One-step implementation of Toffoli gate for neutral atoms based on unconventional Rydberg pumping

Compared with the idea of universal quantum computation, a direct synthesis of a multiqubit logic gate can greatly improve the efficiency of quantum information processing tasks. Here we propose an efficient scheme to implement a three-qubit controlled-not (Toffoli) gate of neutral atoms based on unconventional Rydberg pumping. By adjusting the strengths of Rabi frequencies of driving fields, the Toffoli gate can be achieved within one step, which is also insensitive to the fluctuation of the Rydberg-Rydberg interaction. Considering different atom alignments, we can obtain a high-fidelity Toffoli gate at the same operation time $\sim 7~\mu s$. In addition, our scheme can be further extended to the four-qubit case without altering the operating time.

Another effect called Rydberg antiblockade has been proposed theoretically [33] and demonstrated experimentally [34]. In contrast with Rydberg blockade, Rydberg antiblockade can directly excite two Rydberg atoms from the ground states to the Rydberg states without single-excited Rydberg state. This kind of excitation process provides a method for rapid construction of quantum gates, it can be used to construct controlled-phase gate [35][36][37][38][39][40][41][42] and controlled-not gate [37,41]. However, it is difficult to exactly control the distance between the two Rydberg atoms in the experiment to meet the condition U = 2∆, where U denotes the van der Waals interaction and ∆ is the single-photon detuning parameter. Therefore, once the parameter relationship is disturbed, the scheme will fail due to the rapid decline of fidelity. Recently, our group proposed an unconventional Rydberg pumping (URP) mechanism [43], which is different from Rydberg blockade and Rydberg antiblockade. By driving the same ground state of each atom with a dichromatic classical fields, the evolution of the atoms would be frozen (activated) as two Rydberg atoms are in the same (different) ground states. Based on URP, one can achieve a three-qubit controlled-phase gate, steady-state entanglement, and autonomous quantum error correction.
In this paper, we modify the original URP mechanism by acting on different atoms with different frequencies of the driving field, respectively. Such a modified URP mechanism can directly implement a three-qubit controlled-not (Toffoli) gate without using the synthesis method of Hadamard gate plus controlled-phase gate [44][45][46][47][48][49][50][51][52][53][54][55][56]. Meanwhile, our scheme is not only immune to the variation of RRI strength between the control atoms, but also has a certain robustness to the fluctuation of the interaction strength between the control atoms and the target atom. And our scheme can be further extended to the four-qubit case without altering the operating time. It can be expressed as follows

II. IMPLEMENTATION OF THE TOFFOLI GATE
where ij⊕k = (i×j+k) mod 2. In order to realize this operation, we consider a system composed of three Rydberg atoms, and the relevant configuration of the atomic level is illustrated in Fig. 1(a). Atom 1 and Atom 2 are control qubits which consist of ground states |0 and |1 and an excited Rydberg state |r . There is only one off-resonant transition between |0 and |r with blue detuning ∆ driven by a classical field of Rabi frequency Ω ′ . We label the target qubit as Atom 3, and there are two resonant transitions between |0 ↔ |r and |1 ↔ |r with Rabi frequencies