Error-Resilient Floquet Geometric Quantum Computation

We proposed a new geometric quantum computation (GQC) scheme, called Floquet GQC (FGQC), where error-resilient geometric gates based on periodically driven two-level systems can be constructed via a new non-Abelian geometric phase proposed in a recent study [V. Novi\^{c}enko \textit{et al}, Phys. Rev. A 100, 012127 (2019) ]. Based on Rydberg atoms, we gave possible implementations of universal single-qubit gates and a nontrivial two-qubit gate for FGQC. By using numerical simulation, we evaluated the performance of the FGQC Z and X gates in the presence of both decoherence and a certain kind of systematic control error. The gate fidelities of the Z and X gates are $F_{X,\text{gate}}\approx F_{Z,\text{gate}}\approx 0.9992$. The numerical results provide evidence that FGQC gates can achieve fairly high gate fidelities even in the presence of noise and control imperfection. In addition, we found FGQC is robust against global control error, both analytical demonstration and numerical evidence were given. Consequently, this study makes an important step towards robust geometric quantum computation.

In this paper, we proposed a new geometric computation scheme, called Floquet GQC (FGQC), where universal errorresistant geometric gates can be constructed via a new non-Abelian geometric phase.This non-Abelian geometric phase emerges from a periodically driven quantum systems and was found in a recent study [83].Based on Rydberg atoms, a possible experimental realization of our proposal was provided.By applying this proposal, in the presence of decoherence and a certain kind of systematic control error, numerical simulations of the Z and X gates were given.The relevant results are shown in Figure 2, which indicate that even in the presence of reasonable degree of decoherence and error, FGQC gates can still achieve fairly high fidelities.Furthermore, we found FGQC is robust against global control error.A analytical demonstration was given.In addition, using the a possible set of parameters, we simulated the FGQC X and Z gates in the presence of global control error, then we compared their performances with that of the same gates based on DG and NGQC scenarios [24][25][26][27][28][29][30], the results are shown in Fig. 3.These numerical results provide evidence of the superiority of FGQC gates over NGQC and standard dynamical gate (DG) gates, in terms of solving global control error.This paper is organized as follows.In Sec.II, the general theory of GQC and FGQC is introduced.In Sec.III, a possible implementation of universal single-qubit and a nontrivial two-qubit gate using Rydberg atoms is extensively studied concretely based on the proposed theory in Sec.II.Sec.IV summarizes the findings.

II. GENERAL THEORY
A. General theory of GQC Consider a quantum system exposed to Hamiltonian H(t).For any set of complete basis vectors {|ψ α (0) } at t = 0, the unitary temporal evolution operator can be expressed as is a time-dependent state satisfying the Schrödinger equation.Now, at each moment of time, a different set of time-dependent basis can be selected, {|µ α (t) }, satisfying the boundary conditions at t = 0 and t = τ : The time evolution state can be written as |ψ α (t) = . By substituting this equation into the Schrödinger equation, we have where Consider any unitary once differentiable operator V (t) satisfying the boundary condition V (τ ) = V (0).The operator A(t) is transformed as a proper gauge potential under the basis change: |µ α (t) → γ |µ α (t) V αγ (t), and A(t) will generally contribute a non-Abelian geometric phase to the temporal evolution operator U (τ, 0).
For a GQC scenario, the dynamical phase should be eliminated by engineering H(t) and the ancillary basis {|µ α (t) }.

If the effective Hamiltonian is always diagonal in the initial
Then, the time evolution operator then can be written as follows: where is the sum of the geometric and dynamical phases.In NHQC [32,33], parallel transport conditions µ α (t)|H(t)|µ α (t) = 0 were applied to the Hamiltonian to remove the dynamical phases, while in NHQC+ [31] scenario, the parallel transport condition was replaced with a more wilder condition: τ 0 dt µ α (t)|H(t)|µ α (t) = 0 at the end of the cyclic evolution, making it possible for NHQC+ to be compatible with most of the optimization schemes.

