Effective implementation of nonadiabatic geometric quantum gates of cat-state qubits using an auxiliary qutrit

We propose an effective protocol for the implementation of nonadiabatic geometric quantum gates of cat-state qubits in Kerr-nonlinear resonators driven by two-photon squeezing drives. Coupling the Kerr-nonlinear resonators with an auxiliary qutrit with proper coupling strengths, the selective transition of the auxiliary qutrit is realized. The selective transition can be exploited in the implementation of a set of useful quantum gates, including the phase gates, the NOT gates, the controlled-phase gates, the controlled NOT gates, and the Toffoli gates. Numerical simulations show the implementations of different types of gates are robust against systematic errors, random noise, and decoherence. Therefore, the protocol may be helpful for robust and scalable quantum computation based on cat-state qubits.


Introduction
Quantum computation is one of the most important research fields in quantum information science [1][2][3]. Using quantum coherence and entanglement, quantum computation has shown superiority in solving some specific problems, such as factoring large integers and searching unsorted databases [4][5][6][7]. Generally speaking, a quantum algorithm usually requires a sequence of quantum gates [8][9][10][11]. However, in a real implementation of a quantum gate, the fidelity may be reduced by imperfect factors, e.g. systematic errors, random noise, and decoherence [12][13][14][15]. Thus, it is still challenging to realize a functional quantum computer, because the accuracy of a quantum algorithm is usually spoiled by the errors accumulated in a gate sequence. In order to guarantee the accuracy of quantum computation, efforts have been devoted from two different aspects, improving the fidelity of every single gate against experimental imperfections [16][17][18] and correcting errors accumulated in the gate sequence [19][20][21].
Among the ideas for improving the fidelity of quantum gates, nonadiabatic geometric quantum computation (NGQC) is a promising one [22][23][24] that takes advantage of geometric phases. Because the geometric phases depend on the global properties of evolution paths [25][26][27][28], NGQC is insensitive to the local parameter fluctuation along the evolution paths [29][30][31]. Moreover, since NGQC does not rely on adiabatic evolution, the gate speed is much faster than that in the traditional adiabatic quantum computation [32,33]. Accordingly, the influence of decoherence is suppressed due to the decrease of the interaction time. Recently, NGQC is shown to be compatible with optimal control methods [34][35][36]. Assisted by optimal control methods, NGQC can be implemented with robustness against systematic errors [37][38][39][40][41]. The above results indicate that NGQC holds comprehensive resistance to systematic errors, random noise, and decoherence. Therefore, NGQC is an alternative candidates to produce high-fidelity quantum gates.
On the other hand, to correct errors accumulated in a gate sequence, fault-tolerance quantum codes [42][43][44] are required. To data, there are still some challenges in implementing fault-tolerance quantum

Physical model and Hamiltonian
We consider a physical model consists of N resonators (n = 1, 2, . . . , N), where the resonator modes are exploited as the computational qubits. The quantum information of the resonator mode n is encoded in a pair of orthonormal states |Ψ ± ⟩ n = (|C + ⟩ n ± |C − ⟩ n )/ √ 2. Here, |C + ⟩ n and |C − ⟩ n denote the even and odd cat states. Assuming that | ± α⟩ n are the coherence state of the resonator mode n with the complex amplitude ±α, |C ± ⟩ n are given by with the normalized coefficients N ± = 2[1 ± exp(−2|α| 2 )]. For a large-amplitude cat-state qubit (|α| ≫ 1), we have |⟨Ψ ± | ± α⟩ n | 2 ≃ 1 and |⟨Ψ ± | ∓ α⟩ n | 2 ≃ 0. In fact, one can obtain |⟨Ψ ± | ± α⟩ n | 2 = 1 − 4.85 × 10 −8 and |⟨Ψ ± | ∓ α⟩ n | 2 = 2.81 × 10 −8 even when α = 2. If we apply a resonant two-photon squeezing drive ϵ n with a Kerr nonlinearity K n to the resonator n, the Hamiltonian of the system in the frame rotating at the resonator frequency ω c (denoted by R c ) is (ℏ = 1) [64,68] H c = N n=1 H cn , H cn = −K n a †2 n a 2 n + ϵ n a †2 n + ϵ * n a 2 n , where a n (a † n ) is the annihilation (creation) operator of the resonator mode. |Ψ ± ⟩ n (|C ± ⟩ n ) are two orthonormal degenerate eigenstates of H cn if ϵ n = K n α 2 . When the Kerr nonlinearity is strong enough compared with other couplings, the energy gap E gap between these two states and the nearest eigenstates of H cn can be very large to suppress the leakage from the subspace spanned by {|Ψ ± ⟩ n } [68].
In addition, we consider a three-level auxiliary qutrit q. As shown in figure 1. The levels of qutrit q are denoted as |g⟩ q , |e⟩ q and |f ⟩ q . The transition |g⟩ q ↔ |e⟩ q (|e⟩ q ↔ |f ⟩ q , |g⟩ q ↔ |f ⟩ q ) is resonantly driven by a classical field with Rabi frequency Ω 1 (t) [Ω 2 (t), Ω 3 (t)]. Moreover, the transition |g⟩ q ↔ |e⟩ q (|e⟩ q ↔ |f ⟩ q ) is also resonantly coupled to the resonator mode 1 (ñ > 1) with the coupling strength g 1 (t)[gñ(t)]. The Hamiltonian of the auxiliary qutrit interacting with the classical fields and the resonator modes [69,70] can be written as H q1 (t) = [Ω 1 (t) + g 1 (t)a † 1 ]|g⟩ q ⟨e| + H.c., By exploiting the auxiliary qutrit and the resonant two-photon drive with Kerr nonlinearity, it is possible to implement nonadiabatic geometric single-qubit phase/NOT gate, two-qubit controlled-phase/CNOT, and multi-qubit controlled-phase/Toffoli gate, which will be described in the following sections.

