High-fidelity multiqubit Rydberg gates via two-photon adiabatic rapid passage

We present a robust protocol for implementing high-fidelity multiqubit controlled phase gates $(C^kZ)$ on neutral atom qubits coupled to highly excited Rydberg states. Our approach is based on extending adiabatic rapid passage to two-photon excitation via a short-lived intermediate excited state common to alkali-atom Rydberg experiments, accounting for the full impact of spontaneous decay and differential AC Stark shifts from the complete manifold of hyperfine excited states. We evaluate and optimize gate performance, concluding that for Cs and currently available laser frequencies and powers, a $CCZ$ gate with fidelity $\mathcal{F}>0.995$ for three qubits and $CCCZ$ with $\mathcal{F}>0.99$ for four qubits is attainable in $\sim 1.8$~$\mu$s via this protocol. Higher fidelities are accessible with future technologies, and our results highlight the ultility of neutral atom arrays for the native implementation of multiqubit unitaries.


Introduction
Over the last decade, trapped neutral atoms have emerged as one of the most promising platforms for quantum information processing [1][2][3][4]. Several experiments have demonstrated reliable single atom loading, initialisation and readout in configurable 1D [5], 2D [6,7] and 3D [8,9] optical arrays formed by up to a few hundred sites [10,11], without fundamental limitations to further scaling. Control over high-fidelity single-qubit gates for neutral atoms, with qubits encoded in long-lived hyperfine states, has been demonstrated in several settings [12][13][14][15]. By exciting the atoms to strongly interacting Rydberg states, it is then possible to implement controlled-NOT or controlled-phase two-qubit gates [16,17], which are key for the realisation of a universal gate set for digital quantum computing [18]. Although the coherent excitation of the ground-Rydberg transitions poses several challenges [19] that had limited the fidelity of two-qubit gates for a number of years [20][21][22][23][24], recent experiments have overcome many of these difficulties through suppression of technical noise [25] and incorporation of optimal control techniques [26][27][28], reporting on the preparation of Bell states with fidelities F 0.97 using alkali atoms in 1D arrays [27] and F 0.89 in 2D arrays [29], and F > 0.99 with alkaline-earth atoms [30].
In this work, we show that adiabatic rapid passage (ARP) techniques can be extended to provide high-fidelity multiply-controlled phase gates (C k Z). Our protocol is tailored to the case of two-photon excitation via a rapidly decaying intermediate excited state, as is typical of experiments using alkali atoms. This scheme is robust to experimental imperfections and requires only global addressing of the qubits within the blockade regime. These multi-qubit gates have the potential to substantially optimise quantum circuit implementation [1,4,43,44], as well as quantum error correction [52].
Our gate sequences are designed using an exact model that accounts for the effects of spontaneous decay and AC Stark shifts from the full manifold of hyperfine excited states for experimentally feasible parameters, providing a realistic evaluation of the intrinsic gate fidelities attainable for Cs atoms prepared in the motional ground state using current technologies. Starting from analytical pulse profiles for CCZ on three qubits with F ∼ 0.993, we apply optimal control using a variant of the dressed chopped random basis (dCRAB) method [53,54] to design global laser pulses achieving a fidelity of F ∼ 0.995, outperforming the equivalent circuit decomposition into two-qubit gates for F 2Q < 0.999. Extension of this method to four qubits yields a CCCZ gate with F > 0.99 without optimisation, providing a viable route to demonstrating efficient and scalable digital computing on neutral atom arrays.

