Quantum logic and entanglement by neutral Rydberg atoms: methods and fidelity

Most methods of Rydberg gates use hyperfine-Zeeman ground substates in individual atoms to encode information. Experimental two-qubit entanglement by Rydberg interactions adopted this method with ⁸⁷Rb [53, 101, 102, 109, 113], a combination of ⁸⁷Rb and ⁸⁵Rb [112], or ¹¹³Cs [110, 111, 114, 116]. In these cases, the two qubit states |0⟩ and |1⟩ are chosen from two hyperfine-Zeeman substates of cesium or rubidium, where for ¹¹³Cs the energy gap between the two qubit states is h × 9.2 GHz, and for ⁸⁷Rb it is h × 6.8 GHz, where h is the Planck constant. A schematic is shown in figure 2(a). Besides of heavy alkali-metal atoms, this method is applicable for ¹⁶⁵Ho which has a ground electronic configuration 4f¹¹6s² and eight hyperfine-split levels with hyperfine numbers in the range F ∈ [4, 11]. These hyperfine levels have distinct nearest splittings ∈h × [4.3, 8.3] GHz [156], which means that it is possible to choose qubit settings with much flexibility in the case of ¹⁶⁵Ho.

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The hyperfine-split states not only can allow quantum information encoding with single atoms, but also can be used for qubit encoding in ensembles of single atoms [100] as experimentally demonstrated in references [157, 158]. An example of the ensemble qubit defined by the states of three atoms is shown in figure 2(b). The use of atomic ensemble can also allow a novel method to define a qubit: to load random numbers of atoms in the sites of an atomic array, i.e. as long as there is at least one atom at each site, we will be able to encode quantum information by using each site as a qubit. It is technically achievable but not an easy task to deterministically load a certain number of atoms in one trap, so that loading a random number of atoms in one trap is easier. The difficulty appeared in this method is that because the Rabi frequency for a ground-Rydberg state rotation with K atoms is enhanced by a factor of $\sqrt{K}$ [159, 160], so that the standard Rydberg excitation with a pulse of a certain pulse area can not work when K fluctuates from site to site. Fortunately, it was shown that by using time-dependent or adiabatic pulses one can achieve high-fidelity Rydberg logic gates with the ensemble encoding when K fluctuates from site to site [122, 125, 136].

The third method is to use single-ensemble collective states as qubit states when the atomic ground state of each atom in the ensemble has a multilevel structure [155]. Consider an ensemble of K identical atoms in which each atom has N + 1 stable levels where N ≪ K, where the first state is labeled as |0⟩_P and the remaining N states are labeled as {|1⟩_P, |2⟩_P, ..., |N⟩_P}. When all the atoms are in the state |0⟩_P, the N-qubit state is |000...0⟩_L, where P and L denote the physical and logical states, respectively. When there is a collective 'excitation' in the |1⟩_P states among the K atoms, we have

$$\begin{equation}\vert 100\dots {0\rangle }_{\text{L}}\equiv \frac{1}{\sqrt{K}}\sum\limits _{j=1}^{K}\vert 0\dots {1}_{j}\dots {0\rangle }_{\text{P}},\end{equation} \tag{ 1 }$$

where the number of digits in the ket on the left (right) hand side is N (K). Similarly, when there is a collective 'excitation' in the |2⟩_P states among the K atoms, we have

$$\begin{equation}\vert 010\dots {0\rangle }_{\text{L}}\equiv \frac{1}{\sqrt{K}}\sum\limits _{j=1}^{K}\vert 0\dots {2}_{j}\dots {0\rangle }_{\text{P}},\end{equation} \tag{ 2 }$$

and so on for |001...0⟩_L and other similar states. This method needs K ≫ N so as to employ the Rydberg blockade mechanism with less inhomogeneity, but a schematic with K = N + 1 = 3 is shown in figure 2(c) for brevity. As shown in reference [155], by using the Rydberg blockade mechanism, one can start from equation (1) to prepare the N-qubit logical state

$$\begin{equation}\begin{aligned}\hfill \vert 110\dots {0\rangle }_{\text{L}}& \equiv \frac{1}{\sqrt{K(K-1)}}\sum\limits _{m\ne j,m=1}^{K}\sum\limits _{j=1}^{K}\vert 0\dots {1}_{j}\dots {2}_{m}\dots {0\rangle }_{\text{P}},\hfill \\ \hfill \vert 101\dots {0\rangle }_{\text{L}}& \equiv \frac{1}{\sqrt{K(K-1)}}\sum\limits _{m\ne j,m=1}^{K}\sum\limits _{j=1}^{K}\vert 0\dots {1}_{j}\dots {3}_{m}\dots {0\rangle }_{\text{P}},\hfill \end{aligned}\end{equation} \tag{ 3 }$$

and so on. The distinction between this method from those demonstrated in references [157, 158] lies in that one ensemble is enough to encode an N-bit binary here, while the multi-ensemble approach shown in figure 2(b) needs N ensembles.

Another single-ensemble approach was proposed in reference [156] by taking ¹⁶⁵Ho as an example. The eight hyperfine-split levels of ¹⁶⁵Ho have 128 hyperfine-Zeeman substates when an external magnetic field is applied. Excluding eight 'unpaired' states because the level structure of the 128 states is like an upside-down pyramid, we still have 120 states that can be used for encoding information. Labeling these sates as i ∈ {1, 2, ..., 120}, then 60 qubits can be defined, where the two qubit states in the nth qubit are defined by the (2n − 1)th and (2n)th hyperfine-Zeeman substates, respectively. The single and two-qubit gates in the single-ensemble encoding method was studied in reference [155].