Single-qubit arbitrary phase gate
Let us now describe the implementation of the single-qubit arbitrary phase gate. The outline and flowchart is shown in figure 2(a). To implement the phase gate, Ω 2 (t), Ω 3 (t) and gñ(t) are switched off. The auxiliary qutrit is initially prepared in the state |g⟩ q . In this case, the auxiliary qutrit only couples to the resonator 1. Thus, the Hamiltonian of the system composed of the auxiliary qutrit q and the resonator 1 is H p (t) = H c1 + H q1 (t). If the condition E gap ≫ |g 1 (t)| is satisfied, the evolution of the resonator 1 is restricted in the subspace spanned by {|Ψ ± ⟩ 1 }. Consequently, the Hamiltonian H p (t) can be simplified as with the coefficients β ± = ⟨Ψ ± |a|Ψ ± ⟩ 1 = ±2α/ N + N − and the vectors |g(e), Ψ ± ⟩ = |g(e)⟩ q ⊗ |Ψ ± ⟩ 1 .
Since H res (t) is proportional to exp(−2|α| 2 ), and can be neglected when |α| ≫ 1, it is considered as a remainder term and omitted. Then, the condition Ω 1 (t) = g 1 (t)β − = Ω p (t)/2 is further considered. The Hamiltonian H ′ p (t) is finally simplified asH Here, we design Ω p (t) with two time-dependent parameters η(t) and µ(t) as and setting the parameters as µ(t) = π sin 2 (π t/T), The wave shapes of Re[Ω p (t)] and Im[Ω p (t)] are shown in figure 5(a). Based on such design of Ω p (t), one can realize a geometric Θ p -phase gate for the cat-state qubit of resonator mode 1 as with the robustness against the systematic error H (se) to the second order of error coefficient ϵ (see appendix C for detailed deviations).

Multi-qubit controlled-phase gate
We now start to describe the implementation of the multi-qubit controlled-phase gate. To make the explanations more clear, we first focus on the implementation of the two-qubit controlled-phase gate, where the Hamiltonian is H cp (t) = 2 n=1 H cn + H qn (t), containing the interactions between the auxiliary qutrit and both the resonators 1 and 2. The auxiliary qutrit is initially prepared in the state |g⟩ q . In addition, the condition E gap ≫ |g n (t)| is considered to restrict the evolution of the resonators 1 and 2 in the subspace spanned by {|Ψ ± , Ψ ± ⟩ 1,2 = |Ψ ± ⟩ 1 ⊗ |Ψ ± ⟩ 2 }. Similar to the single-qubit case, omitting the remainder term proportional to exp(−2|α| 2 ) (here, the amplitudes of cat-state qubits 1 and 2 are both set as α for simplicity), we obtain an effective Hamiltonian as To realize the controlled-phase gates, we first set Ω 1 (t) = g 1 (t)β − = Ω cp (t)/2 and Ω 2 (t) = g 2 (t)β + =Ω cp /2, whereΩ cp is a time-independent parameter. Then, the Hamiltonian H ′ cp (t) in equation (9) becomes Noticing that |±, are the eigenvectors ofH cp2 with eigenvalues ±Ω cp . By performing a rotation transform withR 2 WhenΩ cp ≫ Ω cp (t),H cp1 (t) can be omitted as it contains only terms with high-frequency oscillations. Consequently, the final effective Hamiltonian to realize the two-qubit controlled-phase gate isH cp2 (t).
Considering the fact thatH cp2 possesses similar dynamic structure asH p (t) in equation (5), we can design Ω cp (t) with the same parameters ζ(t) and µ(t) as shown in equations (6) and (7) by replacing Ω p (t) and Θ p with Ω cp (t) and Θ cp , respectively. Then, the geometric two-qubit controlled-Θ cp -phase gate can be realized. Now, we extend the implementations of the two-qubit controlled-phase gate to the implementations of the N-qubit controlled-phase gates (N ⩾ 3), where the Hamiltonian is H Similar to the deviations of equation (9), under the conditions E gap ≫ |g n (t)| and |α| ≫ 1, one obtains an effective Hamiltonian as where we define with n 1 ⩽ n 2 and I n = ι=± |Ψ ι ⟩ n ⟨Ψ ι | (the identity operator for the cat-state qubit of the resonator n). By with Here, |Ψ (ȷ) j ⟩ 2,N denotes the ȷ-th permutation state for the resonators {ñ = 2, 3 . . . , N} with total j resonators in the state |Ψ + ⟩ñ and N − 1 − j resonators in the state |Ψ − ⟩ñ, N j = C j N−1 is the aggregate of all possible permutations, and T j (ȷ) is the operator to transform the canonical-permutation state to the state with ȷth permutation. It is easy to find that the state |Ψ ȷ cp2 with eigenvalue jΩ cp . Specially, when j = 0, there exists only one permutation as |Ψ cp2 . Considering the conditionΩ cp ≫ Ω cp (t), similar to the deviations of equation (11), one can derive an effective Hamiltonian as by omitting terms with high-frequency oscillations. Therefore, using the same design for Ω cp (t) in the implementation of the two-qubit controlled-Θ cp -phase gate, one can realize an N-qubit controlled-Θ cp -phase gate as with the HamiltonianH cp (t) in equation (17). To summarize the implementation of the multi-qubit controlled-phase gate, we show the outline and flowchart is shown in figure 2(b).

