Transition slow-down by Rydberg interaction of neutral atoms and a fast controlled-NOT quantum gate

There are exciting advances in Rydberg atom quantum science recently Jaksch et al. (2000); Lukin et al. (2001); Saffman et al. (2010); Saffman (2016); Weiss and Saffman (2017); Firstenberg et al. (2016); Adams et al. (2020); Browaeys and Lahaye (2020) because of the feasibility to coherently and rapidly switch on and off the strong dipole-dipole interaction. Such interaction enables simulation of quantum many-body physics Gurian et al. (2012); Tretyakov et al. (2017); De Léséleuc et al. (2019); Schauß et al. (2015); Labuhn et al. (2016); Zeiher et al. (2016, 2017); Bernien et al. (2017); de Léséleuc et al. (2018); Guardado-Sanchez et al. (2018); Kim et al. (2018); Keesling et al. (2019); Ding et al. (2020); Borish et al. (2020), probing and manipulation of single photons Dudin and Kuzmich (2012); Peyronel et al. (2012); Firstenberg et al. (2013); Li et al. (2013); Gorniaczyk et al. (2014); Baur et al. (2014); Tiarks et al. (2014); Li et al. (2016); Busche et al. (2017); Ripka and Pfau (2018); Liang et al. (2018); Li et al. (2019); Thompson et al. (2017), large-scale entanglement generation Ebert et al. (2015); Omran et al. (2019), and quantum computation Ebert et al. (2015); Omran et al. (2019); Wilk et al. (2010); Isenhower et al. (2010); Zhang et al. (2010); Maller et al. (2015); Zeng et al. (2017); Levine et al. (2019); Graham et al. (2019); Madjarov et al. (2020); Jau et al. (2016); Levine et al. (2018); Picken et al. (2019); Tiarks et al. (2019); Jo et al. (2020). To date, however, most effort focused on the effect of excitation annihilation (EA) proposed in Jaksch et al. (2000); Lukin et al. (2001) and reviewed in Saffman et al. (2010); Saffman (2016); Weiss and Saffman (2017); Firstenberg et al. (2016); Adams et al. (2020); Browaeys and Lahaye (2020), although there are other categories as summarized in Table 1. Belonging to the Rydberg blockade regime, EA involves single-atom Rydberg excitation and hence is not sensitive to the fluctuation of interaction. Besides EA, one can also excite both qubits to Rydberg states Jaksch et al. (2000), or explore the antiblockade regime Ates et al. (2007); Amthor et al. (2010), or use the resonant dipole-dipole flip Thompson et al. (2017); De Léséleuc et al. (2019). These latter processes, however, involve two-atom Rydberg excitation and are sensitive to the fluctuation of qubit separation that can substantially reduces the fidelity of a quantum control by using them Jo et al. (2020). It is an open question whether there is another means other than EA to explore the Rydberg blockade regime for efficient and high-fidelity quantum control.

Here, we study an unexplored transition slow-down (TSD) effect of the dipole-dipole interaction between Rydberg atoms. When the state of one atom oscillates back and forth between ground and Rydberg states, its Rydberg interaction does not block a Rydberg-involved state swap in a nearby atom, but slows it down, and the fold of slow-down is adjustable. The resulted TSD denotes the slow-down of the state transfer of one of the two atoms although the collective Rabi frequency is enhanced due to the many-body effect. Albeit appeared as a slow-down, a controlled TSD on the contrary can drastically speed up certain crucial element in a quantum computing circuit, such as the controlled-NOT (CNOT) which is the very important two-qubit entangling gate in the circuit model of quantum computing in both theory Nielsen and Chuang (2000); Williams (2011); Ladd et al. (2010); Shor (1997); Bremner et al. (2002); Shende et al. (2004) and experiment Peruzzo et al. (2014); Debnath et al. (2016). TSD is a new route toward efficient, flexible, and high-fidelity quantum control over neutral atoms because it is implemented in the strong blockade regime which is robust to the fluctuation of interactions.

