Quantum gates with weak van der Waals interactions of neutral Rydberg atoms

Neutral atoms can be rapidly entangled when they are excited to high-lying Rydberg states, which renders the possibility to use neutral atoms for large-scale quantum computing Saffman et al. (2010); Saffman (2016); Weiss and Saffman (2017); Adams et al. (2020); Wu et al. (2021); Morgado and Whitlock (2021). There have been several experiments demonstrating entanglement between individual neutral atoms by Rydberg interactions 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); Jo et al. (2020); Madjarov et al. (2020) primarily via the blockade mechanism Jaksch et al. (2000) where two nearby Rydberg atoms should possess a strong interaction $V$. The value of $V$ drops quickly when the qubit spacing increases, so it is necessary to place the qubits close enough for the blockade condition to hold. For example, the interatomic distance was in the range $[3.6,~{}5.7]~{}\mu$m in recent experiments of high-fidelity neutral-atom entanglement Levine et al. (2018, 2019); Graham et al. (2019); Madjarov et al. (2020). On the other hand, the yet to be large-scale quantum processor is supposed to host a large number of qubits in which entanglement operations between distant qubits are required in a general computational task. To tackle this issue, Ref. Weimer et al. (2012) suggested a method to entangle two qubits separated by a chain of ancillary qubits by adiabatically following the many-body ground state of the qubit chain coupled via the Rydberg blockade mechanism, and Ref. Cesa and Martin (2017) proposed to entangle distant qubits by coupling two targeted qubits with a group of ancillary atoms via Rydberg interactions. These methods depend on strong interactions of nearby Rydberg atoms. For example, Ref. Weimer et al. (2012) analyzed a model by assuming a large enough $V$ enabled by a small average interatomic spacing $1~{}\mu$m, so that the entanglement of two logic qubits separated by $20~{}\mu$m would require a coherent control over around 19 ancillary qubits.

In this work, we analyze an entangling gate between two well-separated neutral atoms by partially exciting them to Rydberg states. In contrast to the gate protocols dependent on the blockade mechanism, we find that a tiny $V$ can rapidly generate entanglement between ground hyperfine levels via detuned ground-Rydberg Rabi oscillations, and a controlled phase gate with an arbitrary phase $\theta$ can be created accurately. The intrinsic fidelity of the gate is perfect when $V$ is frozen and when the Rydberg-state decay is ignored, but for realistic setups where there will be fluctuations of qubit positions Graham et al. (2019) and Rydberg-state decay, the gate can still have a fidelity $0.99$ with qubit spacing over $20~{}\mu$m. Compared to the Rydberg gate by a dynamical phase shift $Vt/\hbar$ from the Rydberg interactions Jaksch et al. (2000), our gate does not require the Rydberg Rabi frequency $\Omega$ to be much larger than $V/\hbar$ and, thus, a high-fidelity implementation is less technically demanding, where $h(\hbar)$ is the (reduced) Planck constant. Compared to the Rydberg gates in Shi (2017) that work with similar condition $V/\hbar\sim\Omega$ but need five pulses to couple multiple Rydberg states, the gate in this work needs fewer pulses to couple only one Rydberg state. In contrast to previous Rydberg gates based on Förster resonances Shi (2017); Beterov et al. (2016); Petrosyan et al. (2017); Beterov et al. (2018a, b); Khazali and Molmer (2020); Beterov et al. (2020); Stojanovic (2021), our gate depends on van der Waals interactions that naturally appear in well separated atoms without resorting to external fields for tuning a Förster resonance. Since entangling operations between well-separated qubits are necessary for large-scale quantum computing, our method can simplify the quantum circuit in quantum computing because otherwise many standard Rydberg gates are needed to entangle two distant qubits.

The remainder of this article is organized as follows. In Sec. II, we study the controlled-phase gate with an arbitrary phase, and then extend it to a CNOT gate. In Sec. III, we study the achievable gate fidelity with realistic parameters and fluctuation of qubit positions. Section IV gives a comparison between the gate protocols here and previous Rydberg gates with distant qubits. A brief summary is given in Sec. V.