B. Offsetting the dynamical phase using a part of the geometric phase
In this subsection, another way to implement GQC is introduced, where the dynamical phase is offset by a part of the geometric phases [86].
Consider a set of ancillary states {|µ α (t) } dependent on two time-varying real parameters: λ 1 (t) and λ 2 (t).Then, T and that is, the gauge potential can be divided into two parts: By substituting H(t) = i λ1 (t) For a more specific example, consider X(t) in the following form where F λ 1 is a λ 1 -dependent real function and H 0 (λ 2 ) is a λ 2 -dependent hermitian operator.To cancel out the dynamical part of the effective Hamiltonian, H(t) = i λ1 (t) ∂X ∂λ1 = λ1 (t) ∂F (λ1) ∂λ1 H 0 (λ 2 ) was selected, resulting in a pure geometric effective Hamiltonian H eff (t) = A (λ2) (t) = λ2 (t)R † (λ)i∂ λ2 R(λ).Then, Eq. ( 2) can be simplified to The formal solution of Eq. ( 7) can be expressed as follows: c(t) = T exp i t 0 A (λ2) (t )dt .Using the cyclic condition |µ α (τ ) = |µ α (0) and the definition of c αβ (t), |ψ α (τ ) = β c αβ (τ )|µ β (0) , indicating that c(τ ) is just the transformation matrix from the initial states to the final states.Therefore, the temporal evolution operator can be expressed as follows: This cyclic unitary time-evolution operator is geometric because A (λ2) (t) is a gauge potential.

C. Non-Abelian geometric phases in periodically driven systems
In this subsection, a brief introduction of the work reported in Ref. [83] is provided, demonstrating that a non-Abelian geometric phase will form in the adiabatic evolution of a quantum system within a fully degenerate Floquet band.
Consider a driven quantum system with the following Hamiltonian: where f (λ 1 + T ) = f (λ 1 ) is a periodic real function with period T and T 0 f (λ 1 )dλ 1 = 0. To remove the dynamical phase using the strategy introduced in Sec.II B, R(λ) = e X(λ) should be selected, where X(λ) can be expressed as (6).F (λ 1 ) is the primitive function of f (λ 1 ) and the real function f (λ 1 ) satisfies the condition: ) is also periodic with period T , so are both R(λ) and A (λ2) (λ 1 , t).A (λ2) (λ 1 , t) and c(λ 1 , t) can be expanded in a complete set of basis of λ 1 parameter space: {e ilλ1(2π/T ) |l ∈ Z}, with which Eq. ( 7) can be recast into: where Clearly, | l = e il(2π/T )λ1 are the eigenbasis of operator i∂ λ1 with eigenvalues −l(2π/T ).These eigenvalues do not change when the time t is varied; hence, they form bands, which are referred to as Floquet bands [83].In other words, the operator i∂ λ1 is diagonal in basis | l .The operator A (λ2) (λ 1 , t) in general is not diagonal in this basis; its off-diagonal terms describe the coupling between different Floquet bands.However, according to Eq. ( 4), A αγ;km (t) is proportional to λ2 (t).If λ 2 (t) changes slowly in time and λ 1 (t) is a fast timevarying parameter so that A (λ2) αγ;km (t) kω, ∀k, we may ignore all the terms of H F (t) with k = m, i.e., In analogy to the conventional adiabatic approximation, this is the adiabatic approximation of Floquet band.Using Eq. ( 12), the approximate solution of Eq. ( 10) can be obtained as follows: where . By comparing Eq. ( 13) with Eq. ( 8), it can be clearly observed that A (λ2) (λ 1 , t) can be replaced with an average quantity A (λ2) 0 (t), which can be interpreted as a gauge potential resulting from the adiabatic motion in a single degenerate Floquet band.However, it is still not clear what benefits this replacement will provide to quantum computing tasks and whether a universal quantum computation scheme can be constructed in practice using the gauge potential A (λ2) 0 (t).We will address these problems in the next section.

III. FLOQUET GQC BASED ON RYDBERG ATOMS
For universal quantum computation (QC), at least two types of noncommutable single-qubit gates and one nontrivial twoqubit gate are needed [87].On the other hand, the Rydberg atom provides an appealing experimental platform for the implementation of quantum computation because of its long coherence time [88,89].In this section, using the gauge potential A (λ2) 0 (t) in Eq. ( 13), we give a possible physical realizations of universal single-qubit gates and a nontrivial two-qubit gate based on Rydberg atoms.They constitute a universal geometric QC scheme.This geometric QC scheme is referred to as Floquet geometric quantum computation (FGQC).Notably, our theory can be applied to other platforms; only Rydberg atom has been used to show the feasibility.

