The cold atom qubit platform [1–3] has been evolving rapidly since the first experimental demonstrations of the Rydberg blockade gate [4,5]. Recently, the ground-Rydberg excitation laser has improved significantly [6,7], the coherent quantum control of Rydberg atoms has been making steady progress [8–11], and shuttling atoms to form defect-free arrays has become a helpful tool [12–14]. Most interestingly, on the subject of the two-qubit Controlled-PHASE gate as the essential ingredient of quantum computation hardwares, the Rydberg blockade gate has received substantial advancement in fidelity [15,16], thanks to the method of continuously modulated off-resonant pulse replacing the early proposal of discrete resonant pulse sequence [10,17,18]. Nevertheless, the Rydberg dipole-dipole interaction does not naturally extend to very long ranges [2,19,20], and therefore enhancing the connectivity of the Rydberg blockade Controlled-PHASE gate has to resort to special techniques. Now that the number of cold atom qubits loaded in an array has convincingly surpassed the level of 1000 [21], finding a path to realize high-connectivity entangling quantum logic gates for very-large-scale arrays becomes an imminent challenge. We are looking forward to a realistic solution where qubit atoms can stay stationary during the gate operations such as the buffer-atom-mediated (BAM) gate and the buffer-atom framework [22], while mechanically shuttling the qubit atoms across the array seems slow in speed and may cause extra sources of leakage [23]. Like the role of SWAP gates in the superconducting qubit platform [24–26], to simultaneously obtain high connectivity, high speed, and high fidelity for sizable quantum algorithms, the future cold atom qubit platform is calling for a fast SWAP gate that will eventually work together with the buffer atoms and ancilla atoms [22].

With the exciting achievements in the last decade, the Rydberg blockade effect [2–4,19,27] has been extensively adopted in various quantum information processing tasks for its robustness and versatility. So far, the Rydberg blockade effect has repeatedly proved its better experimental compatibility compared with other schemes utilizing the Rydberg dipole-dipole interactions such as Rydberg pumping or Rydberg anti-blockade [28–31], which often require a precisely fixed value of Rydberg dipole-dipole interaction strength or extended time of interaction [32]. In particular, the recent success of applying synthetic continuously modulated driving in constructing high-fidelity Rydberg blockade entangling phase gates inspires us to ask the question of whether a fast Rydberg blockade SWAP gate is possible via similar methods. Meanwhile, we are pursuing robustness against several major adverse effects and compatibility with currently available Rydberg blockade gate experimental apparatuses.

In this paper, we provide a definitive answer that the Rydberg blockade SWAP gate protocol does exist. Moreover, we are going to design the fast Rydberg blockade SWAP gate to meet the current experimental conditions and analyze its principles and characteristics. Not surprisingly, the anticipated gate protocols have close ties with the quantum geometry [33–35]. The rest of the content is organized as follows. At first, we introduce the basic mechanisms of a fast Rydberg blockade SWAP gate and demonstrate the typical routines of how to design proper gate protocols to satisfy the practical requirements. Subsequently we also discuss the influence of typical adverse effects that reduce the fidelity of Rydberg blockade gates, where we keep adopting the method of suppressing the high-frequency components in the driving waveforms [36]. While this content focuses on the one-photon ground-Rydberg transition, we observe that the main idea becomes immediately applicable to continuously modulated pulses in two-photon ground-Rydberg transition [37] at almost no extra burden in designing. We will evaluate the two-qubit SWAP gate fidelity according to the well-established criteria [38,39].