The gates can be fast. The total gate duration is $t_{\text{g}}=2\pi(1/\Omega_{\text{c}}+2/\overline{\Omega_{\text{t}}})$ according to the pulse sequence in Fig. 1 or Fig. 2. For a C_Z or CNOT gate where $\theta=\pi$, we should have the condition

so that the total gate duration is $t_{\text{g}}=2\pi(1/\Omega_{\text{c}}+\sqrt{3}/\Omega_{\text{t}})$. Rydberg Rabi frequencies around $2\pi\times 4.6$ MHz were used for entanglement of individual neutral atoms in Ref. Graham et al. (2019), so the C_Z or CNOT gates can have a short duration $t_{\text{g}}=0.59~{}\mu$s if similar Rabi frequencies are employed.

A high fidelity (about $99\%$) can be realized with our gates. Because the gate sequences for the C_Z and the CNOT gates are similar, we consider the C_Z gate as an example. In order to avoid slow gate speed, a Rydberg interaction around or over $h\times 0.5$ MHz is desirable. Then, higher Rydberg states are preferred so that the Rydberg interaction can still be around $h\times 0.5$ MHz even if the qubits are separated by tens of $\mu$m. Among the previous demonstrations of neutral atom quantum gates, the highest $d$-orbital Rydberg state of ⁸⁷Rb atoms ever used had a principal quantum number $n=97$ Isenhower et al. (2010); Zhang et al. (2010). The excitation of $s$-orbital Rydberg states with $n=102$ was demonstrated in an ensemble of ⁸⁷Rb atoms where coherent many-body phenomena were observed Dudin and Kuzmich (2012). To avoid the anisotropic interaction of $d$-orbital Rydberg states, we consider $s$-orbital Rydberg states whose interaction is highly isotropic in real space Walker and Saffman (2008). For these reasons, we analyze the gate performance with $|r\rangle\equiv|97S_{1/2},m_{J}=1/2\rangle$ and Rydberg Rabi frequencies

so that the duration will be $3.4~{}\mu$s for a C_Z gate. We note that a similar Rydberg Rabi frequency $2\pi\times 0.81$ MHz was used in Zhang et al. (2010) for entanglement of individual ⁸⁷Rb atoms via the excitation of the $97d$ states. The Rydberg interaction of $|rr\rangle$ for well separated qubits is characterized by $C_{6}/\mathcal{L}^{6}$ Walker and Saffman (2008); Saffman et al. (2010), where $C_{6}=h\times 39.5$ THz$\mu$m⁶ Shi et al. (2014) is the van der Waals coefficient and $\mathcal{L}$ is the qubit spacing. To analyze the performance of a C_Z gate where $\theta=\pi$ in Eq. (1), the condition in Eq. (5) requires that the distance between the centers of the traps for the two qubits should be $20.99~{}\mu$m with Eq. (6) [which leads to $V\approx h\times 0.46$ MHz according to Eq. (5)].

There are intrinsic errors from the Rydberg-state decay and the position fluctuations of the atomic qubits when the traps are turned off during the gate sequence (there is also an error due to Doppler dephasing but it is not an intrinsic issue since it is removable by the methods in, e.g., Refs Ryabtsev et al. (2011); Shi (2020b)). The former can be estimated as $E_{\text{decay}}=T_{\text{Ryd}}/\tau$, where $\tau$ is the lifetime of the Rydberg state and $T_{\text{Ryd}}$ is the total duration for the input state to stay in the Rydberg state averaged over the four input eigenstates,

where $|\psi(t)\rangle$ is the wavefunction calculated by unitary time evolution from the initial state $|\psi(0)\rangle$. Numerical simulation shows $T_{\text{Ryd}}\approx 1.52\times 2\pi/\Omega_{\text{c}}$ in general and $T_{\text{Ryd}}\approx 1.91~{}\mu$s with Eq. (6), and $\tau$ is $0.311$ ms (or 1.10 ms) at room temperature (or at $4.2$ K) Beterov et al. (2009), leading to $E_{\text{decay}}=6.14\times 10^{-3}$ (or $1.74\times 10^{-3})$.