C k Z gate via two-photon adiabatic rapid passage
The C k Z gate is implemented on an ensemble of k + 1 alkali atoms, each of which represents a qubit with the computational states |0 and |1 encoded in two hyperfine ground states with frequency separation ω q . As illustrated in figure 1(a), the state |1 is coupled to a Rydberg state |r via a two-photon transition through an intermediate excited state |e , which is resolved into its hyperfine components |f e , m f e . The single-atom Hamiltonian describing this excitation on atom i is given by where Ω f e ,m fe 1 and Ω r f e ,m fe are the Rabi frequencies of the couplings from |1 → |f e , m f e and from |f e , m f e → |r , Δ f e ,m fe = Δ − E(f e , m f e ) represent the intermediate-state detunings composed by the centre of mass laser detuning Δ and the hyperfine splitting E(f e , m fe ), and δ is the two-photon detuning. We consider the far off-resonant limit |Δ| |Ω f e ,m fe 1 |, |Ω r f e ,m fe |, for which the coupling from |1 to |r can be described in terms of an effective two-level system with Rabi frequency and detuning where are the AC Stark shifts of the states |1 and |r , respectively. To account for intrinsic gate errors arising from spontaneous decay from the excited state and Rydberg state with decay rates Γ e and Γ r respectively, we model the system evolution using the dissipative Schrödinger equation by introducing effective single atom terms |f e , m f e f e , m f e | + Γ r 2 |r r| .
This approach provides a lower bound on the estimated fidelity as the non-Hermitian terms in the Hamiltonian cause loss of population from the computational basis, neglecting the small probability of atoms returning to the logical qubit levels.
The Rydberg states experience dipole-induced pairwise interactions described by Hamiltonian where the strength V ij rr depends on the separation d ij between the atoms i and j and their orientation with respect to the quantisation axis. In the fully blockaded regime, which is fulfilled when |V ij rr | |Ω R | for all  (1). Any three nearest-neighbour atoms can be globally illuminated with laser pulses that realise the two-photon ARP CCZ gate protocol by providing the drive from |1 to |r through the intermediate state |e . (b) Quantum circuit representation of the pulses used to implement the CCZ gate. A schematic representation of the Ω R (t) and δ R (t) ARP pulses is also included. (c) Sketches of the effective two-level systems involved in the ARP processes for the CCZ gate. For each two-level system, the time evolution of the ground (black lines) and excited (blue lines) states when the ARP pulses Ω R (t), δ R (t) shown in figure 2(a) are applied is shown. pairs of atoms, the ensemble can only host one Rydberg excitation. Under this condition, we seek to apply a sequence of global pulses which realises a C k Z gate, given by the unitary matrix The main difficulty in finding such a pulse sequence resides in the fact that (k + 1)-body states with different numbers n of atoms in the computational state |1 are coupled to the manifold of singly-excited Rydberg states with different effective Rabi frequencies √ nΩ R . To circumvent this issue we apply two consecutive ARP pulses, which can achieve full population transfer in the resulting effective two-level systems for a wide range of Rabi frequencies. The required π phases at the end of the sequence can be imprinted by inverting the sign of the detuning at the start of the second ARP pulse [55].

CCZ gate
As a specific example of the two-photon ARP C k Z gate protocol, we analyse in detail the implementation of a CCZ gate. As illustrated in figure 1(a), the three atoms involved in the gate are arranged in a triangle with equal side lengths d and experience the same pairwise Rydberg interaction strength V rr . In a first approach, we realise the CCZ gate by applying two consecutive global ARP pulses of duration T 1 and T 2 , with the effective Rabi frequency and detuning varying in time according to the analytical functions [28] with i = 1, 2 for the first and second pulse, respectively. In figure 1(c) we represent schematically the relevant states of the different two-level systems over which the ARP pulses act, and for each of them we plot the time evolution of the ground (black lines) and excited (blue lines) states for the ARP pulses given by equations (8) and (9). According to equation (3), the effective detuning δ R (t) can be controlled through δ(t), which is the two-photon detuning resulting from the sum of the detunings of the lasers driving the |1 ↔ |e and |e ↔ |r transitions. Therefore, this parameter can be modified in time as desired by chirping the frequency of either laser. In order to obtain the desired shape for Ω R (t) for a two-photon transition, we fix the intermediate state detuning Δ and the intensity I r e of the laser driving the |e ↔ |r excitation and vary the intensity of the laser coupling the states |1 and |e as I e 1 (t) ∝ Ω 2 R (t). This time-dependent laser intensity induces a differential AC Stark shift between the states |0 and |1 , which after the application of the ARP pulses results in a relative phase error on each qubit. As shown in figure 1(b), to compensate for this effect after the ARP pulses we apply a global single-qubit rotation Z(φ), where φ is chosen to maximise the gate fidelity. In the following we assume this single-qubit rotation to be perfect as it can be implemented as an instantaneous phase-step on all future qubit pulses or using control pulses with errors below 10 −4 [14].