Ω 1 and Ω 2 , respectively. The atomic alignment is shown in Fig. 1(b), the control qubits are on a dotted circle centered on the target qubit. In the interaction picture, the Hamiltonian of the total system can be written as ( = 1) where U jk is considered as van der Waals interaction strength U vdW = C 6 /R 6 between the jth and kth atom. R is the distance between two Rydberg atoms and C 6 depends on the quantum numbers of the Rydberg state. To embody the effect of the RRI, we extend the Hamiltonian to a three-atom basis form and rewrite H I after moving to a rotating frame with respect to U 0 = exp(−it j =k U jk |rr jk rr|) by using the formula iU † 0 U 0 + U † 0 H I U 0 . Then we have + Ω ′ (e i∆t |0αβ rαβ| + |0αr rαr| where we have utilized the URP condition as U 13 = U 23 = ∆. Under the condition ∆ ≫ {Ω ′ , Ω 1 , Ω 2 }, we can further simplify Eq. (3) by eliminating the high-frequency oscillating terms and obtain IR + H where H (1) The transitions of different subspaces are described from H IR . When we select the condition U 12 ≫ Ω ′ , the transitions from |r0r (|0rr ) to |rrr can be eliminated as the high-frequency oscillating terms in H IR . If we further assume the condition 2|00r ± |r0r ± |0rr )/2 and |Φ 0 = (|r0r − |0rr )/ √ 2. Thus, the effective transitions between the computational basis states {|000 , |001 } and |Φ ± can be represented as Fig. 2(b), from which we can see that the interactions between {|000 , |001 } and {|Φ + , |Φ − } are largely detuned under the condition |λ ± | ≫ {Ω 1 , Ω 2 } owing to Ω ′ ≫ {Ω 1 , Ω 2 }. At the same time, there is neither a transition between the ground states, nor Stark shifts of the ground states, because these transition paths mediated by the independent channels |Φ ± interfere destructively. In addition, the evolutions of other computational basis states IR are similar to the above case as U 12 ≫ Ω ′ , thence these transition processes can also be forbidden.
For a more general case of U 12 which is not much larger than Ω ′ , we can use the effective operator method [57][58][59] to obtain a result similar to the above case, i.e., |000 and |001 will not evolve. The time-independent form of H where H e = Ω ′ (|00r r0r|+|00r 0rr|+|0rr rrr|+|r0r rrr|)+ H.c. + U 12 |rrr rrr| and V − = V † + = Ω 1 |000 00r| + Ω 2 |001 00r|. Under the condition that the Rabi frequencies Ω 1 and Ω 2 are sufficiently weak compared with Ω ′ and U 12 , the effective Hamiltonian of subspace {|000 , |001 } can be described as where the inverse matrix H −1 e can be calculated by replacing 0 on the diagonal element of the matrix H e with an infinitesimal in order to avoid singular value. This equation does not involve the transitions from the ground states to the Rydberg state, but two additional Stark-shift terms, which is weak enough compared to {Ω 1 , Ω 2 } in Eq. (4) to be negligible. As a result, considering the above two magnitude of U 12 , the effective Hamiltonian of the total system can be simplified as Thus the Toffoli gate can be carried out within one step as t = π/Ω by setting Ω = Ω 2 1 + Ω 2 2 and Ω 1 = −Ω 2 . It is worth noting that there is no two-excited Rydberg state in the effective Hamiltonian, so our scheme does not need to strictly control the interaction strength between Rydberg atoms like Rydberg antiblockade. As a result, the energy level shift caused by dipole-dipole force arising from the interaction between two Rydberg states is avoided effectively, and the corresponding Rydberg state will not be influenced by this factor.