For most alkaline-earth-like (AEL) atoms, there are both an electronic ground state and a long-lived electronically excited state. The distinction between alkali-metal atoms and AEL atoms is that the outermost shell in an atom of the former species has only one electron, while in the latter case two electrons are in the open shell. As shown in figure 2(d), the ground state and the long-lived p-orbital clock state of an AEL atom can define a qubit [161]. In the case of ¹⁷¹Yb, the lifetimes of the ³ P₀ and ³ P₂ states are over 20 and 10 s, respectively [162], and similar for other AEL atoms. This essentially makes it possible to define the ground state 6s²¹ S₀ and the metastable clock state 6s6p³ P₀₍₂₎ as the two qubit states for the case of ¹⁷¹Yb. Other well-studied divalent AEL atoms have similar stable clock states that can be used for defining a qubit.

There are other types of neutral-atom quantum registers except those discussed in sections 2.1.1–2.1.4.

First, one can define qubits with Rydberg states, either with two Rydberg states [163] or with one Rydberg and one stable states [164]. The lifetime of low-angular-momentum Rydberg states is only several 100 μs [165], which means that only fast enough quantum operations can make Rydberg-state qubits useful for QC. There have been theoretical entanglement protocols based on this type of qubit via low-angular-momentum Rydberg states [164] or circular Rydberg states [163]. Two experiments were put forward for demonstrating two-atom entanglement between Rydberg and ground (or optical clock) states of individual atoms [67, 113].

Second, nuclear spin states in AEL atoms can also define qubits. In most AEL atoms, the ground state has no hyperfine splitting. When the atom has a nonzero nuclear spin, states with different projections of the nuclear spin onto the quantization axis can be used as qubit states. For example, the two nuclear spin Zeeman levels ±1/2 of the ¹⁷¹Yb ground state can define a qubit. Among the AEL species without ground-state hyperfine splittings, ⁸⁷Sr has the largest nuclear spin quantum number 9/2. In this case, one can choose any two states of nearby spin projections m_I and m_I + 1 (m_I < 9/2) to define a qubit in the presence of external magnetic fields [105].

Third, one can use both the electronic and nuclear spins as register states [166]. Taking ¹⁷¹Yb as an example, the four states |6s²¹ S₀, m_I ± 1/2⟩ and |6s6p³ P₀, m_I ± 1/2⟩ are useful for storing quantum information since they are all long-lived states. If three of these states are selected, a qutrit can be defined; if four are chosen, then a qudit can be defined.

The categories in sections 2.1.1–2.1.5 highlight the versatility to encode quantum information with neutral atoms. There are several entanglement experiments [53, 101, 102, 109–112, 114, 116] by using hyperfine-Zeeman substates for encoding information. It is an open question whether there are experimentally friendly ways to carry out fast, robust, and accurate entanglement suitable for quantum information processing by the other methods of encoding.

An s-orbital ground state of an alkali-metal atom can absorb an ultraviolet (UV) photon and transit to a p-orbital Rydberg state, as demonstrated in [111, 167]. A schematic is shown in figure 3(e). For an AEL atom, the clock state can be excited to an s-orbital Rydberg state via the absorption of an UV photon [67]. To study the one-photon schemes, we use |g⟩ and |r⟩ to denote the qubit and Rydberg states, respectively. By using the dipole approximation in the atom–light coupling and the rotating-wave approximation, the Hamiltonian becomes

$$\begin{equation}{\hat{H}}_{\text{one}-\text{pho}}=\hslash {\Omega}\vert r\rangle \langle g\vert /2+\text{H.c.}\end{equation} \tag{ 4 }$$

Here, in principle, there are nonresonant ac Stark shifts from the laser field that can be calculated by the sum over transitions from relevant states [110]. But by shifting the laser frequency with an appropriate value, the resonance is recovered and in an appropriate rotating frame we arrive at equation (4). Starting from the ground state |g⟩, the state evolves to −i|r⟩ at the moment t = t₁ when ${\int }_{0}^{{t}_{1}}{\Omega}\,\mathrm{d}t=\pi$, i.e. when a π pulse is used. Similarly, a π pulse can excite the atom from the Rydberg state |r⟩ to the ground state. Usually, we can assume quasi-rectangular pulses and constant Rabi frequency during the excitation. Then, a π pulse with duration t₁ = π/Ω can achieve a Rydberg excitation. Because the phase change of the wavefunction in the Rydberg excitation |g⟩ → |r⟩ or its deexcitation is π/2, one Rabi cycle |g⟩ → |r⟩ → |g⟩ results in a π phase change which is a crucial factor enabling the Rydberg blockade gate [99].

A one-photon Rydberg excitation of ¹³³Cs atoms was used in [111] for entanglement. There was also an entanglement experiment using one-photon excitation of the ³ S₁ Rydberg state from the long-lived ³ P₀ clock state of ⁸⁸Sr [67]. The high fidelity achieved in reference [67] partly benefited from the large Rydberg Rabi frequency which led to fast entanglement generation.