The fluctuation of the qubit positions contributes the major error because our gate works perfectly only when the interaction is equal to the desired value. There will be random position fluctuation of the qubits before the traps are switched off, and the atoms fly freely during the gate sequence. This leads to fluctuation of the qubit spacing which results in fluctuation of $V$. A numerical investigation of this matter requires information for realistic r.m.s. qubit-position fluctuations in reference to the trap center. Though most publications on entanglement experiments did not show such details, Ref. Graham et al. (2019) showed that the longitudinal and transverse r.m.s. position fluctuations of the atoms in their traps were $\sigma_{z0}=1.47~{}\mu$m and $\sigma_{\perp 0}=0.27~{}\mu$m, respectively. In order to incorporate the effect of the free flight of the atoms during the gate sequence, we would like to modify the values of r.m.s. fluctuations. The change of the atomic locations during the gate sequence will change the interatomic distance. For an atomic temperature around $T_{a}=10~{}\mu$K (e.g., qubits with $T_{a}=5,~{}10,~{}15~{}\mu$K were studied in the entanglement experiments of Refs. Picken et al. (2019), Levine et al. (2018), and Graham et al. (2019), respectively), the r.m.s. distance for a qubit to fly during the gate sequence is about $\ell=v_{\text{rms}}t_{\text{g}}$, where $v_{\text{rms}}=\sqrt{k_{B}T_{a}/m}$ is the r.m.s. speed of the atom along one of the three directions in the 3D space, $T_{a}$ is the effective temperature of the atom, and $k_{B}$ and $m$ are the Boltzmann constant and the atomic mass, respectively. For a free flight of qubits, the average change of distance will be $v_{\text{rms}}t_{\text{g}}/2$ for a duration $t_{\text{g}}$. So, we analyze the fluctuation of the qubit positions by increasing the r.m.s. fluctuations to

which leads to $(\sigma_{z},\sigma_{\perp})=(1.52,~{}0.32)~{}\mu$m in the condition of Eq. (6).

The schematic of position fluctuation is shown in Fig. 3(a). We suppose that the quantization axis is along $\mathbf{x}$, and $\pi$ polarized Rydberg laser lights travel along $\mathbf{z}$. The Rydberg excitation scheme is shown in Fig. 3(b). The centers of the traps for the control and target qubits are $(0,0,0)$ and $(L,0,0)$, respectively. Because of the finiteness of the trap depths, the positions of the control and target qubits are $(x_{\text{c}},y_{\text{c}},z_{\text{c}})$ and $(L+x_{\text{t}},y_{\text{t}},z_{\text{t}})$, respectively, where the distribution function

characterizes $\zeta\in\{x_{\text{c}},y_{\text{c}},x_{\text{t}},y_{\text{t}}\}$, and

is the distribution function for $\zeta=z_{\text{c}}$ or $z_{\text{t}}$. In the numerical simulation, the value of $V$ is calculated by using the actual distance of the qubits $\mathcal{L}=$$\sqrt{(x_{\text{c}}-x_{\text{t}}-L)^{2}+(y_{\text{c}}-t_{\text{t}})^{2}+(z_{\text{c}}-z_{\text{t}})^{2}}$. The average fidelity is sampled with

where the fidelity $\mathcal{F}$ is defined by Pedersen et al. (2007)

Here, $\mathscr{U}$ is the actual gate matrix evaluated by using the unitary dynamics with the Rydberg-state decay ignored, and $U$=diag$\{1,1,1,-1\}$ is the ideal gate matrix in the basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$. The integral for each $\zeta$ is approximated by taking discrete values; for example, for $x_{\text{c}}$, the sampling is approximated by $\sum\mathscr{G}(x_{\text{c}})\mathcal{F}/\sum\mathscr{G}(x_{\text{c}})$ with $x_{\text{c}}\in\{-1.5,-1.5+\delta,-1.5+2\delta,~{}\cdots,~{}1.5-2\delta,1.5-\delta,1.5\}\sigma_{\perp}$, and the total gate fidelity is sampled by similar approximations for $\{y_{\text{c}},z_{\text{c}},x_{\text{t}},y_{\text{t}},z_{\text{t}}\}$. The numerical simulation is resource-demanding, but we can obtain an estimate about the convergence of the integration via slowly varying the steps. For $\delta$ equal to $0.25,0.2,0.15,0.12$, and $0.1$, the sampled average fidelities are $0.9910,0.9912,0.9914,0.9920$, and $0.9920$ respectively, from which we estimate $\overline{\mathcal{F}}=0.992$. With Rydberg-state decay included, the intrinsic fidelity would be $\overline{\mathcal{F}}-E_{\text{decay}}=0.986$ (or $0.990$) at $300$ K (or 4.2 K).