We sketch the basic ingredients of atom-laser transitions and the relevant linkage structure in Fig. 1. Without loss of generality, we choose two non-degenerate magnetically insensitive states among the ground or metastable ground levels as the two-qubit register states, which can be instantiated in typical alkali and alkali earth atoms. In this SWAP gate protocol, we need two mutually coherent lasers ${\omega }_0,{\omega }_1$ to drive the ground-Rydberg transition $|0⟩\leftrightarrow |r⟩,|1⟩\leftrightarrow |r⟩$, respectively. Assume uniform optical drivings on the control and target atoms as the usual practice in experimental implementations now; then the Hamiltonian $H_a$ for a single control or qubit atom is $$H_a=\frac{1}{2}\hslash {\mathrm{\Omega }}_0{e}^{i{\mathrm{\Delta }}_{0}t}|0⟩⟨r|+\frac{1}{2}\hslash {\mathrm{\Omega }}_1{e}^{i{\mathrm{\Delta }}_{1}t}|1⟩⟨r|+\text{H.c.},$$(1)where we define the detuning terms as ${\omega }_{\text{transition}}-{\omega }_{\text{laser}}$ and let the rotating wave frequency be ${\omega }_{\text{transition}}$. Both the SWAP gate behavior and the Rydberg blockade effect manifest themselves beyond single-body phenomena, and therefore it seems not only natural but also necessary to carry on the analysis under the two-qubit basis. Besides the single-body Hamiltonians of the control and target qubit atoms in Eq. (1), the two-body interaction process also contains the Rydberg blockade effect as modeled by the Föster resonance below: $$H_q=B|rr⟩⟨q{q}^{\prime }|+B|q{q}^{\prime }⟩⟨rr|+{\delta }_q|q{q}^{\prime }⟩⟨q{q}^{\prime }|,$$(2)where $B$ is a real number denoting the strength of the blockade and ${\delta }_q$ is the penalty energy term. Then the four orthonormal basis two-qubit states divide themselves into two mutually independent groups of different dynamics: $\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$ and $|00⟩,\frac{1}{\sqrt{2}}(|01⟩+|10⟩),|11⟩$, which we name as singlet and triplet, respectively. According to the corresponding linkage structures shown in Fig. 1, the dynamics of a singlet does not involve the Rydberg blockade effect due to symmetry reasons. More details of the Hamiltonian and time evolution calculations are provided in Appendix A.

A SWAP gate has many variation forms in practical implementations and we consider two specific cases in this work as shown in Table 1. According to the linear superposition properties and linkage structures of the time evolution under study as shown in Fig. 1, it suffices to examine the gate process of four specific initial states, namely, the singlet and triplet states. For a “standard” format of a SWAP gate as in the left part in Table 1, wave functions of all four different initial states should end up with a full population return, in which $\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$ and $\frac{1}{\sqrt{2}}(|01⟩+|10⟩)$ should receive $\pi$ difference in phase. For an “opposite” format of a SWAP gate as in the right part in Table 1, wave functions starting in the initial state of $|00⟩$ will end up in $|11⟩$ and vice versa for that of $|11⟩$, while $\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$ and $\frac{1}{\sqrt{2}}(|01⟩+|10⟩)$ should receive no phase difference.Table 1.

Two Formats of SWAP Gate under Consideration^a

Previously, we generally interpreted the principles of the entangling phase gates [36,37,41] as letting the qubit atoms’ wave functions receive the correct state-dependent conditional phase shifts, which originate from dynamical phase, geometric phase, or both. On the other hand, the Rydberg blockade SWAP gate inevitably involves net population transfer with respect to the computational basis $|00⟩,|01⟩,|10⟩,|11⟩$. This subtle difference will bring a noticeable change in the design such as that now the SWAP gates $|0⟩,|1⟩$ both experience the ground-Rydberg transition, which also causes non-trivial variation of the underlying dynamics.

With the above principles outlining the mechanisms of the Rydberg blockade SWAP gate, the next question is to find out whether the desired optical driving exists and is compatible with experimental reality. Analytically attaining solutions seems too difficult for now due to the nature of the problem as inverse solving. Alternatively, we seek viable results via a numerical search or optimization like the development of Rydberg blockade entangling phase gates [18,36]. Without loss of generality, we set the criteria on the Rabi frequency waveforms that they start and end at intensity zero with first order derivative zero, for the ease of practicing modulation experimentally.