III. NUMERICAL SIMULATION
To assess the performance of the Toffoli gate, we introduce the trace-preserving quantum-operator-based (TPQO) average fidelity method defined as [60]  where O j is the tensor of Pauli matrices III, IIX, ... , ZZZ, O is the perfect Toffoli gate, d = 8 for a three-qubit quantum logic gate, and ε is the trace-preserving quantum operation achieved through our scheme. By using the TPQO average fidelity method, we plot average fidelities of the Toffoli gate with different Rabi frequencies Ω 1 in Fig. 3(a), where U 13 = U 23 = ∆, U 12 = ∆/8 and ∆ = 50Ω ′ by using the full Hamiltonian H I in Eq. (2) and the effective Hamiltonian H eff in Eq. (6), respectively. And the results of H I are in good agreement with H eff . The average fidelity can reach at 0.998, 0.9972 and 0.9914 with the increase of Rabi frequency Ω 1 from 0.025Ω ′ to 0.075Ω ′ . From this we can see that the larger value of Ω 1 , the worse the approximation. Therefore, without considering any decohenrece, we will get a higher fidelity if the value of Ω 1 is relatively small. Next, we take into account the situation that U 13 (U 23 ) and ∆ are not strictly equal. Compared with Rydberg antiblockade that requires exact control of the interaction strength between Rydberg atoms, our scheme does not need to strictly control the interaction strength between the control atoms and the target atom. In Fig. 3(b), we depict the average fidelities with different mismatching rate η (∆) between U and ∆, where U 13 = U 23 = U and η (∆) = (U − ∆)/∆, U 12 = ∆/8, ∆ = 50Ω ′ and Ω 1 = 0.05Ω ′ . The results show that the fidelities can still be maintained above 0.98 when η (∆) is from −0.02 to 0.02, which is equivalent to the difference between U and ∆ can reach the order of magnitude Ω ′ . Thus, even if the matching relationship between U and ∆ is slightly off, the fidelity is still high and has little impact on the implementation of the gate. Moreover, we discuss the influence of the gate fidelity based on the strengths of Rabi frequencies driven the control qubits. In Fig. 3(c), we suppose that the control qubit Atom 2 is driven by Rabi frequency Ω ′′ and plot the average fidelities with different mismatching rate η (Ω ′ ) = (Ω ′′ −Ω ′ )/Ω ′ between Ω ′ and Ω ′′ when U 13 = U 23 = ∆, U 12 = ∆/8, ∆ = 50Ω ′ and Ω 1 = 0.05Ω ′ . It can be seen that the fidelities with different Ω ′′ are all remained above 0.994. Hence, the Rabi frequencies driven control qubits does not have to be exactly equal when they are much greater than {Ω 1 , Ω 2 }.
In addition, we would like to study the impact of changes in the value of U 12 on the scheme. In Fig 3(d), we plot the average fidelities with different distance between control qubits when Ω 1 = 0.05Ω ′ under the conditions of U 13 = U 23 = ∆ and ∆ = 50Ω ′ . The distance between control atoms 1R corresponds to the relation U 12 = ∆ = 50Ω ′ . As the distance increases from 0.5R to 1.5R, U 12 can be changed from 3200Ω ′ to 4.39Ω ′ , but the corresponding average fidelities are remained above 0.997. Thus, the magnitude of U 12 hardly influences the implementation of the Toffoli gate. However, we notice a special case where the distance is approximately equal to 0.8909R, i.e. U 12 = 2∆. At this point, the average fidelity can only reach 0.6757. The reason is that the two-photon resonance under U 12 = 2∆ results in the direct transition between |000 (|001 ) and |rr0 (|rr1 ), which affects the final average fidelity. And the transitions of |0r0 (|r00 ) ↔ |rr0 and |0r1 (|r01 ) ↔ |rr1 in Eq. (3) are decoupled from ground states and can be omitted in most cases except U 12 = 2∆. Thus, the value of U 12 should keep away from the vicinity of 2∆ when U 12 ≫ Ω ′ . In the presence of the spontaneous emission of Rydberg atoms, the system can be dominated by the following master where γ is the decay rate of the Rydberg state, and we have assumed that the branching ratios of spontaneous emission from Rydberg state |r to the ground states |0 and |1 are both γ/2, σ 1 = |0 1 r|, σ 2 = |1 1 r|, σ 3 = |0 2 r|, σ 4 = |1 2 r|, σ 5 = |0 3 r| and σ 6 = |1 3 r| denote six decay channels of the three atoms, respectively. In Fig. 4, we plot the variation trend of the average fidelities under the influence of decay rate γ with different Ω 