We compute the gate fidelity using the limited tomography prescription [27,56] where U CCZ is the ideal gate matrix, U is the non-unitary operator resulting from time evolution under the Hamiltonian H tot = H + H dd + H and the application of the global rotation Z(φ), and the initial state 1 |ijk is chosen as a symmetric superposition of all possible computational basis states |ijk , making the fidelity sensitive to both population leakage and relative phase errors for each computational state. Note that since each of the computational states evolves in an independent Hilbert space under the application of H tot there are no artificial errors caused by interference between the different states forming the superposition |Ψ .
We consider the explicit case of Cs atoms trapped in a triangular array separated by a distance d = 4 μm at a temperature T = 0 K and with a quantisation axis defined by a 10 G bias field perpendicular to the array plane. The hyperfine ground states |0 = |6S 1/2 , f = 3, m f = 0 and |1 = |6S 1/2 , f = 4, m f = 0 embody the computational basis, and we use circularly polarised laser beams propagating parallel to the quantisation axis to couple |1 via the intermediate state |e = |7P 1/2 to the Rydberg state |r = |82S 1/2 , m j = −1/2 , for which the interaction strength is V rr /2π = 608 MHz. We combine open-source packages to compute the Rydberg-Rydberg interactions [57] and the hyperfine-resolved atom-light couplings [58].
We then optimise gate fidelity for experimentally achievable parameters, applying the realistic constraint of fixed power of ∼290 mW focused to ∼3 μm at the center of the triangle, corresponding to an intensity at the atom positions of I r e = 6.236 mW μm −2 , and numerically optimising intermediate state detuning, pulse parameters T i , τ i and δ i R (i = 1, 2), and intensity of the laser from |1 → |e . To accelerate the optimisation, we use a reduced model where the dissipative contribution of the intermediate states is effectively incorporated into the Rydberg level (see appendix B), and then use the optimal parameters found with this procedure to compute the time evolution with the full model.
In figure 2 we illustrate the CCZ gate sequence corresponding to the optimal parameters Δ/2π = 16.3 GHz, T 1 = 0.852 μs, T 2 = 0.944 μs, τ 1 = 2.59T 1 , τ 2 = 3.19T 2 , δ 1 R /2π = 6.270 MHz, δ 2 R /2π = 5.586 MHz, and I e 1 = 0.390 mW μm −2 , for which Ω 0 R /2π = 4.334 MHz. The fidelity obtained with the full model is F = 0.9932. Figure 2(a) shows the time dependence of the effective Rabi frequency Ω R (t) in the upper plot, while the two-photon detuning δ(t) and the effective detuning δ R (t) seen by the atoms are shown in the lower plot. In figure 2(b) we plot the time dependence of the population of the intermediate states P e and the leakage of population to hyperfine ground states outside of the computational basis P d = 1 − Ψ|Ψ for an initial state consisting of an equal superposition of all computational states, |Ψ = 1/ √ 8 ijk=0,1 |ijk . In figure 2(c) we display the final population P ijk = 8| ijk|U| Ψ | 2 and phase φ ijk = arg{ ijk|U|Ψ } of all the input states other than the uncoupled state |000 after the application of the CCZ gate. The state with the worst fidelity is |111 , for which the output population is P 111 = 0.988.