Here we choose a Fourier series as the basis to express and generate the waveforms, which also naturally helps to eliminate or suppress high-frequency components [36]. More specifically, the function $f$ of Rabi frequency or detuning terms can be expressed as $[a_0,a_1,\dots ,a_N]$, representing $f(t)=2\pi \times (a_0+{\sum }_{n=1}^N{a}_n\exp (2\pi int/\tau )+a_n^{*}\exp (-2\pi int/\tau ))/(2N+1)$ (in MHz) for a given reference time $\tau =0.25\mathrm{\mu s}$.

According to the discussions so far, we need to pay specific attention to the time evolutions of four initial states, $|00⟩,\frac{1}{2}(|01⟩-|10⟩),\frac{1}{2}(|01⟩+|10⟩),|11⟩$, and in particular the wave functions associated with these initial states, which we express as ${\mathrm{\Psi }}_{|00⟩}(t),{\mathrm{\Psi }}_{\frac{1}{2}(|01⟩-|10⟩)}(t),{\mathrm{\Psi }}_{\frac{1}{2}(|01⟩+|10⟩)}(t),{\mathrm{\Psi }}_{|11⟩}(t)$. It turns out that reasonable solutions of a Rydberg blockade SWAP gate do exist after an extensive search in the parameter space. On practical account, many numerical search algorithms can efficiently find appropriate values of $a_n$ to satisfy the requirement of the SWAP gates under study. And not surprisingly, the solutions are not unique as in the case of previous studies of Rydberg blockade Controlled-PHASE gates.

The Rydberg blockade effect provides the necessary interaction between two qubit atoms, enabling the exchange of quantum information embedded in the qubit atoms. As a natural consequence of quantum no-cloning, the quantum information does not receive duplication during the process. Meanwhile, the transition linkage structure plays a special role in making the Rydberg blockade SWAP gate possible.

We first look at the category with both amplitude and frequency modulations, namely, the hybrid modulation. The Rabi frequency waveforms of the two driving lasers ${\mathrm{\Omega }}_0(t),{\mathrm{\Omega }}_1(t)$ can be proportional or even the same, providing certain degrees of extra conveniences to experimental implementation. Here, we let ${\mathrm{\Omega }}_0={\mathrm{\Omega }}_1=[66.26,-20.40,-4.41,-12.61,-1.47,5.76]$, ${\mathrm{\Delta }}_0=[-\mathrm{13.37,}-21.68-19.43i,-10.79-19.20i,-25.28-33.11i,-45.53-20.21i,-8.44+1.27i]$, and ${\mathrm{\Delta }}_1=[\mathrm{54.53,}-9.77-16.65i,-12.84-11.28i,\hspace{0.17em}\hspace{0.17em}46.29-34.26i,-11.51-31.76i,-4.05+2.88i]$, and Fig. 2 shows the waveforms and key features of the atomic wave functions during the time evolution. This result corresponds to a SWAP gate in the standard format as given by the left column in Table 1.

If only amplitude modulation functions are adequate for the task without involving the extra burden of frequency modulation, this will sometimes simplify the experimental implementation of Rydberg blockade gates. We investigate this possibility and obtain such results. In particular, we let ${\mathrm{\Omega }}_0=[\mathrm{199.45,}-\mathrm{59.56,}-\mathrm{34.98,\hspace{0.17em}\hspace{0.17em}32.30,\hspace{0.17em}\hspace{0.17em}6.86,}-\mathrm{11.85,}-\mathrm{6.17,}-\mathrm{17.72,}-8.61]$, ${\mathrm{\Omega }}_1=[\mathrm{206.49,}-\mathrm{55.67,}-\mathrm{48.16,\hspace{0.17em}\hspace{0.17em}27.76,\hspace{0.17em}\hspace{0.17em}11.52,}-\mathrm{3.03,}-\mathrm{2.06,}-\mathrm{25.43,}-8.19]$, and ${\mathrm{\Delta }}_0=[9.05]$, ${\mathrm{\Delta }}_1=[-9.34]$, and Fig. 3 shows the waveforms and key features of the atomic wave functions during the time evolution. Here, we particularly choose a result that does not yield the standard format of SWAP gates, but with an overall $\pi$ phase shift to demonstrate the versatility of our method of design.