Single-qubit NOT gate
Let us now study the implementation of the single-qubit NOT gate, where the outline and flowchart is shown in figure 3. Here, Ω 2 (t) and g 2 (t) are switched off, and the auxiliary qutrit is initially prepared in the state |g⟩ q . Thus, the auxiliary qutrit only couples to the resonator 1. The Hamiltonian of the system composed of the auxiliary qutrit q and the resonator is H NOT (t) = H c1 + H q1 (t). The procedure of the NOT gate can be divided into five steps, where the final time of Step m (m = 1, 2, . . . , 5) is assumed as τ m . We will introduce the implementation of the NOT gate step by step in the following.
The control field Ω s1 (t) is designed in form of equation (6) using parameters µ(t) and ζ(t) (i.e. replacing Ω p (t) in equation (6) by Ω s1 (t)). When the parameters are chosen as one can realize an evolution as with |±⟩ q = (|g⟩ q ± |e⟩ q )/2 and a phase shift α s1 (see appendix C for details).
Step 2: In this step, the control fields are set as Ω 1 (t) = g 1 (t)β − = Ω s2 (t)/2. Then, the effective Hamiltonian becomes We consider that Ω s2 (t) is set in form of equation (6) with parameters µ(t) and ζ(t) (i.e. replacing Ω p (t) in equation (6) by Ω s2 (t)). By designing an evolution as with a phase shift α s2 can be realized (see appendix C for details).
Step 3: In this step, we switch off the drive of the Kerr-nonlinearity K 1 and the two-photon squeezing ϵ 1 for the resonator 1, i.e. K 1 = ϵ 1 = 0. In addition, the classical field Ω 1 (t) is set as a real constant Ω s3 . The Hamiltonian of the system can be described as Considering that |±⟩ q are the eigenvectors ofH s3 , we can rewriteH s3 andH s3 (t) using |±⟩ q as In the rotating frame of R s3 (t) = exp(−iH s3 t), the Hamiltonian H s3 (t) becomes By neglecting the terms with high-frequency oscillations [exp(±2iΩ s3 t)] under the condition g 1 (t) ≪ Ω s3 , one can obtain an effective Hamiltonian as With the Hamiltonian H ′ s3 (t) in equation (27), the evolution operator in the rotating frame R s3 (t) can be calculated as where denote the displacement operators for the resonator mode 1. 1 . From the above result, when the auxiliary qutrit is in the state |+⟩ q , a displacement |α⟩ 1 → | − α⟩ 1 can be achieved for the resonator mode 1. While for auxiliary qutrit in the state |−⟩ q , a displacement | − α⟩ 1 → |α⟩ 1 can be realized.

Combining the results
with λ ± = ( N + + N − )/ 2N + N − , lim α→∞ λ + = 1, and lim α→∞ λ − = 0, one can derive with |α| ≫ 1. This means that the states of the resonator mode 1 can be turned from |Ψ ± ⟩ 1 to |Ψ ∓ ⟩ 1 when the states of the auxiliary qutrit are |±⟩ q . Therefore, after Step 3, we realize the evolution as Step 4: In this step, we switch on the Kerr-nonlinearity K 1 and the two-photon squeezing ϵ 1 for the resonator 1, and set the control fields as When we design the control field Ω s4 (t) in form of equation (6) by using the parameters µ(t) and ζ(t) designed as the evolution in Step 4 is where the phase shift α s4 of the state |ψ 1 (t)⟩ in Step 4 is offset by that in Step 1, i.e. α s4 = −α s1 (see appendix C for detailed derivations).
Step 5: We change the control fields as Ω 1 (t) = g 1 (t)β + = Ω s5 (t)/2, then the effective Hamiltonian becomes Designing Ω s5 (t) in form of equation (6) with the parameters µ(t) and ζ(t) as the evolution can be achieved, where the phase shift α s5 of the state |ψ 2 (t)⟩ in Step 5 is offset by that in Step 2, i.e. α s5 = −α s2 (see appendix C for details). After the total five steps, the evolution in the computational subspace is which is a NOT gate for the cat-state qubit of the resonator 1. We assume |φ with an auxiliary function υ(t) satisfying boundary conditions υ(0) = 0 and υ(τ 5 ) = π. It is easy to find that both the vectors |φ ± (t)⟩ go through cycling evolutions in the time interval [0, is the solution of the Schrödinger equation for the considered system,Ξ ± (t) = |φ ± (t)⟩⟨φ ± (t)| obviously satisfies the von Neumann equation shown in equation (A1). In addition, we can prove that the dynamic phases acquired by |φ ± (t)⟩ in Step 1 (2) are completely offset by that in Step 4 (5), and there are not dynamic phases acquired in Step 3 (see appendix C). Therefore, the implemented NOT gate is a pure geometric gate according to the conditions of NGQC shown in appendix A.

Two-qubit CNOT gate
Let us first consider the implementation of a two-qubit CNOT gate, where the Hamiltonian is The implementation of the two-qubit CNOT gate can be divided into five steps (the final time of each step is supposed asτ m ) as shown in the following. In addition, the outline and flowchart is shown in figure 4 (N = 2 case).