Pulse optimisation
To further enhance the ARP CCZ performance we apply a simplified version of the dCRAB technique [53,54] (see appendix C for details) to optimise the shape of the effective Rabi frequency Ω R (t). We find a pulse shape which leads to an improvement of the fidelity to F = 0.9954. Figure 3(a) shows the time dependence of the dCRAB-optimised effective Rabi frequency and two-photon detuning. Besides yielding an increase in the fidelity, this sequence reduces the peak effective Rabi frequency to Ω max R /2π = 3.38 MHz, which corresponds to a laser intensity I e 1 = 0.235 mW μm −2 . In figure 3(b) we plot the time evolution of P e and P d for the initial state |Ψ = 1/ √ 8 ijk=0,1 |ijk . We observe that the pulse shaping reduces the excitation of the intermediate states and the leakage to hyperfine ground states outside of the computational basis. Finally, figure 3(c) shows the output populations and phases for all of the computational states after    ideal case of a uniform π phase across all input states than the one corresponding to the analytical pulses shown in figure 2(c), the gain in population transfer outweighs this effect. Note that the measure of the fidelity (10) that we employ penalises both population loss and phase errors.

CCZ gate robustness
After finding the optimal pulse sequence assuming ideal conditions, we now move on to evaluate the sensitivity of the CCZ gate ARP protocol to different types of experimental imperfections. We start by considering fluctuations of the optimal parameters presented in the previous sections. In figure 4(a) we show 1 − F obtained when the two-photon detuning δ and laser intensity I e 1 are deviated from the optimal values used in figure 3 by a fixed relative amount throughout the pulse duration. For relative fluctuations of up to ±3% on the laser power and ±1% on the two-photon detuning, the fidelity remains in the range 0.9939 F 0.9954. In figure 4(b) we plot 1 − F and the Rydberg interaction strength V rr as a function of the separation d between the atoms in an equilateral triangle configuration. The ARP pulses are the same as depicted in figure 3, but for each interaction strength we have re-optimised the corrective phase φ. The gate fidelity reduces slightly for increased separation due to finite Rydberg blockade, but remains F > 0.993 even for d ∼ 6 μm. Independent evaluation of gate fidelity for geometries with asymmetric spacings between the atoms confirms that the performance of the protocol is limited by the minimum value amongst all the pairwise Rydberg interaction strengths rather than by the presence of unequal interaction strengths.
Furthermore, the performance of the gate is affected by issues that are common to all neutral atom experiments. One of the most important sources of error is dephasing in the excitation to the Rydberg state caused by the fact that the excitation lasers have a finite linewidth. We have performed numerical calculations accounting for this effect by allowing for random phases φ 1 (t) and φ 2 (t) on the Rabi frequencies associated with the |1 ↔ |e and |e ↔ |r lasers, i.e., we have modified them as , Ω r f e ,m fe → e iφ 2 (t) Ω r f e ,m fe . (11) According to equation (8), the effective Rabi frequency is modified as Ω R → Ω R e i(φ 1 (t)+φ 2 (t)) . In order to speed up the calculations needed to obtain significant statistical samples, we have computed the gate fidelity under the presence of laser phase noise using the effective model described in appendix B. The phases φ 1 (t) and φ 2 (t) are stochastic quantities generated according to a laser phase noise spectral density S φ (f ), where f denotes the Fourier frequency. The phase noise spectral density is related to the frequency noise spectral density as S δν (f ) = f 2 S φ (f ), where δν = ν − ν 0 and ν 0 is the main laser frequency. The spectral function S δν (f ) can be used to compute the auto-correlation function, which is the Fourier transform of the laser line shape S E (δν), where E denotes the electric field amplitude. Assuming that the lasers are cavity-locked, we can approximate S E (δν) by a Lorentzian distribution. Under this assumption, the frequency noise spectral function is S ν (f ) = constant = h 0 /π (white noise), where h 0 is the full width at half maximum (FWHM) of S E (δν) [59].