While the above two typical categories of modulation styles suffice for the purpose of experimental realizations, we want to emphasize that even for the case of single-photon ground-Rydberg transitions there exist more degrees of freedom in the method of synthetic continuously modulated driving. For example, we can apply only amplitude modulation with resonant conditions of ${\omega }_1,{\omega }_2$, where this and extra examples will be provided in Appendix A. Furthermore, this category of the gate process allows the design of many variation forms of SWAP gates, which basically reduces to determining appropriate parameters of the optical driving.

While the abstract concept and computational formulation of a Rydberg blockade SWAP gate seem plausible in the above analysis, its potential performance under realistic experimental conditions needs further consideration and examination. Here we proceed to estimate and discuss this subject with respect to a few typical and commonly encountered adverse effects. One key question is whether the gate process still holds at a reasonable quality against certain degrees of deviations from the idealized theoretical condition. Fortunately and interestingly, the synthetic continuously modulated driving, when employed in the Rydberg blockade SWAP gate, also implicitly triggers the two-atom dark state under the presence of a Rydberg blockade effect [39], which offers substantial help to the robustness of the gate. Meanwhile, we also hope to identify and analyze the major difficulties if gate error on the order of ${10}^{-4}$ or smaller is desired. If the experiment works with the two-photon ground-Rydberg transition instead of the single-photon transition, then one needs to use extra care in suppressing the population of the intermediate level, which usually has a much larger decay rate than the ground and Rydberg levels.

First of all, we would like to calculate the influences of possible errors in the Rabi frequencies and detunings of driving laser fields. This type of adverse effect occurs commonly in typical cold atom qubit experiments, and it makes sense to require the SWAP gate to maintain reasonable performance as long as these errors stay within a technically possible range. We show a typical result in Fig. 4 by studying the gate errors of waveforms in Fig. 2 under these adverse effects. It suggests that the experimental calibration of Rabi frequency needs to reach a certain precision for a high-fidelity gate. With the more and more advanced laser lock and noise filtering techniques, the laser phase noise mostly exists in the relatively low-frequency range $⪅0.1\mathrm{MHz}$, such that the effective phase noise seems relatively slow compared to the fast gate operation, which roughly acts like a slowly varying detuning term. Then Fig. 4 can also help to estimate the gate performance with respect to laser phase noise.

Generally speaking, when designing cold atom qubit arrays, we hope to accurately fix the positions of qubit atoms all across the board and henceforth the distances between qubit atoms as well. Nevertheless, sometimes the deviations of the inter-atomic distances from design can become significant and relatively difficult to accurately track. This easily causes the varying of Rydberg dipole-dipole interaction strength, and sometimes the Rabi frequencies seen by the atoms too. The waveforms in Fig. 3 are designed for an idealized Rydberg blockade effect, and on its basis we further develop a variation form specifically tailored for finite Rydberg blockade strength $B=2\pi \times 125\mathrm{MHz}$, ${\delta }_q=0$. And then we proceed to examine the influences of varying the overall ratio of Rabi frequencies ${\mathrm{\Omega }}_0,{\mathrm{\Omega }}_1$ and the Rydberg blockade strength $B$ to emulate the experimental adverse effects. The numerical result presented in Fig. 5 indicates the inherent advantage of a Rydberg blockade gate that retains relatively high-quality performance over a reasonable range of $B$. We observe that the Rydberg blockade SWAP gate here also has the versatility to adapt to the actual condition of Rydberg dipole-dipole interaction, similar to the case of the Controlled-PHASE gate [37]. More interestingly, the results of our calculations suggest that the Rydberg dipole-dipole interaction strength on the order of $\sim 100\mathrm{MHz}$ suffices for satisfying gate fidelity with respect to known experimental capabilities and from the practical point of view this indicates fast gate operation [41].