A two-photon Rydberg excitation can excite an s- or d-orbital Rydberg state |r⟩ from the ground state |g⟩. Form the s-orbital ground state, a two-photon Rydberg excitation can proceed via an intermediate p-orbital state |p⟩ which should be largely detuned to avoid dissipation from it. Most Rydberg-mediated entanglement experiments were based on two-photon excitations [53, 101, 102, 109, 110, 112–116]. With similar approximations as used in equation (4), the Hamiltonian for a two-photon Rydberg excitation is

$$\begin{equation}{\hat{H}}_{\text{two}-\text{pho;0}}=\hslash [{{\Omega}}_{1}\vert p\rangle \langle g\vert /2+{{\Omega}}_{2}\vert r\rangle \langle p\vert /2+\text{H.c.}]+\hslash {\Delta}\vert p\rangle \langle p\vert +\hslash {{\Delta}}_{r}^{(\text{ac})}\vert r\rangle \langle r\vert +\hslash {{\Delta}}_{g}^{(\text{ac})}\vert g\rangle \langle g\vert ,\end{equation} \tag{ 5 }$$

where Δ is the frequency mismatch between the laser fields and the atomic transition, and ${{\Delta}}_{r}^{(\text{ac})}$ and ${{\Delta}}_{g}^{(\text{ac})}$ are the Stark shifts at the Rydberg and ground states, respectively. According to reference [110], they are given by

$$\begin{equation}{{\Delta}}_{r}^{(\text{ac})}=-\frac{{e}^{2}}{4{m}_{e}\hslash }\left(\frac{{E}_{1}^{2}}{{\omega }_{1}^{2}}+\frac{{E}_{2}^{2}}{{\omega }_{2}^{2}}\right),\end{equation} \tag{ 6 }$$

$$\begin{equation}{{\Delta}}_{g}^{(\text{ac})}=-\frac{1}{4\hslash }\left({\alpha }_{1}{E}_{1}^{2}+{\alpha }_{2}{E}_{2}^{2}\right),\end{equation} \tag{ 7 }$$

where e is the elementary charge, m_e is the mass of the electron, and ω_j and E_j are the frequency and electric field amplitude of the laser field, where j = 1 and 2 for the lower and upper transitions, respectively, and α₁ and α₂ are the nonresonant polarizabilities that can be calculated via the sum over transitions to states other than the intermediate state |p⟩.

When Δ is large compared to the decay rate of |p⟩, the intermediate state can be adiabatically eliminated [172–175], leading to

$$\begin{equation}{\hat{H}}_{\text{two}\text{-}\text{pho}}=\hslash \left\{[{{\Omega}}_{\text{eff}}\vert r\rangle \langle g\vert /2+\text{H.c.}]+{{\Delta}}_{r}\vert r\rangle \langle r\vert +{{\Delta}}_{g}\vert g\rangle \langle g\vert \right\},\end{equation} \tag{ 8 }$$

where Ω_eff = −Ω₁Ω₂/(2Δ) and

$$\begin{equation}\begin{aligned}\hfill {{\Delta}}_{r}& =-\frac{{{\Omega}}_{2}^{2}}{4{\Delta}}+{{\Delta}}_{r}^{(\text{ac})},\hfill \\ \hfill {{\Delta}}_{g}& =-\frac{{{\Omega}}_{1}^{2}}{4{\Delta}}+{{\Delta}}_{g}^{(\text{ac})}.\hfill \end{aligned}\end{equation} \tag{ 9 }$$

Here, note that Ω₁₍₂₎ ∝ E₁₍₂₎, the effective detunings Δ_r and Δ_g are functions of Δ, E₁, and E₂ for any given set of atomic levels. As studied in reference [110], resonant conditions with Δ_g = Δ_r can be recovered, and it is also possible to recover the conditions so that the phase change to |g⟩ after a 2π rotation is equal to π. The establishment of the latter condition requires efforts because the comparable magnitudes of the resonant and the nonresonant ac Stark shifts can lead to any phase shift in the ground state upon the completion of a 2π rotation; more details can be found in [110].

It is possible to suppress Doppler dephasing in the ground-Rydberg transition in a two-photon Rydberg excitation. Two theories were introduced in reference [168], among which one is briefly shown in figure 3(e). The $2\mathcal{N}\pi$ pulse causes a transition between the Rydberg state |r₁⟩ (i.e. the $n{S}_{\frac{1}{2}}$ state in (b)) and a nearby Rydberg state |r₂⟩ via a low-lying intermediate state |e₂⟩. The choices of the intermediate state |e₁⟩ during the Rydberg excitation shall yield a wavevector k that is more than 2 times smaller than the wavevector k_w of the transition |r₁⟩ ↔ |r₂⟩. By the sequence shown in figure 3(e), the motional dephasing of the ground-Rydberg transition can be suppressed. The wavelengths shown in figure 3 are calculated by using the data in reference [171]. See reference [168] or a case study in reference [176] for more details about the suppression of the Doppler dephasing of the ground-Rydberg transition via the $\pi -2\mathcal{N}\pi -\pi$ method; we note, however, that there is another theory to suppress the motional dephasing in reference [168] which uses a strategy different from the one shown in figure 3(e).