A. Single-qubit gates
Consider a Rydberg atom with the ground state and the Rydberg state encoding |0 and |1 respectively.By applying appropriately selected laser light, the ground state can be directly coupled to the Rydberg state directly using one photon process or via multi-photon process using one or two intermediate (non-Rydberg) states [90].After performing an adiabatic elimination of the intermediate state, the Hamiltonian in rotating frame can be written as follows: This Hamiltonian drives the transition |0 → |1 with Rabi frequency (t) exp[iϕ(t)], as shown in Fig. 1 (a), where ∆(t) is the effective detuning.In an actual experiment, the laser parameters can be modulated with an acousto-optic modulator (AOM) driven by an arbitrary waveform generator (AWG).By combining with some feed-forward approaches, the experimentally applied pulse can be ensured to be a faithful representation of the desired profile [88][89][90].By taking |0 , |1 as the computational basis, an arbitrary one-qubit nonadiabatic geometric gate can be achieved as fol- lows: where F = 1 2 (σ x , σ y , σ z ) T , n = (sin θ cos φ, sin θ sin φ, cos θ) is an arbitrary unit vector and γ is an arbitrary phase with geometric feature.Eq. ( 15) describes a rotational operation around the axis n by an angle γ.If the pulse shapes of the following form are selected: where ω and θ 0 are time-independent parameters; ∆ 0 (t) and 0 (t) are in general time-dependent real functions.Then, the Hamiltonian can be rewritten as follows: where r(t) = ( 0 (t) cos ϕ(t), 0 (t) sin ϕ(t), ∆ 0 (t)) T .Eq. ( 18) has the same form as Eq. ( 9) if λ 1 (t) = ωt + θ 0 , f λ 1 (t) = cos[λ 1 (t)] and H 0 (t) = F • r(t).According to Eq. ( 6), for a FGQC scenario, the ancillary basis is given by the transformation: R(ωt, t) = exp − i sin(ωt+θ0) ω F • r(t) .This transformation will result in the following approximated effective Hamiltonian: where Ω(t)n(t) ≡ r(t) × ṙ(t)/|r(t)| 2 with n(t), a unit vector; J 0 (a) is the zero-order Bessel function.If the plane determined by vectors r and ṙ is fixed, then n(t) does not change, and the effective Hamiltonian (19) at different times can commutable with each other.The time evolution operator (13) can be simplified into ) has the same form as (15).As n and γ(τ ) can take any value, a universal single-qubit gate can be applied using FGQC scenario.
Taking the Z and X gates as examples, we will give a possible set of pulse shapes and parameters for each gate.Moreover, to evaluate the performance of these gates with the given parameters, numerical results in the presence of both decoherence and a certain kind of systematic error will be presented (we also presented similar contents for the Hadamard and T gates in Appendix B).To implement a single-qubit Z gate using FGQC, we selected ∆ 0 (t) = 0, 0 (t) = Ω 0 and ϕ(t) = N t, with Ω 0 and N are real constants which satisfy N [1 − J 0 (|r(t)|/ω)]τ = π, and τ is the run time of the gate.Here, |r(t)| = Ω 0 ; the pulse shapes are shown in Fig. 2 (a).In the numerical simulation, we set Ω 0 = 2 × 2π MHz, ω ≈ 0.513 × 2π MHz, N ≈ 45.728 × 2π KHz, and the run time τ ≈ 7.797 µs.For the X gate, 0 (t) = Ω 0 sin M t, ∆ 0 (t) = Ω 0 cos M t, and ϕ(t) = π/2 with M = N real timeindependent constants.The corresponding pulse shapes are shown in Fig. 2 (b).Other parameters are the same as the Z gate.
The noisy gate operation in the presence of global control error is described by the master equation as follows: where H (t) = (1 + δ)H(t) with H(t) the ideal Hamiltonian, δ represents the amplitude of global control error, is the Lindblad superoperator which acts on the density matrix ρ of the quantum system; {a, b} = ab + ba is the anticommutator.In the case of a two-level Rydberg atom, the incoherent processes, including decay and dephasing: A 1 = σ − , A 2 = σ z , where σ − is the spin ladder operator and σ z is the Pauli z matrix; thus, γ 1 and γ 2 are the decay and dephasing rates, respectively.Here, we selected γ 1 = 12.83 Hz, γ 2 = 128.3Hz and δ = 0.05.
In Figs. 2 (c) and (d), for the X and Z gates, the temporal evolution of state populations is shown for |0 and |1 with a given initial state |ψ(0) = |1 .The corresponding fidelities between the target states and the temporal evolution states are also shown in these figures; the final state fidelities for the X and Z gates at τ are as follows: F X (τ ) ≈ 0.9997 and F Z (τ ) ≈ 0.9998 respectively.The gate fidelity was also investigated, which is defined by F = (1/2π) 2π 0 ψ I |ρ|ψ I dΘ, for the initial states of the form |ψ = cos Θ|0 + sin Θ|1 , where a total of 101 different values of Θ were uniformly selected in the range [0, 2π].The results are shown in Fig. 2 (e).The gate fidelities at τ are F X,gate ≈ F Z,gate ≈ 0.9992 for the X and Z gates, respectively.These fairly high state and gate fidelities indicate two significant factors: firstly, the run time of our proposal is short enough to lighten the destructive effect of decoherence; secondly, other parameters in the proposal match well with the requirements of FGQC theory.
With respect to the reachability of our proposal, a recent experimental study in Ref. [90] was selected for comparison.In this study, the amplitude of (t) can be changed from 0 to 5 × 2π MHz within 0.5 µs, and the detuning ∆(t) can be changed from −15 × 2π MHz to a value larger than 7.5 × 2π MHz within 1 µs.Besides, the pulses in this study were given by a numerical optimization method known as RedCRAB; their shapes are not that smooth.While in our proposal, taking the X gate as an example, the parameters (t) and ∆(t) should be changed smoothly within the range [−2, 2] × 2π MHz in about 7.797 µs; clearly, the pulses in this proposal are more smoother and changes slower within a smaller range.In other words, by comparing with recently experimental results, this proposal needs a series of smoother, more slowly varying pulses with smaller ranges.Therefore, the control in implementing FGQC X gate is achievable The control of FGQC Z gate is also slow and smooth, despite ∆(t) = 0, it also needs ϕ(t) = N t, that is ϕ(t) increase linearly at a slow rate, to our knowledge, it is not clear weather this kind of control is achievable.