Though Rydberg atoms have kindled the flame for the ambition to large-scale quantum computing Jaksch et al. (2000); Lukin et al. (2001); Saffman et al. (2010); Saffman (2016); Weiss and Saffman (2017), and recent experiments demonstrated remarkable advances Wilk et al. (2010); Isenhower et al. (2010); Zhang et al. (2010); Maller et al. (2015); Jau et al. (2016); Zeng et al. (2017); Levine et al. (2018); Picken et al. (2019); Levine et al. (2019); Graham et al. (2019); Madjarov et al. (2020); Tiarks et al. (2019); Jo et al. (2020), further progress toward neutral-atom quantum computing is hindered by the difficulty to prepare a fast and accurate CNOT. This is partly because each of those CNOT gates was realized via combining an EA-based controlled-Z ($C_{Z}$) and a series of single-qubit gates Isenhower et al. (2010); Zhang et al. (2010); Maller et al. (2015); Zeng et al. (2017); Graham et al. (2019); Levine et al. (2019), leading to CNOT durations dominated by single-qubit operations (e.g., over $4~{}\mu$s in Levine et al. (2019); Graham et al. (2019)). In contrast, an TSD-based CNOT has a duration about $2\pi/\Omega+\epsilon$ by only two Rydberg pulses, i,e., needs no single-qubit rotations, where $\epsilon$ is a transient moment to implement a phase change in the pulses and $\Omega$ the Rydberg Rabi frequency. Using values of $\Omega$ and $\epsilon$ from Graham et al. (2019); Levine et al. (2019), the TSD-based CNOT (duration $\sim 0.3~{}\mu$s) would be orders of magnitude faster than those in Graham et al. (2019); Levine et al. (2019), which means that a neutral-atom CNOT can be much faster than the CNOT (or ground Bell-state gate) by trapped ions Cirac and Zoller (1995); Sørensen and Mølmer (1999); Ballance et al. (2015, 2016); Gaebler et al. (2016) (notice that the fast ion-trap gates in Schäfer et al. (2018); Zhang et al. (2020) are phase gates). Although still inferior to superconducting circuits You and Nori (2005); Devoret and Schoelkopf (2013) where CNOT gate times can be around $50$ ns Barends et al. (2014), the TSD-based CNOT is applicable in scalable neutral-atom platforms that are ideal for long-lived storage of quantum information in room temperatures.

The simplest model of TSD consists of two nearby atoms, each pumped by a laser pulse that induces a transition between a ground state $|1\rangle$ and a Rydberg state $|r\rangle$. The Rabi frequency is $\Omega_{\text{c(t)}}$ for the control (target) qubit, and the dipole-dipole interaction $V$ in $|rr\rangle$ is assumed large compared to $\Omega_{\text{c(t)}}$ so that $|rr\rangle$ is not populated. For Rydberg interaction of the van der Waals type, $V$ is limited because in this regime the native dipole-dipole interaction should be much smaller than the energy gaps between nearby two-atom Rydberg states Walker and Saffman (2008), and thus the qubit spacing should be large enough. On the other hand, a direct dipole-dipole interaction can be huge for high-lying Rydberg states although it is no longer a pure energy shift. In this sense, the residual blockade error of the order $\sim\Omega_{\text{c(t)}}^{2}/V^{2}$ Saffman and Walker (2005) can be negligible in the strong dipole-dipole interaction regime as long as the two-atom spacing is beyond the LeRoy radius.

Figure 1 shows a contrast between EA and a two-state TSD, where the pulse sent to the control atom is applied during $t\in[t_{0},~{}t_{0}+t_{1})$, and that to the target atom is during $t\in[0,~{}2t_{1})$ with $t_{0}<t_{1}\equiv\pi/\Omega_{\text{t}}$. Starting from an initial two-atom state $|11\rangle$, the wavefunction at $t=t_{0}$ becomes

$\displaystyle|\psi(t_{0})\rangle$

$\displaystyle=$

$\displaystyle\cos(\Omega_{\text{t}}t_{0}/2)|11\rangle-i\sin(\Omega_{\text{t}}t_{0}/2)|1r\rangle.$

(1)

During $t\in[t_{0},~{}t_{0}+t_{1})$, the system Hamiltonian is

$\displaystyle\hat{H}$

$\displaystyle=$

$\displaystyle[(\Omega_{\text{c}}|r1\rangle+\Omega_{\text{t}}|1r\rangle)\langle 11|/2+\text{H.c.}]+\hat{H^{\prime}},$