Rydberg excitation can be realized with a three-photon excitation via two intermediate states. Like the two-photon method, the intermediate states in the three-photon transition shall be largely detuned to avoid dissipation from them. For the excitation of single atoms, the attraction of a three-photon transition lies in, as proposed in reference [169], that it can offer a complete suppression of the Doppler broadening due to the atomic motion; moreover, it can suppress the recoil effect, i.e. the momentum change of atoms during the Rydberg excitation, as shown in figures 3(c) and (f) where (θ₁, θ₂) = (1.37, 1.21) [169]. The configuration in figure 3(f) ensures that when the atomic state is excited by absorption of the three photons, the total momentum of the three photons is zero, leading to no momentum change in the atom. Likewise, there is nearly no position-dependent phase change to the atomic state since the three wavevectors of the three absorbed photons add up to exactly zero. Until now, there is no experimental demonstration of Rydberg excitation for entanglement generation via the three-photon excitation. But in an experimental study of collective Rydberg excitations in atomic gas, the three-photon method was used for the suppression of Doppler dephasing [177].

In reference [170], it was shown that a four-photon excitation chain via three low-lying intermediate states can suppress the Doppler dephasing, where the wavelengths for the four fields λ_j with j = 1–4 satisfy ${\sum }_{j}{(-1)}^{j}{\lambda }_{j}^{-1}\approx 0$, as shown in figures 3(d) and (g). This means that by sending the first and third lasers in a direction opposite to that for the second and fourth lasers, Doppler dephasing as well as photon recoil can be avoided. In this four-photon method, one does not need to focus the laser fields on one spot from three directions as in [169]. To suppress control error from the position fluctuation of the qubit, the control fields can be sent along the longitudinal axis of the optical dipole trap (if this type of trap is used). Experiments have not explored the four-photon method for Rydberg-mediated entanglement.

To simplify, we call an excitation with more than one photons exchanged between the atom and the field a multi-photon excitation. Both the one-photon and multi-photon methods have their strength.

First, high-power laser fields are required in the one-photon Rydberg excitation because the dipole matrix element between the ground and a high-lying Rydberg state is small, but for the multi-photon case a large Rydberg Rabi frequency can be achieved with fields of lower powers. So, a multi-photon excitation is more favorable when fast rotations are desirable if laser powers are limited.

Second, a multi-photon Rydberg excitation suffers from loss via the scattering from the intermediate states, which is absent in a one-photon Rydberg excitation. When ignoring the hyperfine structure of |p⟩ and when Ω₁ = Ω₂ for the two-photon case in equation (5), the decay probability of the state via |p⟩ is π/(2τ_p Δ) [116] in a two-photon Rydberg excitation with a π pulse, where τ_p is the lifetime of the state |p⟩. This essentially means that by using large detuning at |p⟩ the coherence can be preserved. For example, the scattering through the intermediate state in [116] was much less detrimental compared to the Doppler dephasing of the ground-Rydberg transition.

Third, the Doppler dephasing in the ground-Rydberg transition is not removable in one-photon Rydberg excitations, but can be suppressed in other schemes. For example, the wavevector in the one-photon Rydberg excitation in reference [167] is 2.5 times larger than that of typical two-photon excitations [116], one may expect a strong Doppler dephasing in the former case. As shown in figures 3(e)–(g), it is possible to suppress the Doppler dephasing in the two, three, and four-photon methods, respectively. Comparing the three schemes for suppressing the Doppler dephasing, the three-photon method in figure 3(f) requires arranging the three laser fields so that they focus on a specific spot in the three-dimensional space where the atom (or ensemble of atoms) is supposed to reside, while for the two and four-photon methods shown in figures 3(e) and (g) the fields should be nearly collinear.

Fourth, in a colder environment the p-orbital Rydberg states excited in the one-photon and three-photon methods can have longer lifetimes than the s or d-orbital Rydberg states prepared in the two-photon methods. The Rydberg-state decay is dominated by the blackbody radiation at room temperature, so that the lifetimes of Rydberg states of different angular momenta are similar. For example, the room-temperature lifetimes of an s, p, and d-orbital rubidium Rydberg states of a principal quantum number 100 are 0.33, 0.38, and 0.32 ms, respectively [165]. At lower temperatures, the Rydberg-state decay is dominated by the spontaneous emission which strongly depends on the angular states of the Rydberg atom. For example, the lifetimes are 1.2, 2.1, and 0.9 ms for an s, p, and d-orbital rubidium state with a principal quantum number around 100 at 4.2 K; at 77 K these lifetimes are 0.73, 0.99, and 0.63 ms, respectively. In this sense, there is an advantage to use the one and three-photon excitations of p-orbital Rydberg states so that Rydberg-state decay is less severe when the atomic qubits are hosted in a cryogenic chamber.

Fifth, as in the two-photon case, there can be Stark shifts for the ground and Rydberg states when the intermediate states are adiabatically eliminated. It is a delicate issue to tune to conditions in three and four-photon Rydberg excitations for the induction of π phase change in a 2π Rydberg pulse that is critical in the Rydberg blockade gate [99]. Because there are multiple terms in the ac Stark shifts in the two, three, and four-photon schemes, it is in principle possible to recover a π phase change in a 2π Rydberg pulse [110]. For one-photon Rydberg excitation the Hamiltonian shown in equation (4) can easily lead to a π phase change to the state with a 2π pulse.

