and phase ϕ p ) is applied to induce linear transitions between eigenlevels. The Hamiltonian of this setup reads [93,95,97] wheren is the number of Cooper pairs andφ is the overall phase across the junction array.
Here, E C is the resonator charging energy, E J is the Josephson energy of a single SQUID, and N is the number of SQUIDs in the array. The Josephson energy is periodically modulated (with frequency ω 2p and phase ϕ 2p ) by the external magnetic flux Φ(t), leading to After the Taylor expansion of cos(φ/N ) to fourth order, we obtain whereX = (φ/N ) 2 /2. The quadratic time-independent part of the Hamiltonian can be diagonalized by defininĝ where n 0 = [E J /(32N E C )] 1/4 and φ 0 = 1/2n 0 are the zero-point fluctuations.
By dropping the constant terms, the Hamiltonian H 0 becomes where ω c = 8E C E J /N . For simplicity, we assume ω 2p = 2ω p and ϕ 2p = −2ξ. Then, moving into a rotating frame at frequency ω p and neglecting all of the fast oscillating terms, the approximate Hamiltonian under the rotating wave approximation (RWA) can be written as Then, by defining a 1 ≡ a, we recover the total Hamiltonian H tot (t) given in Eq. (9) for the single-qubit case. Note that, as shown in [98], the Kerr nonlinearity (which is a rescaled third-order susceptibility of a nonlinear medium) can be exponentially enhanced by applying quadrature squeezing, which interaction with a medium is proportional to its second-order susceptibility. This means that one can exponentially amplify higher-order nonlinearities by applying lower-order nonlinear effects. Other methods of applying quadrature squeezing to increase nonlinear interactions are described in, e.g., [99][100][101].

B. Errors in the sperconducting circuit
According to the above analysis, the control drives depend on the parameters E C , E J , and V p . Assuming that the parameters with errors respectively become The errors in the control parameters can be approximately calculated by Therefore, when considering the errors in the control parameters, the Hamiltonian becomes where δH cat induces an error on the amplitude of the coherence states |± α as and δH c influences the control drives designed by invariant-based reverse engineering. For |δE C /E C |, |δE J /E J | ≤ 0.1, and α = 0.5, the absolute error of the coherent-state amplitude approximately satisfies |δα| ≤ 0.0375. Because of withα ± = 0.5 ± 0.0375, the cat qubit can be near perfectly stabilized in the preset states by a small fluctuation of the amplitude α.
Moreover, according to Eq. (14) and δH c in Eq. (43), the errors in Ω x (t), Ω y (t), and Ω z (t) can be approximately described by the linear superpositions of δE J , δE C , and δV p . Therefore, when E J , E C , and V p are subjected to AWGN or 1/f noise, the wave shapes of Ω x (t), Ω y (t), and Ω z (t) are also mixed with the same type of noise. According to the analysis in Sec. IV, the fidelities of the gates are insensitive to these types of noise in Ω x (t), Ω y (t), and Ω z (t). Consequently, our proposal is robust against AWGN and 1/f noise in E J and E C of the considered superconducting circuits.

C. Kitten states
We also note that the Kerr nonlinearity enables the generation of not only conventional (i.e., two-component) Schrödinger cat states, i.e., superpositions of two macroscopically distinct states, but also the generation of superpositions of a larger number of macroscopically distinct states. The states are referred to as Schrödinger kitten states or multi-component cat-like states, as predicted in [102] and experimentally generated via the Kerr interaction in superconducting quantum circuits in [103].
These kitten states, which are examples of Gauss sums, have been used in an unconventional algorithm for number factorization, i.e., to distinguish between factors and nonfactors. Implementations of the Gauss-sums algorithm include NMR spectroscopy [104,105] and Ramsey spectroscopy using cold atoms [106].

D. Two-qubit geometric entangling gates
To realize two-qubit geometric gates, we consider two superconducting circuits with the same structure shown in Fig. 8. The two circuits are coupled with each other through a Josephson-junction coupler with Josephson en-ergyĒ J [87,107]. The coupler provides a coupling between the two circuits as withφ m = φ 0m (a m + a † m ) (m = 1, 2). Here,Φ(t) is the external flux applied to the loop of the coupler, and Φ 0 is the flux quantum. By modulatingΦ(t) at the frequency of the second circuit, the required cross-Kerr nonlinearity χ ab (t)a † 1 a 1 a † 2 a 2 and the longitudinal interaction can be realized with the Taylor expansion of cos[φ 2 −φ 1 + 2πΦ(t)/Φ 0 ] to fourth order [87]. The coupling between two circuits changes the detuning and strength of the two-photon drive of each circuit. However, to diagonalize the quadratic time-independent part of the Hamiltonian, the coefficients ofn 2 m andφ 2 m should be set equal, wheren m = −in 0m (a m − a † m ). Consequently, one can derive that the strength of the self-Kerr nonlinearity for each circuit is still expressed by K = E C /2N 2 .