(2)

where $\hat{H^{\prime}}$ includes excitation between $\{|1r\rangle,~{}|r1\rangle\}$ and $|rr\rangle$ and the dipole-dipole flip from $|rr\rangle$. Focusing on the strong interaction regime, $\hat{H^{\prime}}$ can be discarded because $|rr\rangle$ is not coupled Jaksch et al. (2000) and hence $\hat{H}=\bar{\Omega}|\mathbb{R}\rangle\langle 11|/2+\text{H.c.}$, where $\bar{\Omega}\equiv\sqrt{\Omega_{\text{c}}^{2}+\Omega_{\text{t}}^{2}}$ and $|\mathbb{R}\rangle\equiv(\Omega_{\text{c}}|r1\rangle+\Omega_{\text{t}}|1r\rangle)/\bar{\Omega}$. When $\alpha\equiv\Omega_{\text{c}}/\Omega_{\text{t}}=\sqrt{15}$, we have $\bar{\Omega}t_{1}=4\pi$, so that the wavefunction at $t=t_{0}+t_{1}$, given by $e^{-it_{1}\hat{H}}|\psi(t_{0})\rangle$, is equal to Eq. (1), and it is like that nothing happens to the target qubit upon the completion of the drive in the control qubit. Then, because the pulse for the control qubit ends at $t=t_{0}+t_{1}$, the continuous pumping on the target qubit drives the state to $|1r\rangle$ at $t=2t_{1}$, i.e., a $2\pi$ pulse, instead of a $\pi$ pulse, completes the transition $|11\rangle\rightarrow|1r\rangle$, which corresponds to a 2-fold slow-down.

To show the strength of TSD in quantum information, we would like to consider TSD in three states. This is because the two-state TSD ends up in a Rydberg state which is not stable, while quantum information shall be encoded with two qubit states $|0\rangle$ and $|1\rangle$ in the more stable ground manifold. For frequently used rubidium and cesium atoms, $|0\rangle$ and $|1\rangle$ can be chosen from the two hyperfine-split ground levels with a frequency difference of several gigahertz.

Consider a ground-Rydberg-ground transition chain $|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$, i.e., a state swap between the two long-lived qubit states via a metastable Rydberg state $|r\rangle$. The three-state TSD is implemented by pumping the control and target qubits with respective Rabi frequencies $\Omega_{\text{c}}$ and $\Omega_{\text{t}}$ for the same duration, with transition $|1\rangle\xleftrightarrow{\Omega_{\text{c}}}|r\rangle$ for the control qubit and $|0\rangle\xleftrightarrow{\Omega_{\text{t}}}|r\rangle\xleftrightarrow{\Omega_{\text{t}}}|1\rangle$ for the target qubit. This requires a setup capable of pumping both qubit states to a common Rydberg state in the target qubit, which is feasible as experimentally demonstrated for two-atom entanglement about a decade ago Wilk et al. (2010) and for $W$-state preparation in atom ensembles later Ebert et al. (2015). Because $|0\rangle$ is not pumped in the control qubit, the Hamiltonian is $\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}})=\Omega_{\text{t}}(|00\rangle+|01\rangle)\langle 0r|/2+$H.c. for the input states $|00\rangle$ and $|01\rangle$, where the subscript $c0$ ($c1$) denotes that the Hamiltonian applies when the input state for the control qubit is $|0(1)\rangle$. For the remaining input states, we consider the ordered basis $\{|1r\rangle,|r1\rangle,|r0\rangle,|11\rangle,|10\rangle\}$ for the following Hamiltonian

$\displaystyle\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}},\Omega_{\text{t}})$

$\displaystyle=$

$\displaystyle\frac{1}{2}\left(\begin{array}[]{ccccc}0&0&0&\Omega_{\text{t}}&\Omega_{\text{t}}\\ 0&0&0&\Omega_{\text{c}}&0\\ 0&0&0&0&\Omega_{\text{c}}\\ \Omega_{\text{t}}&\Omega_{\text{c}}&0&0&0\\ \Omega_{\text{t}}&0&\Omega_{\text{c}}&0&0\end{array}\right),$

(8)

where we ignore the two-atom Rydberg state $|rr\rangle$ with reasons shown below Eq. (2). By diagonalizing Eq. (8) one has

	$\displaystyle|10\rangle$	$\displaystyle=$	$\displaystyle(|\mathscr{R}_{4}\rangle-|\mathscr{R}_{3}\rangle+|\mathscr{R}_{2}\rangle-|\mathscr{R}_{1}\rangle)/2,$
	$\displaystyle|11\rangle$	$\displaystyle=$	$\displaystyle(|\mathscr{R}_{4}\rangle-|\mathscr{R}_{3}\rangle-|\mathscr{R}_{2}\rangle+|\mathscr{R}_{1}\rangle)/2,$		(9)

where $|\mathscr{R}_{k}\rangle$ with $k=1-4$ are four eigenvectors of Eq. (8), with eigenvalues $(\Omega_{\text{c}},~{}-\Omega_{\text{c}},~{}\overline{\Omega},-\overline{\Omega})/2$, respectively [the fifth eigenvector does not enter Eq. (9)], where $\overline{\Omega}\equiv\sqrt{\Omega_{\text{c}}^{2}+2\Omega_{\text{t}}^{2}}$ and