In figure 4(c) we plot the average value of the fidelity F and the corresponding standard deviation σ F obtained by running the optimal gate protocol 150 times using Lorentzian laser line shapes S E (δν) with different values of the FWHM linewidth. Although for a FWHM of 1 kHz the average fidelity drops considerably, for widths 100 Hz (which are routinely achieved experimentally e.g. [60]) it approaches the value of F = 0.9954 that we obtained without considering phase noise. Besides reducing the laser bandwidth further through filtering using a high-finesse cavity [25], another possible approach to mitigate the effects of laser phase noise is to engineer a frequency noise spectral density S δν (f ) which instead of having a flat profile is minimal for Fourier frequencies f around the effective Rabi frequency Ω R , in such a way that the phase fluctuations have a very different frequency than the population transfer and therefore the excitation to the Rydberg manifold experiences little dephasing [19].
Besides laser phase noise, the proposed gate is highly sensitive to differences in laser intensity at each atom due to relative positioning and alignment with respect to the global beam and the delocalisation of the atomic wavefunction within the traps. These issues are common to gates relying on global driving, and they can be mitigated by utilising a spatial light modulator or holographic optical element to convert the beam profile to a top-hat rather than Gaussian distribution (as demonstrated e.g. in [61]), to achieve uniform intensity across the atoms making the scheme robust to small misalignments and the delocalisation of the atoms. An alternative strategy is to correct for small differences in the field intensities by applying independent phase corrections Z(φ) to each atom which would allow correction for accumulated phase errors, but in practice this would be challenging to tune and optimise experimentally when targeting fidelities of the order of the ones reported in this paper.

Conclusion and outlook
In conclusion, we have shown that ARP pulses can be implemented in a two-photon Rydberg excitation scheme with alkali atoms to realise a high fidelity CCZ gate. Using a detailed model that takes into account all the relevant atomic levels and AC Stark shifts for Cs atoms, we have obtained F = 0.9954 with optimally shaped pulses found through dCRAB. This gate is equivalent to six CNOT gates (plus additional single-qubit rotations) [62], meaning that matching the performance of our native CCZ ARP gate would require the ability to perform two-qubit gates with a fidelity F (0.9954) 1/6 = 0.9992 which so far remains elusive using two-photon excitation in alkali atoms (see appendix D).
The two-photon ARP controlled-phase protocol could in principle be generalised to any number of qubits by re-optimizing the pulse shapes. Using the analytical pulses of figure 2 in a four-qubit system formed by a square of size d = 4 μm as shown in figure 5(a), we have found a gate fidelity F = 0.9879 using as a measure the generalisation of equation (10) to a CCCZ gate. In this case the relatively weak interaction V rr /2π = 119 MHz between atoms placed in opposite corners of the square leads to significant phase errors due to finite blockade. In order to minimise this undesired effect, the ensemble could be arranged in a three-dimensional pyramidal configuration with three atoms in the x-y plane and a fourth one equally separated from the triangle vertices, as shown in figure 5(b). With all the inter-atomic distances set to d = 4 μm in this geometry, the Rydberg interactions with the out-of-plane qubit are V rr /2π = 612 MHz, and the CCCZ gate fidelity increases to F = 0.9910. Even though this fidelity would already be difficult to outperform with a decomposition of the CCCZ gate into two-and three-qubit gates, further optimisation would be likely to lead to an improved performance. The main challenge in scaling the two-photon ARP gate scheme to an even larger number of qubits would be to ensure the condition that all the pairwise interactions are strong enough to provide total Rydberg blockade in the ensemble. These results provide a viable route to scalable quantum computing on neutral atom arrays by enabling implementation of high-fidelity multi-qubit unitiaries using parameters realistic for current experimental platforms. In future, higher fidelities may be attainable through power scaling to increase the coupling Rabi frequency from |e → |r to permit faster gate speeds and higher intermediate state detuning. description of their dissipative effect. Using the adiabatic elimination procedure, we can estimate the total population of the intermediate states as where P 1 2 are the contributions to the intermediate-state population coming from the laser couplings from |1 to |e and from |e to |r , respectively. Thus, we can incorporate the effects of spontaneous decay from |e into the effective single-atom Hamiltonian describing the coupling from |1 to |r by introducing effective dissipation terms from |1 and |r weighed by P 1 e and P r e , respectively. This effective Hamiltonian, the parameters of which are depicted in figure 6(a) (note that δ R = δ + Δ 1 − Δ r ), reads This effective model yields almost identical results to the full model explicitly taking into account all the intermediate excited states. In figure 6(b) we show the population transfer and phase imprinting computed with the effective model for the dCRAB-optimised pulses discussed in the main text and illustrated in figure 3 of the main text. For this specific pulse sequence, the relative difference in the gate fidelities computed with the exact and effective model is |(F − F eff )/F| = 4 × 10 −6 . This approach thus provides a robust method to accurately model gate fidelities in a reduced Hilbert space to enable efficient optimisation and permit scaling to larger numbers of qubits.