Step 1: We consider the conditions E gap ≫ |g n (t)| and |α| ≫ 1. Similar to the derivations in section 3.2, an effective Hamiltonian can be obtained as by setting with ε± = ±1. ConsideringΩ s1 (t) as a positive real constant satisfying Ω 1 (t) ≪Ω s1 , one can derive an effective Hamiltonian as If we design Ω 1 (t) in form of equation (6) with two time-dependent parameters ζ(t) and µ(t), i.e. replacing Ω p (t) in equation (6) by Ω 1 (t), the evolution of the systems as can be realized with parameters chosen as with a phase shift α s1 (see appendix C for details). Step 2: The control fields are set as Step 2 is still given in form of equation (6) with ζ(t) and µ(t). When the parameters are selected as the evolution of the systems is (see appendix C for details) with |ψ 1 (τ 2 )⟩ = |ψ 1 (τ 1 )⟩ and |ψ 2 (τ 2 )⟩ = |ψ 2 (τ 1 )⟩.
Step 4: In this step, the control fields are set as Ω 1 (t) = Ω 2 (t) = g 1 (t) = g 2 (t) = 0, giving the Hamiltonian H s4 (t) =H q (t). Similar to Step 2, we design Ω 3 (t) in form of equation (6) with ζ(t) and µ(t) selected as The evolution of the system in Step 4 is Step 5: We consider the conditions and Ω 1 (t) ≪Ω s1 . Similar to the derivations for Step 1, an effective Hamiltonian can be obtained as Designing Ω 1 (t) in form of equation (6) with ζ(t) and µ(t) set as the evolution of the system is After the total five steps, the evolution operator of the system is (50) is a two-qubit CNOT gate with the cat-state qubit in resonator 1 being the controlled qubit and that in resonator 2 being the control qubit. Similar to section 3.3, we define a set of auxiliary vectors as Step 2 and Step 4, as the vectors |φ 1 (t)⟩ and |φ 2 (t)⟩ do not evolve, they do not acquire dynamic phases. In addition, according to the results of section 3.3, the total dynamic phases acquired by |φ 1 (t)⟩ and |φ 2 (t)⟩ in Step 3 are both zeros. For vectors |φ 3 (t)⟩ and |φ 4 (t)⟩, they do not evolve in Step 1, Step 3, and Step 5, thus acquiring zero dynamic phases in these steps. Moreover, the dynamic phases acquired by |φ 3 (t)⟩ and |φ 4 (t)⟩ in Step 2 are completely offset by those in Step 4 (see appendix C for details). Therefore, the dynamic phases acquired by the vectors {|φ℘(t)⟩} in the whole five steps are all zeros, and the implemented CNOT gate is a geometric quantum gate.

Multi-qubit Toffoli gate
Let us now extend the implementations of two-qubit CNOT gate to the implementations of the N-qubit Toffoli gates (N ⩾ 3), where the Hamiltonian is H (N) Similar to the implementation of the CNOT gate, the implementation of the N-qubit Toffoli gate can be also divided into five steps shown as follows. The outline and flowchart is shown in figure 4.
Step 1: We consider the conditions E gap ≫ |g n (t)|, |α| ≫ 1, gñ(t)β + =Ω s1 /2, Ω 2 (t) = (N − 1)Ω s1 /2, and g 1 (t) = Ω 3 (t) = 0. Similar to the derivation of equation (15), one can obtain an effective Hamiltonian of the system asH Under the conditionΩ s1 ≫ Ω 1 (t), similar to the deviations of equation (17), one can derive an effective Hamiltonian as by omitting terms with high-frequency oscillations. Using the same design of Ω 1 (t) for Step 1 of the two-qubit CNOT gate, one can realize an evolution as Step 2: We set the control fields as Ω 1 (t) = Ω 2 (t) = g n (t) = 0. The Hamiltonian for Step 2 is H Using the same design of Ω 3 (t) for Step 2 of the two-qubit CNOT gate, the evolution of the system can be described by Step 3: The operation in Step 3 of the N-qubit Toffoli gate is the same as that in Step 3 of the two-qubit CNOT gate. If the system is in the state |ψ (N) j ′ ,ȷ ′ ,ι (t)⟩, as the qutrit is in the state |f ⟩ q , the system dose not evolve. While for the system is in the state |ψ (N) 0,1,ι (t)⟩, according to the results of section 3.3, a single-qubit NOT gate is applied to cat-state qubit of resonator 1. Therefore, the evolution of the system in this step is Step 4: The control fields are set as Ω 1 (t) = Ω 2 (t) = g n (t) = 0, and the Hamiltonian for Step 4 is H . Therefore, using the same design of Ω 3 (t) for Step 4 of the two-qubit CNOT gate, one can realize an evolution as withῑ = ∓ when ι = ±.
Step 5: We consider the conditions E gap ≫ |g n (t)|, |α| ≫ 1, gñ(t)β + =Ω s1 /2, Ω 2 (t) = (N − 1)Ω s1 /2, g 1 (t) = Ω 3 (t) = 0, andΩ s1 ≫ Ω 1 (t). Similar to the the results in Step 1 of the N-qubit Toffoli gate, the effective Hamiltonian is equal to H (N) s1 (t) in equation (52). By using the same design of Ω 1 (t) for Step 5 of the two-qubit CNOT gate, the evolution of the system can be written as After the whole five steps, the evolution of the system can be described by the evolution operator which is an N-qubit Toffoli gate with the cat-state qubit of the resonator 1 being the controlled qubit and the cat-state qubit of the cavitiyñ (ñ > 1) being the control qubit. Defining the auxiliary vectors {|φ

Numerical simulations and discussions
In this section, let us estimate the performance of different kinds of geometric quantum gates under the systematic errors, random noise, and decoherence factors with some specific examples. For simplicity, the amplitudes of the cat-state qubits are all set as α = 2 in the following discussions.