	$\displaystyle|\mathscr{R}_{1,2}\rangle$	$\displaystyle=$	$\displaystyle\frac{1}{2}[|r1\rangle-|r0\rangle\pm(|11\rangle-|10\rangle)],$		(10)
	$\displaystyle|\mathscr{R}_{3,4}\rangle$	$\displaystyle=$	$\displaystyle\frac{1}{2\overline{\Omega}}[\Omega_{\text{c}}(|r1\rangle+|r0\rangle)\pm\overline{\Omega}(|11\rangle+|10\rangle)+2\Omega_{\text{t}}|1r\rangle].$

We proceed to describe an accurate and exceedingly fast CNOT by a spin-echo assisted TSD. The sequence consists of two pulses, each with duration $\pi/\Omega_{\text{c}}$ and condition $|\Omega_{\text{t}}|/\Omega_{\text{c}}=\sqrt{6}/2$, shown in Fig. 2(a). A $\pi$ phase change (requiring a time $\epsilon$) is inserted between the pulses sent to the target qubit so as to induce spin echo, where $\epsilon$ can be around $10$ ns Levine et al. (2019). The spin echo suppresses the state swap of the target qubit if the control qubit is initialized in $|0\rangle$ Shi (2018a, 2020a); but when the control qubit is initialized in $|1\rangle$, it is pumped to $|r\rangle$ which results in TSD in the target qubit, i.e., the transition $|0\rangle\rightleftharpoons|r\rangle\rightleftharpoons|1\rangle$ in the target qubit, which can occur with a pulse duration $\sqrt{2}\pi/\Omega_{\text{t}}$ if no TSD is used Shi (2018b), will be slowed down by a fold of $\sqrt{3}$. The mechanism is understood in two steps. First, during $t\in[0,~{}t_{\pi})$ with $t_{\pi}=\pi/\Omega_{\text{c}}$, the input state $|\psi_{c0}(0)\rangle=|00\rangle$ or $|01\rangle$ evolves according to $e^{-it\mathcal{H}_{c0}(\Omega_{\text{t}})}|\psi_{c0}\rangle$, and after a $\pi$ phase change to the Rydberg Rabi frequency which may require a finite transient time $\epsilon$, the state evolution becomes $e^{-i(t-t_{\pi}-\epsilon)\mathcal{H}_{c0}(-\Omega_{\text{t}})}|\psi_{c0}(t_{\pi})\rangle$ during the second pulse, leading to $e^{-it\mathcal{H}_{c0}(-\Omega_{\text{t}})t_{\pi}}|\psi_{c0}(t_{\pi})\rangle=e^{-it\mathcal{H}_{c0}(-\Omega_{\text{t}})t_{\pi}}e^{-it\mathcal{H}_{c0}(\Omega_{\text{t}})t_{\pi}}|\psi_{c0}(0)\rangle=|\psi_{c0}(0)\rangle$ at the end of the sequence. Second, according to Eq. (10), the first pulse during $t\in[0,~{}t_{\pi})$ evolves the input state $|\psi_{c1}(0)\rangle=|10\rangle$ according to

$\displaystyle\frac{1}{2}(e^{\frac{it\overline{\Omega}}{2}}|\mathscr{R}_{4}\rangle-e^{\frac{-it\overline{\Omega}}{2}}|\mathscr{R}_{3}\rangle+e^{\frac{it\Omega_{\text{c}}}{2}}|\mathscr{R}_{2}\rangle-e^{-\frac{it\Omega_{\text{c}}}{2}}|\mathscr{R}_{1}\rangle),$

(11)

which becomes

$\displaystyle|\psi_{c1}(t_{\pi})\rangle$

$\displaystyle=$

$\displaystyle\frac{1}{2}(i|\mathscr{R}_{1}\rangle+i|\mathscr{R}_{2}\rangle+|\mathscr{R}_{3}\rangle-|\mathscr{R}_{4}\rangle)$