Appendix C. Optimal pulse parameters
We have applied a variant of the dCRAB quantum optimisation technique to find an improved shape for the effective Rabi frequencyΩ R = g(t)Ω R (t), where Ω R (t) is the optimised analytical pulse described in the main text and depicted in figure 2. The envelope function g(t) has the form , ( C 1 ) Table 2. Numerical values of the optimal parameters of the dCRAB envelope function given by equation (C1).
Parameter with ω i k = 2πkr k /T i . We have set a frequency cut-off at N = 6 and performed the numerical optimisation with respect to the variables A k , B k and r k . The optimal parameters that we have found with this method and then used in the calculations of figures 3 and 4 of the main text are summarised in table 2.

Appendix D. Decomposition of CCZ gate into CZ and single-qubit gates
In figure 7 we depict the circuit representation of the optimal decomposition of the CCZ gate into six CZ gates and single-qubit gates [62]. The single-qubit rotation is given by T = exp(iπ/8σ z ), and H represents the Hadamard gate. In order for this decomposition to work with the definition of the C k Z gates that we have employed in this paper U C k Z = 2 ⊗ k+1 |0 ⊗ k+1 0| − I, the Hadamard gate needs to be defined with the roles of the |0 and |1 states swapped, H = 1/ √ 2(−|0 0| + |0 1| + |1 0| + |1 1|). Note that the CCZ could in principle be implemented with five two-qubit gates [63], but this would require realizing a CV = (1 − i)|00 00| − I gate which is not native to Rydberg systems.
Using the ARP protocol outlined in the paper we optimise the two-qubit CZ gate pulse sequence for equivalent parameters to those used for the CCZ gate. The two-qubit gate fidelity is calculated as above using where U † CZ is the ideal CZ matrix, U is the non-unitary operator implemented by the actual pulses and |Ψ is a symmetric superposition of the computational input states. We obtain an optimal two-qubit gate fidelity F 2Q ∼ 0.9985 for a total pulse time of τ ∼ 1.3 μs. We compare this performance to the sequential CZ pulse sequence presented in [17,21] which requires time τ = 4π/Ω R ∼ 0.5 μs and the global addressing CZ gate scheme of [27] which requires τ ∼ 2.73π/Ω R ∼ 0.3 μs, and obtain comparable performance with F 2Q ∼ 0.9985 for all three gate schemes, which exceed the performance of two-photon protocols based on STIRAP [28].
Using these optimal two-qubit CZ gates we simulate the CCZ gate decomposition circuit depicted in figure 7 assuming perfect single-qubit gates, and find CCZ gate fidelities F ∼ 0.991, which are significantly lower than the fidelity F = 0.9954 obtained with the optimal two-photon ARP pulses discussed in the main text. In order for the two-qubit gates to outperform the our native CCZ ARP gate, their fidelity would need to be F 2Q ∼ (0.9954) 1/6 ∼ 0.9992. This fidelity exceeds any known protocol based on two-photon excitation, and are only predicted for direct single-photon transitions using UV wavelengths [28,64] which introduces technical challenges due to power scaling and increased Doppler sensitivity. However, even if these excellent fidelities were achievable, the native CCZ would still have the advantage of requiring only a single-qubit rotation and to have a total duration shorter than the quantum pulse sequence associated with the quantum circuit depicted in figure 7.