Single-qubit phase gate
To demonstrate the implementation of the phase gates, a π/2-phase gate is given as an example. By setting Θ p = π/2, time variations of the real part Re[Ω p (t)] and the imaginary part Im[Ω p (t)] of the control field Ω p (t) with T = 1 µs are shown in figure 5(a).
In order to estimate the performance of the π/2-phase gate with different initial states, let us consider the average fidelity over all possible initial states defined as [71,72] being the target π/2-phase gate, U p (t) being the evolution operator of the system under the control of H p (t), P p = |g, Ψ + ⟩⟨g, Ψ + | + |g, Ψ − ⟩⟨g, Ψ − | being the projection operator onto the computational subspace, and N = 2 for the two-dimensional computational subspace.
Because the cat-state qubit should be stabilized in the two-dimensional subspace spanned by {|Ψ ± ⟩ 1 } in the implementation of the π/2-phase gate, we should first select a suitable value of the Kerr nonlinearity K to provide enough energy gap E gap . To this end, we plotF p (T) versus K/2π in figure 5(b). According to figure 5(b), the final average fidelityF p (T) is beyond 0.9998 when K ⩾ 2π × 2 MHz. When we consider an available Kerr nonlinearity as K = 2π × 8 MHz [64,68], the average infidelity is only 1 −F p (T) = 9.9328 × 10 −7 . The result indicates that K = 2π × 8 MHz can provides enough energy gap (E gap = 658 MHz) to implement accurate geometric phase gate. Therefore, we fix K = 2π × 8 MHz in the following simulations.
With the selected value of the Kerr nonlinearity, we also investigate the time-variation of the average fidelity. The average fidelityF p (t) of the π/2-phase gate versus t is plotted in figure 5(c), which shows the π/2-phase gate is close to 1 at the final time. Therefore, the control fields designed in section 3.1 can be successfully used in the implementation of the π/2-phase gate.
Due to operational and instrumental imperfections, the systematic errors may appear in experiments. As shown in figure 5(b), the fidelity of the π/2-phase gate is insensitive to the deviation of K. This is because the Kerr nonlinearity is used to form a large energy gap E gap between eigenvectors of H c , and it is not necessary for us to accurately control the value of K. Therefore, we only consider the system error of the control field as Ω p (t) → (1 + ϵ)Ω p (t), where ϵ is the coefficient of the systematic error. The final average fidelityF p (T) of the π/2-phase gate versus the error coefficient ϵ is plotted in figure 5(d), where we can find thatF p (T) is higher than 0.993 when ϵ ∈ [−0.2, 0.2]. This proves that the implementation of the π/2-phase gate is very robust against the influence of the systematic error of the control field.
Apart from the systematic errors, the random noise is also a disturbing factor in experiments. As a good model to characterize the influence of random noise, additive Gaussian white noise (AWGN) [15,73,74] is considered here to study the influence of the random noise. The control field Ω p (t) subjected to AWGN can be described as Ω p (t) → Ω p (t) + awgn[Ω p (t), R SN ], where awgn denotes a function generating AWGN for the signal field Ω p (t) with signal-to-noise ratio (SNR) R SN . Because the effect of AWGN is different in two individual experiments, we perform the numerical simulations 50 times to estimate the average effect of AWGN. The final average infidelities 1 −F p (T) of the π/2-phase gate under the influence of AWGN versus simulation counts with SNR R SN = 10 is shown in figure 5(e). The results indicate that the final average infidelities 1 −F p (T) in the fifty simulations are all less than 6.16 × 10 −6 . Accordingly, the implementation of the π/2-phase gate in the protocol is robust against AWGN.
The decoherence factors, due to interactions between the systems and the environment, also influence the gate fidelities. In the implementation of the π/2-phase gate of the present protocol, the most significant decoherence factors are the resonator decay (decay rate κ), the qutrit energy relaxation (relaxation rate γ), and the qutrit dephasing (dephasing rate γ ϕ ). In the implementation of the π/2-phase gate, when the system is subjected to these decohrence factors, the evolution is gorvened by the master equatioṅ with S − = |g⟩ q ⟨e|, S z = |e⟩ q ⟨e| − |g⟩ q ⟨g|, and L[X]ρ p = 2Xρ p (t)X † − X † Xρ p (t) − ρ p (t)X † X for an arbitrary operator X. Here, ρ p (t) means the density operator of the system in the implementation of the π/2-phase gate. As the evolution governed by the master equation in equation (59) is not unitary, the average fidelitȳ F p (T) cannot be used here. Consequently, we consider the system in the initial state |ψ 0 p ⟩ = |g⟩ q ⊗ (|Ψ + ⟩ 1 + |Ψ − ⟩ 1 )/ √ 2 to perform the numerical simulations. In this case, the fidelity of the π/2-phase gate can be calculated by F p (t) = ⟨ψ 0 p |Ũ † p ρ p (t)Ũ p |ψ 0 p ⟩. The final fidelity F p (T) of the π/2-phase gate with the initial state |ψ 0 p ⟩ versus different decoherence rates is shown in figure 5(f). We can find in figure 5(f) that, the implementation of the π/2-phase gate has good robust against qutrit energy relaxation. The fidelity F p (T) of the π/2-phase gate is higher than 0.9777 when the energy relaxation rate satisfies γ ⩽ 0.1 MHz. Considering the qutrit energy relaxation time 20-35 µs (corresponding to energy relaxation rate 0.0286-0.05 MHz) for a superconducting qutrit, reported in [65], the fidelity of the π/2-phase gate in the present protocol can range in [0.9887, 0.9935]. Accordingly, under the influence of the qutrit energy relaxation, the π/2-phase gate can still be well implemented using the present protocol.