From the above comparison, one can see that although most experiments on Rydberg atoms used two-photon excitations [53, 101, 102, 109, 110, 112–114, 116] and some used one-photon excitations [111, 167], three-photon excitation methods can suppress the recoil and Doppler dephasing as shown in [169], and can excite p-orbital Rydberg states that possess longer lifetimes in cryogenic chambers. The one-photon method can also excite p-orbital Rydberg states, but it is an open question whether there is any method to eliminate the prevailing motional dephasing so as to realize a highly coherent single-photon excitation of a p-orbital Rydberg state.

We have discussed one-photon to four-photon Rydberg excitations. In this sense only a low-orbital-angular-momentum Rydberg state is excited. An atomic Rydberg state with the maximal orbital angular momentum (equal to n − 1 with n the principal quantum number) is called a circular Rydberg state. The excitation of circular Rydberg states can be achieved by adiabatic passage [178] or by a fast scheme using microwave and radio-frequency field [179]. Because of the long lifetimes of circular Rydberg states, the latter method for exciting circular Rydberg states can preserve the quantum coherence of the system [180, 181] and may be useful for quantum information processing [182], but it is still based on first exciting a low-orbital-angular-momentum Rydberg state [183–185].

Rydberg interactions usually refer to interactions between the electric dipoles of two nearby atoms. There are also three-body Förster resonances among three nearby Rydberg atoms [186–188] and even four-body Förster resonances can exist [189], where the interaction can be calculated starting from the fundamental dipole–dipole interaction process. Below, we briefly review the basics for calculating the two-body interactions between alkali-metal Rydberg atoms when the distance between their nuclei is larger than the Le Roy radius [190, 191] so that their Rydberg electrons do not overlap.

The interaction between two Rydberg atoms arises from the electric dipole–dipole interaction while multipole interactions are negligible for most cases [192]. The van der Waals interaction denotes the dipole–dipole interaction in the second-order perturbation theory when two Rydberg atoms A and B are well separated. For a highly-excited atom, we use the principal quantum number n, the electron angular momentum quantum number l, the total angular momentum quantum number j, and the magnetic quantum number m to characterize its state. We assume that the quantization axis is along z. The dipole–dipole interaction between two atoms with quantum numbers (n_A, l_A, j_A, m_A) and (n_B, l_B, j_B, m_B) is an electrostatic interaction given by

$$\begin{equation}{\hat{V}}_{\text{dd}}={\hat{\mathbf{s}}}_{\text{A}}\cdot \left(3\hat{\mathbf{r}}\hat{\mathbf{r}}/{r}^{2}-\hat{\mathbf{I}}\right)\cdot {\hat{\mathbf{s}}}_{\text{B}}/(4\pi {{\epsilon}}_{0}{L}^{3}),\end{equation} \tag{ 10 }$$

where ${\hat{\mathbf{s}}}_{\text{A}(\text{B})}$ is the dipole moment of atom A(B), L is the distance between the two atoms, $\hat{\mathbf{I}}$ is the identity operator, and ₀ is the dielectric permittivity in free space. Using the representation of spherical harmonic rank 2 tensor, the matrix element

$$\begin{equation}{V}_{{m}_{\text{A}}{m}_{\text{B}};{m}_{a}{m}_{b}}=\langle {n}_{\text{A}}{l}_{\text{A}}{j}_{\text{A}}{m}_{\text{A}};{n}_{\text{B}}{l}_{\text{B}}{j}_{\text{B}}{m}_{\text{B}}\vert {\hat{V}}_{\text{dd}}\vert {n}_{a}{l}_{a}{j}_{a}{m}_{a};{n}_{b}{l}_{b}{j}_{b}{m}_{b}\rangle \end{equation} \tag{ 11 }$$

can be written as

$$\begin{equation}-\frac{\sqrt{6}}{4\pi {{\epsilon}}_{0}{L}^{3}}\langle {n}_{\text{A}}{l}_{\text{A}}{j}_{\text{A}}{\Vert}{D}_{\text{A}}^{[1]}{\Vert}{n}_{a}{l}_{a}{j}_{a}\rangle \langle {n}_{\text{B}}{l}_{\text{B}}{j}_{\text{B}}{\Vert}{D}_{\text{B}}^{[1]}{\Vert}{n}_{b}{l}_{b}{j}_{b}\rangle \sum\limits _{M=-2}^{2}{{\Psi}}_{-M}^{[2]}(\theta ,\phi )\sum\limits _{\alpha ,\beta =-1}^{1}{C}_{\alpha \beta M}^{112}{C}_{{m}_{a}\alpha {m}_{\text{A}}}^{{j}_{a}1{j}_{\text{A}}}{C}_{{m}_{b}\beta {m}_{\text{B}}}^{{j}_{b}1{j}_{\text{B}}},\end{equation} \tag{ 12 }$$

where ${D}_{k}^{[1]}$ is the dipole moment of atom k = A or B in terms of rank-1 tensor, C is a Clebsh–Gordan coefficient [193], Ψ [53] is a rank-2 tensor given by the standard spheric harmonics multiplied by a factor of $\sqrt{4\pi /5}$ [194], and (θ, ϕ) gives the angular position of atom B with respect to atom A as schematically shown in figure 4(a). Because all the second-order spheric harmonics, except Ψ₀, are proportional to sin θ, the dipole coupling will conserve the total projection of the magnetic angular momentum of the two atoms when θ = 0 (i.e. along the quantization axis). Here, the dipole matrix element $\langle {n}_{\text{A}}{l}_{\text{A}}{j}_{\text{A}}{\Vert}{D}_{\text{A}}^{[1]}{\Vert}{n}_{a}{l}_{a}{j}_{a}\rangle$ can be calculated using the Wigner–Eckart theorem [193].