(12)

at $t=t_{\pi}$ because $(t_{\pi}\Omega_{\text{c}},~{}t_{\pi}\overline{\Omega})=(\pi,~{}2\pi)$. In Eq. (12), the eigenvectors $|\mathscr{R}_{j}\rangle$, $j=1-4$, are defined in Eq. (10) with Rabi frequencies $(\Omega_{\text{c}},~{}\Omega_{\text{t}})$ during $t\in[0,~{}t_{\pi})$. During $t\in t_{\pi}+\epsilon+[0,~{}t_{\pi})$, the Hamiltonian has Rabi frequencies $(\Omega_{\text{c}},~{}-\Omega_{\text{t}})$ with the eigenvectors $|\mathscr{R}_{1,2}\rangle$ the same as in Eq. (10), but the expressions for $|\mathscr{R}_{3,4}\rangle$ become

$\displaystyle|\mathscr{R}_{3,4}^{\prime}\rangle$

$\displaystyle=$

$\displaystyle\frac{1}{2\overline{\Omega}}[\Omega_{\text{c}}(|r1\rangle+|r0\rangle)\pm\overline{\Omega}(|11\rangle+|10\rangle)-2\Omega_{\text{t}}|1r\rangle].$

Using Eqs. (10) and (LABEL:eq08) we cast Eq. (12) into

$\displaystyle|\psi_{c1}(t_{\pi})\rangle=e^{-it_{\pi}\mathcal{H}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})}|\psi_{c1}(0)\rangle,$

(14)

i.e., the state at $t=t_{\pi}$ is like if Rabi frequencies $(\Omega_{\text{c}},~{}-\Omega_{\text{t}})$ were used during the first pulse. Equation (14) is the key feature enabling a fast CNOT. After the second pulse, the state at $t=2t_{\pi}+\epsilon$ becomes $e^{-it_{\pi}\mathcal{H}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})}|\psi_{c1}(t_{\pi})\rangle$ that can be rewritten as $e^{-2it_{\pi}\mathcal{H}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})}|\psi_{c1}(0)\rangle$ thanks to Eq. (14), which further reduces to $|11\rangle$ according to Eqs. (9) and (11). A similar analysis shows that if the initial state is $|11\rangle$, it maps to $|10\rangle$ upon the completion of the pulse. So the following map is realized

$\displaystyle\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}\rightarrow\{|00\rangle,|01\rangle,|11\rangle,|10\rangle\}$

(15)

which is the standard CNOT. A numerical simulation of the population evolution by $\mathcal{H}_{c0}$ and $\mathcal{H}_{c1}$ in Eq. (8) is shown in Figs. 2(c) and 2(d) for the input states $|00\rangle$ and $|10\rangle$, respectively, where the transient time $\epsilon$ is ignored for brevity.

High fidelity is possible with the TSD-based CNOT. To estimate the intrinsic fidelity limited by Rydberg-state decay and Doppler dephasing, we consider atomic levels and Rydberg laser Rabi frequencies in recent experiments Graham et al. (2019); Levine et al. (2019), and numerically found that the fidelity of the TSD-based CNOT or Bell state would reach $99.9\%$ ($99.8\%$) with experimentally affordable effective temperatures $T_{a}=5~{}(15)~{}\mu$K of qubit motion Picken et al. (2019); Zeng et al. (2017); Graham et al. (2019). As detailed in Appendix A.2, this estimate assumes that the field for $|0\rangle\rightarrow|r\rangle$ and that for $|1\rangle\rightarrow|r\rangle$ in the target qubit copropagate, leading to opposite phases in the two transitions of the chain $|0\rangle\xrightarrow{\Omega_{\text{t}}e^{itkv_{t}}}|r\rangle\xrightarrow{\Omega_{\text{t}}e^{-itkv_{t}}}|1\rangle$, where $k$ is the wavevector and $v_{t}$ the speed of the target qubit along the propagation direction of light. Then, the state transfer from $|0\rangle$ to $|1\rangle$ (or the reverse) picks up two opposite phases $\pm tkv_{t}$ which partly suppresses the dephasing. Fortunately, our method is not strictly dependent on the above dephasing-resilient configuration. For example, for the worst case of counterpropagating fields so that the sequential state transfer from $|0\rangle$ to $|1\rangle$ picks up a total phase $2tkv_{t}$, simulation in Appendix A shows that fidelity over $99.6\%$ is achievable for $T_{a}\sim 15~{}\mu$K in room temperatures. The robustness against the Doppler dephasing benefits from the avoidance of shelving the control atom on the Rydberg level in free flight during pumping the target atom as required in an EA-based CNOT Müller et al. (2009); Shi (2018b).