B. Two-qubit gate
It is shown that an arbitrary one-qubit FGQC gate can be obtained by addressing an individual Rydberg atom with laser pulses.To achieve universal GQC, a nontrivial two-qubit gate is needed beside one-qubit gates.We here demonstrate how to achieve a nontrivial two-qubit FGQC gate using the Rydberg-Rydberg interaction (RRI).
Consider two two-level Rydberg atoms with RRI V , as shown in Fig. 1 (b).The αth atom is resonantly driven by a microwave pulse to achieve the transitions |0 α → |1 α , with Rabi frequency Ω α (t).In the rotating frame, the Hamiltonian of two-Rydberg-atom system can be expressed as follows [27]: where ) is a single-atom Hamiltonian describing the interaction between the αth atom and laser pulses; I α is the identity operator acting on the αth Rydberg atom.
where Ω 0 is a real constant.Both f (ωt) and ϕ(t) are real functions.The Hamiltonian (24) can be rewritten as follows: where F ≡ (σ x , σy , σz ) T /2 and r (t) = Ω 0 (cos ϕ(t), − sin ϕ(t), 0) T with σx ≡ |B 00| + |00 B|, σy ≡ −i|B 00| + i|00 B| and σz ≡ |B B| − |00 00|.To be more specific, we considered f (ωt) = cos ωt; then, The corresponding effective Hamiltonian can be written as follows: Similar to the derivation of Eq. ( 13), if ω φ(t), then at t = τ , the time evolution operator can be approximated as follows: where In terms of the basis {|00 , |01 , |10 , |11 }, the matrix form of average effective Hamiltonian can be expressed as follows: where The time evolution operator is given by where Clearly, when e iCτ = −1 and φ = π/2, we have This operation is a SWAP-like gate because: this is a nontrivial two-qubit gate ensuring that universal quantum computation can be applied using FGQC.