According to figure 5(f), we have the fidelity F p (T) of the π/2-phase gate higher than 0.9616 when the resonator decay rate κ less than 0.01 MHz. Considering the resonator decay time 480-692 µs (resonator decay rate 1.4-2.08 kHz) of the three-dimensional (3D) resonator in [65], the fidelity of the π/2-phase gate is about 0.9917-0.9944. Thus, the current technology of superconducting 3D resonator can provide excellent protection for the cat-state qubit of the resonator mode from the influence of resonator decay.
Finally, let us estimate the performance of the π/2-phase gate subjected to multiple decoherence factors. Considering a set of reported parameters [65,75] as κ = 2 kHz, γ = 0.05 MHz, γ ϕ = 0.02 MHz, the fidelity of the π/2-phase gate is F p (T) = 0.9401. Therefore, the protocol can still produce acceptable fidelity of the π/2-phase gate in the presence of all the three decoherence factors.

Two-qubit controlled phase gate
In this section, let us make some brief discussions about the performance of the two-qubit controlled phase gate. As an example, we here consider the implementation of a two-qubit controlled-π-phase gate. In the numerical simulations of the controlled-π-phase gate, we retain the total interaction time and the Kerr nonlinerity as T = 1 µs and K = 2π × 8 MHz, respectively. In addition, we set Ω 2 (t) = 2g 2 (t) = 80 MHz. The average fidelity of the controlled-π-phase gate can be calculated through [71,72] where we have M cp (t) = P cpŨ † cp U cp (t)P cp , with the evolution operator U cp (t) of the system under the control of H p (t), the ideal controlled-π-phase gate operation as and the projection operator onto the four-dimensional computational subspace (N ′ = 4) of two-qubit gates as We plot the average fidelityF cp (t) of the controlled-π-phase gate versus t in figure 6(a). At the final time t = T, the average infidelity of the controlled-π-phase gate is 1 −F cp (t) = 1.0163 × 10 −5 . This result proves that the controlled-π-phase gate can be perfectly performed using the present protocol.
Next, let us consider the influence of the systematic errors to the control fields, where the control field becomes Ω cp (t) → (1 + ϵ)Ω cp (t). The final average fidelityF cp (T) of the controlled-π-phase gate versus the systematic error coefficient ϵ is shown in figure 6(b). We can find from the result that the final average fidelityF cp (T) of the controlled-π-phase gate is higher than 0.993 when ϵ ∈ [−0.2, 0.2]. Therefore, the implementation of the controlled-π-phase gate is also robust against the systematic error.
We also perform fifty individual numerical simulations to test the robutness of the controlled-π-phase gate against AWGN. The control field mixed with AWGN becomes Ω cp (t) → Ω cp (t) + awgn[Ω cp (t), R SN ]. The final average fidelitiesF cp (T) of the controlled-π-phase gate subjected to AWGN with SNR R SN = 10 versus simulation counts are shown in figure 6(c). The results show that the final average fidelitiesF cp (T) of the controlled-π-phase gate is higher than 0.993 when AWGN is taken into account. Accordingly, the implementation of the controlled-π-phase gate is also insensitive to the influence of AWGN.
Finally, the influence of decoherence factors are considered. In the implementation of the controlled-π-phase gate, the main decoherence factors are the resonator decays of resonators 1 and 2 (decay rate κ), the qutrit energy relaxations of relaxation paths |e⟩ q → |g⟩ q and |f ⟩ q → |e⟩ q (energy relaxation rate γ), qutrit dephasing of dephasing paths between levels |e⟩ q and |g⟩ q , and between levels |f ⟩ q and |e⟩ q (dephasing rate γ ϕ ). Hence, the evolution of the system can be calculated through the master equatioṅ with S ′ − = |e⟩ q ⟨ f |, S ′ z = |f ⟩⟨ f | − |e⟩⟨e|, and the density operator ρ cp (t) of the system in the implementation of the controlled-π-phase gate.

Single-qubit NOT gate
In this section, we make some brief discussions about the implementation of the single-qubit NOT gate. The operation times of different steps are set as τ 5 − τ 4 = τ 4 − τ 3 = τ 2 − τ 1 = τ 1 = 0.125 µs, τ 3 − τ 2 = 3 µs. Accordingly, the total interaction time is T = 3.5 µs. In addition, the Kerr nonlinearity takes the value as K = 2π × 8 MHz, and the control field Ω 1 is set as 2π × 65 MHz in Step 3. The average fidelityF NOT of the NOT gate can be calculated by [71,72] with M NOT (t) = P pŨ † NOT U NOT (t)P p , the ideal operation of the NOT gateŨ NOT = (|g, Ψ + ⟩⟨g, Ψ − |+ |g, Ψ − ⟩⟨g, Ψ + |), the evolution operator U NOT (t) of the system under the control of H NOT (t). Here, as the computational subspace of the single-qubit NOT gate is the same as that in the single-qubit π/2-phase gate, we also use the projection operator P p in M NOT (t). The average fidelityF NOT (t) of the NOT gate versus t is plotted in figure 7(a). We can find that the average fidelity of the NOT gate finally increases toF NOT (T) = 0.9999. Therefore, the NOT gate can be successfully implemented based on the present protocol.