Within the scope of our discussions, we assume uniform laser drivings such that $|{\mathrm{\Omega }}_0|,|{\mathrm{\Omega }}_1|$ are the same on both qubit atoms, and this corresponds to the preferred experimental style especially when executing parallel operation on many atom pairs. Even if the control and target qubit atoms have different driving lasers, the design of Rydberg blockade SWAP gates still applies. In particular, for the case that control and target qubit atoms have different driving Rabi frequencies, the singlet type and the triplet type linkage structures as in Fig. 1 do not necessarily remain, but the method of synthetic continuously modulated driving can still help to find the appropriate SWAP gate waveform if it exists. On the other hand, the spontaneous emission of Rydberg levels imposes a fundamental limit on the practical gate fidelity. For the Rydberg blockade SWAP gate time $T$ and Rydberg level decay rate ${\gamma }_r$, it causes a gate error on the order of $\sim 0.5{\gamma }_{r}T$, as in Rydberg blockade Controlled-PHASE gates extensively studied previously. If other experimental conditions allow, a faster gate seems to be the future direction as long as the higher-order frequency components in the waveforms have been handled carefully [36]. For the rest of this article, we will omit relevant discussions about decays of Rydberg levels in order to focus on the main theme. In other words, we can view the gate fidelity calculations here as conditioned on the post-selection of decays of Rydberg levels that do not occur.

The recent innovations of buffer-atom-mediated quantum logic gates and buffer atom framework [22,41] aim at enhancing connectivity and suppressing cross-talk without mechanically shuttling the qubit atoms. Our Rydberg blockade SWAP gate is readily compatible with the buffer atom framework at the cost of extra lasers and extra control systems. In particular, we can apply the Rydberg blockade SWAP gate between qubit and buffer atoms such that the buffer atom or buffer atom relay can effectively help to operate SWAP gates between qubit atoms, with buffer and qubit atoms of different elements so that cross-talk does not become an issue. When the two driving lasers ${\omega }_0,{\omega }_1$ are parallel and co-propagating, the net atom-photon momentum exchange reduces to zero on average and henceforth the Rydberg blockade SWAP gate with synthetic continuously modulated driving does not cause heating. On the other hand, if the qubit atoms are heated up during computational quantum logic gate operations for other reasons, the Rydberg blockade SWAP gate with a nearby buffer atom provides a tool to cool down the qubit atoms in the run. With these analyses, we anticipate the future large-scale cold atom qubit array can maintain fast gate speed, high connectivity, and high fidelity simultaneously by combining the buffer atom framework and Rydberg blockade SWAP gate with synthetic continuously modulated driving.

In conclusion, we design and analyze the fast SWAP Rydberg blockade gates with synthetic continuously modulated driving. This category of SWAP gates can work with one-photon or two-photon ground-Rydberg transitions, and is readily compatible with the currently available hardwares in the research field of cold atom qubits. According to the results of the Rydberg blockade Controlled-PHASE gate, synthetic continuously modulated driving leads to competitive performance in practical implementation, and here we observe that the Rydberg blockade SWAP gate inherits many of the previously known advantages and features of Controlled-PHASE gates. With the BAM gate mediating the connection between non-interacting qubit atoms, we anticipate that the combination of fast Rydberg blockade SWAP gates and BAM gates will enhance the connectivity of very-large-scale cold atom qubit arrays in the near future, such that an entangling gate between two relatively distant qubit atoms can operate with high speed and high fidelity. While a fast and high-fidelity SWAP gate is known as an indispensable ingredient of superconducting quantum processors, a lot of recent findings together with our results hint that the superconducting qubit platform and the neutral atom qubit platform will find more and more common ground. In the future, we are looking forward to many of the previously established quantum coherent control methods, quantum algorithms, and quantum error correction codes of superconducting qubits extending to the neutral atom qubits.

Acknowledgment. The authors thank Ning Chen, Lingling Lao, Junjie Wu, and Peng Xu for many discussions.