C. The robustness of FGQC against global control error
Since only a gauge potential A (λ2) 0 (t) can be attributed to the cyclic unitary evolution operator (13), FGQC is a geometric scenario.However, as shown in Ref. [44], a geometric scenario is not necessarily resistant to global control error.FIG. 4. The fidelity F vs. the amplitude δ of global control error for the same two-qubit gate using FGQC (red solid line) and NGQC (blue dashed line), respectively.The two-qubit gate is given by Eq. (30).
Here, we will demonstrate that the FGQC is a resistant GQC scenario to global control error.To prove this, we consider a quantum system which is subject to global control error.Given the initial state |ψ m (0) , the corresponding time evolution state can be expressed as Normally, δ is a small quantity in experiment and it is reasonable to assume that δH αγ (t) kω.Similar to the derivation of Eq. ( 13), U (τ, 0) can be approximately expressed as follows: where H(t) = (1/T ) 2π 0 H(λ 1 , t)dλ 1 ; then, using perturbation theory up to O(δ): where |ψ m (τ ) = U (τ, 0)|ψ m (0) is the ideal time evolution state and Eq. ( 34) clearly shows that to eliminate the second term at the right side, Q mn (τ ) should be equal to zero: Eq. ( 35) maintains the robustness of FGQC against global control error to the first order in δ.In a FGQC scenario, H is given by where f (λ 1 ) = (1/T ) T 0 f (λ 1 )dλ 1 and f (λ 1 ) is a periodic function with period T .In this study, f (λ 1 ) = cos(λ 1 ), then, f (λ 1 ) is equal to zero, H(t) = 0, and Eq. ( 35) is satisfied.The above analysis is based on single-qubit FGQC theory, but it can also be applied to the two-qubit gate case.For twoqubit case, one only need replace A (λ2) 0 (t) and H(t) with H (0) eff (t) and H rot (t) respectively.Therefore, FGQC is resistant to global control error.
Numerical evidence of the robustness of FGQC against static global control error.In Fig. 3 and Fig. 4, numerical results of the single-qubit X, Z gates and the two-qubit gate described in Eq. ( 30) are given.They show the singleand two-qubit gate fidelities vs. δ, respectively; clearly, the FGQC scenario is more robust than the standard dynamical scenario and the NGQC scenario in solving global control errors (see Appendix A for the detail of the standard dynamical and NGQC gates).
The single-qubit FGQC T and Hadamard gates were also simulated, the results are shown in Appendix B When the error become time-dependent.For a time dependent error δ = δ(t), we should modify the unitary time evolution operator in Eq. ( 33) by replacing δ H(t) in the exponent with the following expression: In our proposal, λ 1 = ωt and H(λ 1 , t) = cos(ωt)H 0 (t) with ω a time-independent real number and H 0 (t) is a slowly varying operator, the expression Eq. ( 37) can be rewritten as (1/T ) 2π 0 dt cos(ωt)δ(t)H 0 (t).Clearly, when δ(t) is slow enough, this expression is approximately equal to zero.Therefore, FGQC is still a robust scenario when δ(t) is slow enough.
A numerical demonstration was also given through simulating a Z gate in the presence of δ(t).A stochastic process called Ornstein-Uhlenbeck (O-U) process was considered to describe the time-dependence of δ(t).To be specific, O-U process is a stochastic process with power spectral density (PSD) S(ω) = Γ π(Γ 2 +ω 2 ) , where 1/Γ is the correlation time.The Hamiltonian is given by: where H(t) is the ideal Hamiltonian, M is a zoom factor.In the simulation, we set Γτ = 0.001.For each M ; we firstly sampled 100 such process randomly: then, we calculate the gate fidelity for each δ i (t) and obtain the average gate fidelity for these 100 processes.The results are shown in Fig. 5.