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When the coupling strength between the two electric dipoles is much smaller than the energy difference between the dipole-coupled initial and final two-atom states, the two-atom state can barely be excited away from the degenerate manifold of (n_A, l_A, j_A; n_B, l_B, j_B) and (n_B, l_B, j_B; n_A, l_A, j_A). Even when the principal quantum numbers of the two atoms do not exchange, their magnetic angular momenta m_A and m_B can change. In this case, the van der Waals interaction is given by

$$\begin{equation}{\hat{H}}_{\text{vdW}}=-\sum\limits _{{n}_{a}{l}_{a}{j}_{a}}\sum\limits _{{n}_{b}{l}_{b}{j}_{b}}{\hat{V}}_{\text{dd}}{\hat{V}}_{\text{dd}}^{{\dagger}}/{\delta }_{ab}.\end{equation} \tag{ 13 }$$

Here the matrix elements of ${\hat{V}}_{\text{dd}}$ are given in equation (11), and the energy defect is [192]

$$\begin{equation}{\delta }_{ab}=E({n}_{a}{l}_{a}{j}_{a})+E({n}_{b}{l}_{b}{j}_{b})-[E({n}_{\text{A}}{l}_{\text{A}}{j}_{\text{A}})+E({n}_{\text{B}}{l}_{\text{B}}{j}_{\text{B}})],\end{equation} \tag{ 14 }$$

where E(nlj) is the atomic energy of a Rydberg atom with quantum numbers n, l, j. A remarkable example is that the state labeled by (n_A, l_A, j_A, m_A; n_B, l_B, j_B, m_B) can go to the state (n_A, l_A, j_A, m_B; n_B, l_B, j_B, m_A) where m_A and m_B exchange. This spin-exchange process is of special interest when the initial and the final two-atom states can be optically coupled to different two-atom ground states [164], giving rise to entanglement of neutral atoms and a quantum gate similar to the SWAP gate. One can also easily find that the state (n_A, l, j, m; n_B, l, j, m) and (n_B, l, j, m; n_A, l, j, m) are degenerate when n_A ≠ n_B, which means that two atoms in different Rydberg principal levels can exchange their states. This feature can be used to simulate magnetic fields that are useful to realize exotic many-body states [195].

There are many useful references about the calculation of interactions between two Rydberg atoms, including a detailed analysis on the consequences of Zeeman degeneracies [192], a study about two-atom Förster resonance [196], a classification of the interaction according to symmetry [197], and a detailed introduction about the method for the calculation as well as an open-source software [191]. Besides, reference [198] provides a comprehensive object-oriented Python library for calculating properties of highly-excited Rydberg states of alkali-metal atoms, including single-atom dipole matrix elements, Rydberg-state lifetimes, Stark maps of atoms in external electric fields, and two-atom interactions accounting for dipole and quadrupole couplings. An update of reference [198] was given in reference [199] which treats interactions of several species of AEL Rydberg atoms. Other features in reference [199] include the calculation of valence electron wave functions, dynamic polarizabilities, atom-surface interactions, and wavefunctions of atoms in optical traps.

As briefly mentioned in section 1.3, there are mainly three classes of Rydberg gates. These three different methods depend on different characters of the Rydberg interactions.

The Rydberg gates by the blockade mechanism need Rydberg interaction V to be much larger than the laser Rabi frequency Ω times the Planck constant for the excitation of Rydberg states [99]. The distance between the two atoms we want to entangle can be short so that V is large, but should be larger than the Le Roy radius within which the electronic wavefunctions of the two atoms overlap. This is easily satisfied in the experiments [67, 101, 102, 109–116] where qubits were separated beyond 3.6 μm for Rydberg states of principal quantum numbers below 100 [198]. Two-atom Rydberg states are barely populated in this class of Rydberg gates, so that these gates require a minimal condition on the Rydberg interactions: there shall not be accidental zero interactions due to Förster resonance or highly anisotropic interactions of, e.g. d-orbital states [192]; this is in principle not a demanding task since the Förster resonance is rare and it requires efforts to find a Förster resonant dipole–dipole process [200]. Moreover, there is in general no specifically required relation between the quantization axis (usually set by an external magnetic field) and the two-atom orientation for frequently employed Rydberg states.

For the methods to entangle neutral atoms via the antiblockade or via Rydberg excitation of two or more atoms, the requirement on the Rydberg interactions is either that (i) V shall represent one or several isolated dipole–dipole interactions or (ii) V is a van der Waals interaction in the form a pure energy shift. To achieve (i), usually external fields are required, as studied in references [201–208]. Here, we note that a pure energy shift is required in the basic method in regimes ② and ③ of figure 1(c), but for the extensions of the basic methods which will be reviewed later on in sections 4 and 5, isolated dipole–dipole interactions were usually used. Below, we detail how to achieve (ii).