E. Comparison of gate times
We now compare the gate time of our proposal with previous experiments. Relevant information is also listed in Table II. The gate time in our proposal is comparable with that in previous experiments using superconducting systems. For example, the gate time of a controlled-π-phase gate in Ref. [108] using superconducting qubits is about 113 ns, with gate fidelities ∼ 0.99. In Ref. [109], the gate time for implementing the Hadamard gate and the NOT gate with fidelities ∼ 0.99 (by using a transmon qubit coupled to a transmission-line resonator) is ∼ 62.5 ns. In our proposal, to implement the NOT and Hadamard gates with fidelity higher than 0.99, the gate time can be selected as T = 210 ns. To implement the CNOT gate with fidelity higher than 0.99, the gate time in the present scheme is about 220 ns.
Moreover, compared with some previous methods using dipole interactions between neutral atoms, the gate time for two-qubit gates in the present approach is generally shorter. For example, in Ref. [90], the gate time to realize the controlled-π-phase gate with about 10 −4 infidelity is 376.16 µs using the available dipole interaction strengths V = 20 MHz. In the scheme of Ref. [40], the gate time to implement the CNOT gate for two neutral atoms with infidelity about 10 −3 is 85.94 µs, using the reported dipole interaction strengths V = 2π × 50 MHz.
In the present protocol, the gate times for implementing the controlled-π-phase gate and the CNOT gate, with infidelities about 10 −4 to 10 −3 , can be both less than 1 µs with the available Kerr nonlinearity K = 2π × 12.5 MHz. The longer gate times of the controlled-π-phase gate and the CNOT gate in the schemes of Refs. [40,90] is because these protocols should work in the Rydberg blockade regime, which limits the strength of the control fields.
Recently, some modified nonadiabatic geometric quantum computation methods [110][111][112][113], based on the dipole interactions of neutral atoms, have been proposed, which work in different regimes and relax the limitation of strong amplitudes of control fields. The gate times of the controlled-π-phase gate and the CNOT gate in these schemes can be improved to about 1-10 µs. In the present protocol, as the Kerr nonlinearity can provide much bounder energy gap between eigenvectors of H cat , the control fields with stronger amplitudes can be adopted.
In addition, as reported in the scheme of Ref. [114], to implement multi-atom geometric quantum gates in cavity quantum electrodynamics system with the amplitude of laser pulses about 50-200 MHz with fidelity over 0.999, the gate time is about 0.5-3 µs, which is also close to the gate time in the present approach.
Compared with the present method, the gate time of the nonadiabatic geometric quantum computation schemes [115,116] in trapped-ion systems is slower. Because these schemes work in the Lamb-Dicke limit, the amplitudes of driving pulses should be much less than the frequency (1-10 MHz) of the vibration modes of trapped ions, and the gate time is about 1-2 ms.

VII. CONCLUSION
In conclusion, we have proposed a method to realize nonadiabatic geometric quantum computation using cat qubits with invariant-based reverse engineering. The evolution of the cavity mode is restricted to a subspace spanned by a pair of Schrödinger cat states assisted by a Kerr nonlinearity and a two-photon squeezing drive, so that one can generate photonic cat qubits. We add a coherent field to linearly drive the cavity mode, inducing oscillations between dressed cat states. When designing the control fields by invariant-based reverse engineering, the system can evolve quasiperiodically and acquire only pure geometric phases. Thus, one can realize nonadiabatic geometric quantum computation with a cat qubit. By amplifying the amplitudes of different cat states, the input and output states can be easily detected in exper-iments.
Two-qubit quantum gates for cat qubits are also considered with couplings between two cavity modes. As we have shown, the controlled two-qubit geometric quantum entangling gates can also be implemented with high fidelities. The influence of systematic errors, AWGN, 1/f noise, and decoherence (including photon loss and dephasing), was studied here using numerical simulations.
The results indicate that our approach is robust against these errors. Therefore, our protocol can provide efficient high-fidelity quantum gates for nonadiabatic geometric quantum computation in bosonic systems.