The gates in this work have several advantages compared to other gates Jaksch et al. (2000); Shi (2017, 2019); Beterov et al. (2016) that can also work with weak interactions in distant Rydberg atoms.

The first advantage is that our gate does not have rotation errors in the weak interaction regime. It was proposed in Ref. Jaksch et al. (2000) that phase gates can be created via the dynamical phase shift of the van der Waals interaction between two weakly interacting Rydberg atoms as experimentally demonstrated in Ref. Jo et al. (2020). The method of Jaksch et al. (2000) involves the excitation $|11\rangle\rightarrow|rr\rangle$ which has an intrinsic error $2V^{2}/(\hbar^{2}\Omega^{2})$ due to the failure for exciting the state $|rr\rangle$ from $|11\rangle$ as analyzed in Ref. Saffman and Walker (2005). In contrast, the intrinsic accuracy of the gate here is limited only by the Rydberg-state decay in the weak interaction regime.

The second advantage is that our gate depends on van der Waals interaction that naturally appears between two distant Rydberg atoms. As has been experimentally tested Jo et al. (2020), such a strategy allows an easy access to an experimental demonstration. In comparison, by using Förster resonance one can also design entangling gates between well-separated qubits Beterov et al. (2016, 2020, 2018a, 2018b) but fine tuning via external fields is required Safinya et al. (1981); Anderson et al. (1998); Mourachko et al. (1998); Westermann S. et al. (2006); Nipper et al. (2012); Richards and Jones (2016); Kondo et al. (2016); Ryabtsev et al. (2010); Tretyakov et al. (2014); Faoro et al. (2015); Tretyakov et al. (2017); Cheinet et al. (2020). However, it will be useful to investigate, if possible, an extension of our method to the regime of Förster resonance. If it is indeed possible, high-fidelity entanglement should be reachable with qubits separated even longer than the estimate in this work. This is because the Förster resonant interaction falls off as $1/\mathcal{L}^{3}$, while the van der Waals interaction falls off as $1/\mathcal{L}^{6}$ when $\mathcal{L}$ increases.

The third advantage is that our method can more easily realize a C_Z gate compared to previous high-fidelity gates based on weak van der Waals interactions Shi (2017, 2019). Compared to Ref. Shi (2017) which uses five pulses for coupling two different Rydberg states, the gate here needs only three steps to couple one Rydberg state. Compared to Shi (2019) which proposes a quantum entangling gate that should be repeated several times to form a C_Z gate when assisted by single-qubit gates, our method can realize a controlled-phase gate (such as C_Z) with any desired phase by only three laser pulses as shown in Fig. 1.

We study a controlled-phase gate based on weak van der Waals interactions between neutral Rydberg atoms. The gate is realized via phase accumulations in detuned Rabi oscillations enabled by weak Rydberg interactions. The gate is easily realizable and can have a high fidelity because it only needs three steps to couple one type of Rydberg state, has no intrinsic rotation errors in the weak interaction regime, and does not depend on fine tuning of Förster resonant processes. We use practical parameters to show that it is possible to create a C_Z gate in two qubits separated by $21~{}\mu$m with a fidelity about $0.990~{}($or $0.986)$ at 4.2 K (or at room temperature). We also show that a similar pulse sequence can lead to a CNOT gate which is realizable with similarly small van der Waals interactions. The theory brings hope to simplify the quantum circuit for large-scale quantum computing in which entanglement between distant qubits is required.

This work is supported by the National Natural Science Foundation of China under Grants No. 12074300 and No. 11805146, the Natural Science Basic Research plan in Shaanxi Province of China under Grant No. 2020JM-189, and the Fundamental Research Funds for the Central Universities.