The appearance of a pure energy shift depends on two conditions. First, the two-atom separation should be large so that the interaction can be analyzed by a second-order perturbation theory. Second, the choice of principal quantum numbers and angular momenta in |r⟩ shall lead to no state-flip van der Waals interaction. For certain conditions the two-atom separation axis shall be parallel to the quantization axis so that there is no state-flip van der Waals interaction. The first condition can be understood with section 1.3. To explain the other condition in detail, we take the state ⁸⁷Rb |nS_1/2, m_J = ±1/2⟩ ≡ |r_±⟩ as an example. The two-atom state |r₊; r₊⟩ is coupled with

$$\begin{equation}\begin{aligned}\hfill & \vert {n}_{1}{P}_{1/2},{m}_{J}=\pm 1/2;{n}_{2}{P}_{1/2},{m}_{J}=\pm 1(3)/2\rangle ,\enspace \hfill \\ \hfill & \vert {n}_{3}{P}_{1/2},{m}_{J}=\pm 1/2;{n}_{4}{P}_{3/2},{m}_{J}=\pm 1(3)/2\rangle ,\enspace \hfill \\ \hfill & \vert {n}_{5}{P}_{3/2},{m}_{J}=\pm 1(3)/2;{n}_{6}{P}_{1/2},{m}_{J}=\pm 1/2\rangle ,\enspace \hfill \\ \hfill & \vert {n}_{7}{P}_{3/2},{m}_{J}=\pm 1(3)/2;{n}_{8}{P}_{3/2},{m}_{J}=\pm 1(3)/2\rangle ,\hfill \end{aligned}\end{equation} \tag{ 15 }$$

where n_j with j = 1–8 is a principal quantum number near n. Here, a semicolon is used between the symbols for the states of the two atoms in the ket so as to indicate that the ket represents a two-atom state. The above equation shows 40 states coupled with |r₊; r₊⟩ for each set of {n₁, n₂, ..., n₈}, and there are many sets of n_j . Since the Coulomb interaction is strong, the energy separation between states of different n is large. When the energy gap between |r₊; r₊⟩ and another two-atom state in equation (15) is too large compared to the dipole–dipole coupling, the coupling barely takes effect and it can be ignored. Then, it is usually sufficient to restrict |n_j − n| ⩽ 6 in the evaluation of Rydberg interactions [164, 198], i.e. there will be 13 values for n_j . So, there are 40 × 13² two-atom Rydberg states coupled with |r₊; r₊⟩. Some states in equation (15) are mainly coupled with |r₊; r₊⟩, but some states in equation (15) can couple both |r₊; r₊⟩ and |r₋; r₋⟩. Likewise, when we consider the interaction involving the state |r₊ r₋⟩, the two states |r₊ r₋⟩ and |r₋ r₊⟩ are degenerate and can be coupled. Because of these couplings, it demands efforts to realize a Rydberg interaction in the form of a pure energy shift. We show several examples to clarify this. For n = 100 and with basis states {|r₊; r₊⟩, |r₊; r₋⟩, |r₋; r₊⟩, |r₋; r₋⟩}, the second-order perturbation leads to the following van der Waals interaction

$$\begin{equation*}{\hat{H}}_{\text{vdW}}^{(100s,100s)}=h\times \left(\begin{matrix}\hfill 562\,00\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 569\,80\hfill & \hfill 1573\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1573\hfill & \hfill 569\,80\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 562\,00\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}},\end{equation*}$$

when θ = 0, where L is the two-atom distance. One can see that the two-atom states |r₊; r₋⟩ and |r₋; r₊⟩ are coupled with a strength that is about 3% of the diagonal energy shift. When θ = π/4, the interaction becomes,

$$\begin{equation*}{\hat{H}}_{\text{vdW}}^{(100s,100s)}=h\times \left(\begin{matrix}\hfill 567\,90\hfill & \hfill 590\hfill & \hfill 590\hfill & \hfill -590\hfill \\ \hfill 590\hfill & \hfill 564\,00\hfill & \hfill 983\hfill & \hfill -590\hfill \\ \hfill 590\hfill & \hfill 983\hfill & \hfill 564\,00\hfill & \hfill -590\hfill \\ \hfill -590\hfill & \hfill -590\hfill & \hfill -590\hfill & \hfill 567\,90\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}}\end{equation*}$$

which shows that the state |r₊; r₊⟩ is coupled to {|r₊; r₋⟩, |r₋; r₊⟩, |r₋; r₋⟩} with a strength that is about 1% of the diagonal energy shift. When θ = π/2, the interaction becomes,

$$\begin{equation*}{\hat{H}}_{\text{vdW}}^{(100s,100s)}=h\times \left(\begin{matrix}\hfill 573\,80\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1180\hfill \\ \hfill 0\hfill & \hfill 558\,00\hfill & \hfill 393\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 393\hfill & \hfill 558\,00\hfill & \hfill 0\hfill \\ \hfill -1180\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 573\,80\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}}\end{equation*}$$

which shows that |r₊; r₊⟩ is coupled to |r₋; r₋⟩ with a strength that is about 2% of the diagonal energy shift. If the Rydberg atoms have different principal quantum numbers, the issue can be more delicate. Consider ⁸⁷Rb |97S_1/2, m_J = ±1/2⟩ ≡ |R_±⟩, then with the basis