Next, we consider the influence of the systematic errors in the control fields, where the Hamiltonian in Step m becomes H sm (t) → (1 + ϵ)H sm (t). The final average fidelityF NOT (T) of the NOT gate versus the systematic error coefficient ϵ is plotted in figure 7(b). The NOT gate is more insensitive to the systematic error compared with the π/2-phase gate, because the systematic-error-sensitivity nullified method [34] is not applied to Step 3 of the NOT gate. When the error coefficient is restricted in ±5%, the final average fidelityF NOT (T) of the NOT gate can be higher than 95.76%. Accordingly, the implementation of the NOT gate has robustness against systematic errors in a certain range.
We also consider the influence of AWGN to the implementation of the NOT gate with thirty individual simulations. In the simulations, the control field mixed with AWGN in Step m (m = 1, 2, 4, 5) is Ω sm (t) → Ω sm (t) + awgn[Ω sm (t), R SN ], and those in Step 3 are Ω 1 (t) → Ω 1 (t) + awgn[Ω 1 (t), R SN ] and g 1 (t) → g 1 (t) + awgn[g 1 (t), R SN ]. The final average fidelitiesF NOT (T) of the NOT gate subjected to AWGN with SNR R SN = 10 versus simulation counts are plotted in figure 7(c). The lowest fidelity in the thirty simulations is 99.33%. Therefore, the implementation of the NOT gate is also insensitive to AWGN.
Finally, we take into account the influence of decoherence factors to the implementation of the NOT gate. The decoherence factors in the implementation of the NOT gate is the same as those of π/2-phase gate. Hence, the evolution of the system can be calculated through equation (59) with H p (t) and ρ p (t) replaced by H NOT (t) and ρ NOT (t). Here, ρ NOT (t) denotes the density operator of the system in the implementation of the NOT gate. Assuming the initial state as |g⟩ q ⊗ |Ψ + ⟩ 1 , the fidelity of the NOT gate can be calculated by F NOT (t) = ⟨ψ 0 NOT |Ũ † NOT ρ NOT (t)Ũ NOT |ψ 0 NOT ⟩. The final fidelity F NOT (T) of the NOT gate versus decoherence rates is shown in figure 7(d). We can find from the result that the implementation of the NOT gate is very robust against resonator decay. When κ ⩽ 0.1 MHz, F NOT (T) ⩾ 0.9982 can be obtained. This is because the NOT gate with the initial state |g⟩ q ⊗ |Ψ + ⟩ 1 is insesitive to the phase-flip errors between states {|Ψ ± ⟩ 1 }. According to the theory of [64], the resonator decay leads phase-flip errors between states {|Ψ ± ⟩ 1 } when the resonator mode is stabilized in the basis {|Ψ ± ⟩ 1 }. Consequently, the influence of the resonator decay to the implementation of the NOT gate with the initial state |g⟩ q ⊗ |Ψ + ⟩ 1 is small.
The average fidelityF CNOT (t) of the CNOT gate versus t is plotted in figure 8(a), where the average fidelity of the CNOT gate finally reachesF CNOT (T) = 0.9993. Thus, the present protocol can also be successfully applied in the implementation of the CNOT gate.
We also examine the influence of systematic errors of the control fields to the implementation of the CNOT gate. Here the control field in Step 1 and Step 5 (Step 2 and Step 4) becomes Ω 1 (t) → (1 + ϵ)Ω 1 (t) [Ω 3 (t) → (1 + ϵ)Ω 3 (t)]. In the Step 3, the operation is a single-qubit NOT gate, and the erroneous control fields can be found in the discussions of section 4.3. The final average fidelityF CNOT (T) of the CNOT gate versus the systematic error coefficient ϵ is plotted in figure 8(b). The result shows that the average fidelitȳ F CNOT (T) of the CNOT gate is higher than 97.81% when ϵ ∈ [−0.05, 0.05]. Therefore, the implementation of the CNOT gate possesses robustness when the systematic errors is controlled in a certain range.
To estimate the influence of AWGN to the implementation of the CNOT gate, we perform thirty individual simulations with control fields mixed with AWGN. In Step 1 and Step 5 (Step 2 and Step 4), the noisy control field becomes The final average fidelitiesF CNOT (T) of the CNOT gate subjected to AWGN with SNR R SN = 10 versus simulation Figure 9. The possible implementation of the protocol using the superconducting system. The auxiliary qutrit in the protocol can be constructed by a superconducting artificial atom, which is coupled with the 3D resonator n though the coupling capacitance CC n . Each 3D resonator couples with a transmon, inducing Kerr-nonlinearity and the two-photon squeezing drive by using the flux modulation.
counts is plotted in figure 8(c), where the lowest value ofF CNOT (T) is 98.51% in the thirty simulations, and most values ofF CNOT (T) is higher than 99.5%. This result indicates that the implementation of the CNOT gate is robust against AWGN.