IV. CONCLUSION
A new geometric scheme called FGQC is reported.In a FGQC scenario, error-resilient geometric gates based on periodically driven two-level systems can be constructed via a new non-Abelian geometric phase.This non-Abelian geometric phase emerges from a periodically driven quantum systems and was found in a recent study [83].To construct FGQC, possible implementations of universal single-qubit gates and a nontrivial two-qubit gate using Rydberg atoms were proposed.To numerically evaluate its realistic performance, the X and Z gates were simulated in the presence of decoherence and global control error using recent experimental parameters.The noisy X and Z gates acting on a given initial state |ψ(0) = |1 reached state fidelities as high as: F X (τ ) ≈ 0.9997 and F Z (τ ) ≈ 0.9998; the gate fidelities for the X and Z gates are F X,gate ≈ F Z,gate ≈ 0.9992.The robustness of FGQC against global control error was analytically demonstrated using perturbation method.By comparing the numerical results of FGQC with NGQC and DG, the superiority of FGQC in solving global control error was confirmed.Because FGQC is based on two-level systems, compared with NHQC, it has the advantage of not requiring complex quantum control on a multilevel structure.Therefore, this study makes a step towards the experimental realization of errorresilient quantum computation and quantum information processing.
with Ω 0 = 2.0 × 2π MHz, ω ≈ 0.513 × 2π MHz N T ≈ 45.728 × 2π KHz and the run time τ = 1.95 µs.The Hamiltonian of the FGQC T and Z gate has the same form, and the parameters of these two gates are almost the same except the run time of the T gate is shorter.The pulse shapes of the Hadamard and T gates are shown in Fig. 6 (a) and (b) respectively.In Fig. 6 (c) and Fig. 6 (d), for the Hadamard and the T gate respectively, we show the we show the temporal evolution of state populations for |0 and |1 with a given initial state |ψ(0) = |1 .In the presence of the global control error (δ = 0.1) and the decoherence (γ 1 = 8Hz, γ 2 = 80 Hz), the corresponding fidelities between the target states and the temporal evolution states are also shown in these figures (red solid lines), at the end of the run time, F H (τ ) ≈ 0.999338 and F T (τ ) ≈ 0.999263.These fairly high fidelities of noisy gates provide evidence for the possibility of FGQC scenarios in real experiment.We have also investigated the gate fidelity of the FGQC Hadamard and T gates defined by F = (1/2π) In terms of the robustness against the global control error, we compare the performance of the FGQC Hadamard and T gates with those of other typical scenarios based on two-level systems: the nonadiabatic geometric quantum computation (NGQC), the standard dynamical gates (DG).The results are shown in Fig. 7.It is readily to see from these figures that the FGQC scenarios perform much better than other three scenarios.
The detail of other scenarios in simulating the T gate.We set Ω 0 = 2 × 2π MHz hereinafter.The NGQC T gate: The Hamiltonian of this scenario can be written as

FIG. 2 .
FIG. 2. (a) Detuning ∆(t) (blue dashed line) and Rabi frequency (t) (black solid line) for the numerical simulation of the X gate using FGQC.(b) Detuning ∆(t) (blue dashed line), Rabi frequency (t) (black solid line), and phase ϕ(t) (red dotted line) for the numerical simulation of the Z gate using FGQC.Temporal evolution of populations (blue dashed line for state |1 , black dot dashed line for state |0 ) and fidelities (red solid line) with a given initial state |ψ(0) = |1 for the FGQC X (c) and Z gates (d).(e) Temporal evolution of gate fidelities for the X (black solid line) and Z gates (blue dashed line).

FIG. 3 .
FIG. 3. Fidelities of the Z (a) and X (b) gates vs. the amplitude of the global control error (without considering decoherence in the simulation).The results of three different protocols are shown: FGQC (red solid line), NGQC (blue dashed line), and standard DG (black dot dashed line).

FIG. 5 .
FIG.5.The average gate fidelity FS vs. M for the Z gate based on FGQC (blue dots), NGQC (yellow rectangle) and DG (green diamond), respectively.

ψ
I |ρ|ψ I dΘ for initial states of the form |ψ = cos Θ|0 + sin Θ|1 , where a total of 101 different values of Θ were uniformly chosen in the range [0, 2π], the results are shown in Fig. 6 (e), the average fidelities at τ for the Hadamard and T gates are approximately 0.999338 and 0.99984 respectively.

FIG. 6 .
FIG. 6.The pulse shapes in numerically simulating the Hadamard (a) and T gates(b).The temporal evolution of populations (blue dashe line for state |1 , black dotdashed line for state |0 ) and fidelities (red solid line) with a given initial state |ψ(0) = |1 for the FGQC Hadamard gate (c) and T gates (d).(e) The temporal evolution of gate fidelities for the Hadamard (blue dashed line) and T gates (black solid line).

FIG. 7 .
FIG. 7. Fidelities of the Hadamard (a) and T (b) gates versus the amplitude of the global control error (without considering the decoherence in the simulation), results of four different protocols are shown: FGQC (red solid line), NGQC (blue dashed line) and standard DG (black dotdashed line).
FIG. 1.(a) Setup for two-level Rydberg atom.The two-level system is driven off-resonantly with time-dependent detuning ∆(t), Rabi frequency (t), and phase ϕ(t).(b) Illustration of control of twoqubit gate based on RRI between two identical two-level Rydberg atoms, where V is the RRI strength.For individual Rydberg atoms, a resonant microwave pulse with Rabi frequency Ωα can be applied to facilitate the transition |0 → |1 .