The above analysis assumes $V/2\pi=500$ MHz corresponding to an interatomic distance $l=3~{}\mu$m. When $V/2\pi=100$ MHz with larger $l=4.6~{}\mu$m (where crosstalk can be negligible Graham et al. (2019)), the fidelity would be $99.8\%$ ($99.7\%$) with $T_{a}=5~{}(15)~{}\mu$K in room temperatures (see Appendix B for details). This benefits from that our CNOT is not sensitive to the change of interaction $V$ in the strong blockade regime. Finally, with fluctuation (of relative Gaussian width $\sigma$) of the two Rabi frequencies for the two transitions $|0(1)\rangle\leftrightarrow|r\rangle$ in the target qubit, very small extra error $\in(0.02,~{}0.4)\%$ arises for $\sigma\in(1,~{}5)\%$ as detailed in Appendix C. So, high fidelity is achievable since the power fluctuation of Rydberg lasers can be well suppressed Levine et al. (2018, 2019).

The CNOT in Eq. (15) is implemented within a short time $2\pi/\Omega_{\text{c}}=\sqrt{6}\pi/\Omega_{\text{t}}$ plus a transient moment $\epsilon$ for a phase change between the two pulses, where $\epsilon$ can be negligible as in Levine et al. (2019). Although a phase twist is used in both Levine et al. (2019) and here, our TSD method is drastically different in physics from the method of Levine et al. (2019) as discussed in Appendix D. Note that Eq. (15) does not depend on any single-qubit gates if one starts from Eq. (8). If $s$ or $d$-orbital Rydberg states are used, two-photon transitions can be applied which leads to ac Stark shifts for the ground and Rydberg states. As shown in Appendix E, these shifts are annulled in Eq. (8) which is achievable by choosing appropriate ratio between the detuning at the intermediate state and the magnitudes of the fields for the lower and upper transitions Maller et al. (2015).

Our CNOT gate duration can be $\lesssim 0.3~{}\mu$s with Rydberg Rabi frequencies ($2\pi\times(3.5\sim 4.6)$ MHz) like those realized in Graham et al. (2019); Levine et al. (2019) that showed CNOT gate durations orders of magnitude longer than here. Previous CNOT gates Zhang et al. (2010); Maller et al. (2015); Zeng et al. (2017); Graham et al. (2019); Levine et al. (2019); Isenhower et al. (2010) required two or more single-qubit rotations to convert $C_{Z}$ to CNOT, which was usually achieved with two-frequency Raman light Isenhower et al. (2010); Zhang et al. (2010); Zeng et al. (2017); Levine et al. (2019), or with microwave driving assisted by Stark shift of laser fields Graham et al. (2019); Maller et al. (2015). High-fidelity realizations of such gates came with long gate durations Xia et al. (2015); Wang et al. (2016); Graham et al. (2019); Levine et al. (2019), although they can also be carried out rapidly in proof-of-principle experiments Yavuz et al. (2006); Jones et al. (2007) where high fidelity is not concerned. The atom traps have finite lifetimes Saffman (2016) and a faster protocol can lead to more CNOT cycles before the qubit arrays should be reloaded. Because a practical quantum processor tackling real problems uses a series of quantum gates including CNOT Nielsen and Chuang (2000); Williams (2011); Ladd et al. (2010); Shor (1997); Bremner et al. (2002); Shende et al. (2004); Peruzzo et al. (2014); Debnath et al. (2016), certain computation tasks can be executed only with fast enough CNOT in each cycle of laser cooling and loading of atom array. The studied speedup of the minimal functional quantum circuit shows a possibility to confront the problems of finite lifetime of atom traps which limits the capability of a neutral-atom quantum processor. Moreover, concerning the ratio between the coherence time and the CNOT gate time, the TSD method makes neutral-atom systems competitive with superconducting and ion-trap systems as compared in detail in Appendix F.

In conclusion, we have explored another effect of Rydberg blockade by dipole-dipole interaction in neutral atoms, namely, the effect of transition slow-down (TSD). We show that TSD can speed up a neutral-atom quantum computer by proposing an exceedingly fast TSD-based CNOT realized by two pulses separated by a short transient moment for changing the phase of the pulse. The exotic TSD is an excellent character of dipole-dipole interactions other than the well-known excitation annihilation effect that can push Rydberg-atom quantum science to another level.

The author thanks Yan Lu for useful discussions. This work is supported by the National Natural Science Foundation of China under Grant No. 11805146 and Natural Science Basic Research plan in Shaanxi Province of China under Grant No. 2020JM-189.