Finally, let us consider the influence of decoherence factors to the implementation of the CNOT gate. Because the decoherence factors in the implementation of two-qubit CNOT is the same as those of two-qubit controlled-π-phase gate, the evolution of the system can be calculated using equation (61) with the replacements H cp (t) → H CNOT (t) and ρ cp (t) → ρ CNOT (t). Here, we use ρ CNOT (t) to describe the density operator of the system in the implementation of the CNOT gate. Considering the initial state |g⟩ q ⊗ (|Ψ + , Ψ + ⟩ 12 + |Ψ − , Ψ − ⟩ 12 )/ √ 2, the fidelity of the CNOT gate can be calculated by

Conclusion
In conclusion, we have proposed a protocol to realize nonadiabatic geometric quantum gates of cat-state qubits in a Kerr nonlinear resonator driven by a two-photon squeezing drive. Coupling the resonators with an auxiliary qutrit, we realized selective transition for the auxiliary qutrit. More specifically, when the resonators are in the chosen states, a geometric phase can be obtained after the auxiliary qutrit going through a cycling evolution. The obtained geometric phase can be used in constructing single-qubit phase gate and multi-qubit controlled-phase gate. On the other hand, if we turn off the two-photon squeezing drive and the Kerr nonlinearity, a selective evolution for the resonator modes can be realized by applying a strong drive to the auxiliary qutrit. This dynamics can be used to implement the single-qubit NOT gate, two-qubit CNOT gate, and the multi-qubit Toffoli gate of the resonator modes.
The single and two-qubit gates in the protocol can be realized in superconducting systems using the technique of [65]. As shown in figure 9, the auxiliary qutrit in the protocol can be constructed by a superconducting artificial atom, which is coupled with the 3D resonator n (n = 1, 2) though the coupling capacitance C Cn . The Kerr nonlinearity and the two-photon squeezing drive can be respectively realized by the Josephson junction (transmon) nonlinearity and four-wave mixing [65,67,[76][77][78]. To realize multi-qubit gate with more than three resonator modes, one can couple more 3D resonators with the superconducting artificial atom in the center.
Numerical simulations showed that the protocol can produce acceptable gate fidelities in the presence of systematic errors, AWGN, and decoherence factors (including single-photon loss, energy relaxion, and dephasing) with the reported parameters of previous protocols [65,75]. Therefore, the protocol may show some useful perspectives for the fast, robust and scalable geometric quantum computation.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). Therefore, recalling the conditions shown in appendix A, to exploit the vector |ξ l (t)⟩ in NGQC, one should design the parameters to make |ξ l (t)⟩ to complete a cycling evolution |ξ l (T)⟩ = |ξ l (0)⟩. Besides, the dynamic part of the Lewis-Riesenfeld phase α l (t) acquired in [0, T] should be eliminated as In this way, one can obtain a unitary evolution operator as with pure geometric phase If the initial state of the considered system is |A 1 ⟩, to realizing a cycling evolution along the path |ϕ − (t)⟩, the boundary condition of µ(t) can be set as µ(0) = µ(T) = 0. To eliminate the dynamic phase and acquire a nonzero geometric phase θ A , we consider µ(T/2) = π and ζ(t) = −θ A ε(t) +ζ(t), with ε(t) being a step function shown in equation (7) andζ(t) being an undetermined analytic function. In this way,ζ(t) can be described byζ(t) = −θ s δ(t − T/2) +ζ(t) in form, with δ(t) being the Dirac delta function. Moreover, we assume the variation ofζ(t) and µ(t) are symmetrical about t = T/2, i.e.ζ(t) =ζ(T − t) and µ(t) = µ(T − t). It is easy to deriveζ witht = T − t. Then, the dynamic phase acquired in the time interval [0, T] is Besides, the geometric phase acquired in the time interval [0, T] is Therefore, with the parameter settings above, we can acquire a pure geometric phase Θ A along the path |ϕ − (t)⟩ in a cycling evolution during the time interval [0, T].
In the discussions above, µ(t) andζ(t) still remain undetermined parameters with some certain conditions. Generally speaking, there are still a lot of choices for the specific expressions of µ(t) andζ(t). However, not every choice of µ(t) andζ(t) can derive robust control fields against systematic errors. In a real implementation of quantum gates, systematic errors of control fields due to imperfections of devices are unavoidable, and may decrease the gate fidelities. Therefore, it is worth considering the parameter design to enhance the robustness against systematic errors. The systematic-error-sensitivity nullified optimal method [34] is a good method that can make the robustness against systematic errors to the second order of error coefficient ϵ. Here, let us apply the systematic-error-sensitivity nullified optimal method to find proper expressions for parameters µ(t) andζ(t).
With the results shown above, we can get the control field to realize arbitrary single-qubit phase gate in section 3.1 by setting Ω(t) = Ω p (t), Θ A = Θ p , |A 1 ⟩ = |g, Ψ − ⟩ and |A 2 ⟩ = |e, Ψ − ⟩. If the initial state is |g, Ψ − ⟩, after a cycling evolution, a geometric phase Θ p is acquired. If the initial state is |g, Ψ + ⟩, the system remains in the initial state. Thus, the evolution operator can be described by equation (8).
In addition, by setting (11). Therefore, a system in the initial state |g, Ψ − , Ψ − ⟩ can acquire a geometric phase Θ cp after a cycling evolution, but the system does not evolve when the initial state is |g, Ψ + , Ψ ± ⟩ or |g, Ψ − , Ψ + ⟩. Hence, the evolution operator U (2) cp in equation (12) can be obtained.
In Step 3, according to the analysis in section 3.3, the evolution operator is U s3 (t, τ 2 ) shown in equation (28). The time derivatives of the dynamic phases of acquired by |φ ± (t)⟩ arė Noticing the condition arg[g 1 (t)] − π/2 = arg[α] set in section 3.3, the last line of equation (A26) is equal to zero. Therefore, |φ ± (t)⟩ both acquire zero dynamic phases in Step 3.