1 from 0.025Ω ′ to 0.075Ω ′ under the conditions of U 13 = U 23 = ∆, U 12 = ∆/8 and ∆ = 50Ω ′ . It can be seen that although the fidelity of Ω 1 = 0.075Ω ′ is lower than others when γ = 0.001Ω ′ , and the decreasing of the fidelity is less than others with the increasing of the decay rate. Thus, with the appropriate increase of Rabi frequency, the interaction time is shortened, thereby further reducing the influence of dissipation on the scheme. In addition, there is a probability that atoms in the excited Rydberg states will spontaneously decay into an external level out of {|0 , |1 }. To study this situation, we incorporate an uncoupled state |d in Eq. (8), and suppose the branching ratio of spontaneous emission from |r to |d is equal to the branching ratio to the computational basis states |0 and |1 . In Fig. 4, we find the variation trend of the fidelities under the influence of γ is consistent with the case without considering state |d when Ω 1 = 0.05Ω ′ . Therefore, it has little influence on our scheme whether the state outside computational basis states exists or not. In experiment, we choose Rydberg atom with the appropriate principal quantum number to maintain a relatively long radiative lifetime τ for suppressing the influence of spontaneous emission. Therefore, we use the Rydberg state |97d 5/2 , m j = 5/2 (|C 6 | = 37.6946 Thz·µm 6 ) of 87 Rb atom with the lifetime ∼ 320 µs and decay rate γ ≃ 3.125 kHz (γ ≃ 1/τ ) referring to prior works [36,44,[61][62][63][64][65]. And the required strength of Rabi frequency excited to the Rydberg state can be continuously adjusted to 2π × 10 MHz [38,66]. According to the atomic alignment in Fig. 1(b) (U 12 = 2π × 50/8 MHz), we consider the influence of different U and Ω ′′ on the scheme, and the corresponding values are listed in Table 1, U =2π × (49.5, 50, 50.5) MHz and Ω ′′ =2π × (1, 2) MHz. If the arrangement of the three atoms is an equilateral triangle (U 12 = 2π × 50 MHz) or a straight line (U 12 = 2π × 50/64 MHz), we can also obtain the corresponding fidelities listed in Table 1. Other same parameters are selected as (Ω 1 , Ω ′ , ∆) = 2π × (0.05, 1, 50) MHz and γ= 3.125 kHz. Based on the above analysis, our scheme can be directly extended to implement the four-qubit Toffoli gate. Here we consider two structures as shown in Fig. 5(a) and 5(b) where Atom 1, 2, 3 are control qubits. The target qubit is labeled as Atom 4. The Hamiltonian of the four-atom system can be represented as In order to assess the performance of the four-qubit Toffoli gate, we select |ψ = ( 1 i,j,k,l=0 |ijkl − 2|1110 )/4 as the initial state, |φ = ( U 12 = U 23 = U 13 = ∆/27 in Fig. 6(a), and the population of |ψ with H ′ eff in Eq. (10) under condition U 14 = U 24 = U 34 = ∆, U 13 = ∆/64, U 12 = U 23 = ∆/8 in Fig. 6(b). These results show that the populations are almost completely transferred, and the tendencies of the populations governed by the full Hamiltonian and the effective Hamiltonian are identical. Therefore, as long as the corresponding URP conditions are satisfied, the four-qubit Toffoli gate can be implemented in one step without altering the operating time. However, it is currently difficult to extend the scheme to more than four qubits since the interactions become complicated with increase of the number of qubits. We look forward to coming up with a simpler scheme to implement multiqubit Toffoli gates in the follow-up work.

V. SUMMARY
In summary, our work has provided a scheme to implement a Toffoli gate within one step based on unconventional Rydberg pumping (URP) mechanism. Compared with Rydberg antiblockade, our scheme does not need to strictly control the interaction strength between atoms. The fluctuations of parameters U 12 , U and Ω ′′ are allowed within a certain range for maintaining a high fidelity of gate. Meanwhile, the scheme can be directly extended to the four-qubit case without changing the operation time. We hope that our scheme can provide a new approach for the implementation of scalable Rydberg quantum gates.

VI. ACKNOWLEDGEMENTS
This work is supported by National Natural Science Foundation of China (11774047); Fundamental Research Funds for the Central Universities (2412020FZ026); Natural Science Foundation of Jilin Province (JJKH20190279KJ).