$$\begin{equation*}\left\{\vert {R}_{+};{r}_{+}\rangle ,\vert {R}_{-};{r}_{+}\rangle ,\vert {R}_{+};{r}_{-}\rangle ,\vert {R}_{-};{r}_{-}\rangle ,\vert {r}_{+};{R}_{+}\rangle ,\vert {r}_{-};{R}_{+}\rangle ,\vert {r}_{+};{R}_{-}\rangle ,\vert {r}_{-};{R}_{-}\rangle \right\}\end{equation*}$$

the two-atom van der Waals interaction of the atoms is given by

$$\begin{equation*}{\hat{H}}_{\text{vdW}}^{(97s,100s)}=\left(\begin{matrix}\hfill {\hat{V}}_{1}\hfill & \hfill {\hat{V}}_{2}\hfill \\ \hfill {\hat{V}}_{2}\hfill & \hfill {\hat{V}}_{1}\hfill \end{matrix}\right),\end{equation*}$$

when initially one atom is in the n = 97 state and the other is in the n = 100 state, where

$$\begin{align}\hfill {\hat{V}}_{1}& =h\times \left(\begin{matrix}\hfill -891\,80\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -597\,80\hfill & \hfill 588\,00\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 588\,00\hfill & \hfill -597\,80\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -891\,80\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}},\hfill \\ \hfill {\hat{V}}_{2}& =h\times \left(\begin{matrix}\hfill -537\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -375\hfill & \hfill 324\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 324\hfill & \hfill -375\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -537\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}},\hfill \end{align} \tag{ 16 }$$

when θ = 0. Equation (16) shows that the two-atom states |R₋ r₊⟩ and |R₊ r₋⟩ are coupled with each other with a strength that is almost equal to the diagonal energy shift; similarly, the two states |r₋ R₊⟩ and |r₊ R₋⟩ are coupled in a similar way which results in zero energy shift. This is analogous to the well studied Förster resonance [192]. Because the angular selection rules lead to different couplings for different orbitals, the issue is more complexed if p-orbital Rydberg states are employed. We take $\vert {r}_{\pm }\rangle =\vert 100{P}_{\frac{1}{2}},{m}_{J}=\pm 1/2\rangle$ as an example. Then, with the basis {|r₊; r₊⟩, |r₊; r₋⟩, |r₋; r₊⟩, |r₋; r₋⟩}, the van der Waals interaction is given by

$$\begin{equation*}{\hat{H}}_{\text{vdW}}^{(100p,100p)}=h\times \left(\begin{matrix}\hfill 2108\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -3492\hfill & \hfill -112\,00\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -112\,00\hfill & \hfill -3492\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2108\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}}\end{equation*}$$

when θ = 0. For θ = π/2 we have

$$\begin{equation*}{\hat{H}}_{\text{vdW}}^{(100p,100p)}=h\times \left(\begin{matrix}\hfill -6292\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 8400\hfill \\ \hfill 0\hfill & \hfill 4908\hfill & \hfill -2800\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2800\hfill & \hfill 4908\hfill & \hfill 0\hfill \\ \hfill 8400\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -6292\hfill \end{matrix}\right)\frac{\mu {\text{m}}^{6}\enspace \text{GHz}}{{L}^{6}}\end{equation*}$$

which shows that the states |r₊; r₊⟩ and |r₋; r₋⟩ are strongly coupled; similarly, the states |r₊; r₋⟩ and |r₋; r₊⟩ are strongly coupled. The above examples show that to use the phase shift from a van der Waals interaction in the weak interaction regime, the two atoms shall be placed along the quantization axis, and the total angular momentum of one atom shall be equal to that of the other atom, and we'd better excite both atoms to the same principal quantum level. If the atoms fly freely during the operation of the quantum gate, the relative angular position of the two atoms can change, which means that it is also necessary to suppress the atomic motion so that the off-diagonal (i.e. state-flip) coupling is not significant.

The most known gate protocol was the original Rydberg blockade gate proposed in reference [99]. For two atoms each with two ground qubit states |0⟩ and |1⟩, external laser fields are applied to induce the ground-Rydberg transition |1⟩ ↔ |r⟩. The energy separation between |0⟩ and |1⟩ is E_hf = h × 9.2 GHz for ¹¹³Cs, and h × 6.8 GHz for ⁸⁷Rb. So the Rabi frequency for |1⟩ ↔ |r⟩ shall be much smaller than E_hf/h to avoid exciting |0⟩ to Rydberg state. Of course, one shall also be careful not to excite some Rydberg state |r'⟩ where the frequency separation between |r'⟩ and |0⟩ is near to that between |r⟩ and |1⟩. The error from these unwanted transitions can be suppressed to the order of 10⁻⁵ for typical setups as analyzed in reference [219]. The method in reference [99] is schematically shown in figure 5(a) which follows as (i) apply a π pulse to the control atom to induce the transition |1⟩ → −i|r⟩; (ii) apply a 2π pulse to the target atom to induce the transition |1⟩ → −i|r⟩ → −|1⟩ if there is no Rydberg blockade; (iii) apply a π pulse to the control atom to induce the transition −i|r⟩ → −|1⟩. The first and third pulses will change the states only if the input state of the two qubits is |1α⟩ where α is 0 or 1, but the second pulse will change the input state |01⟩, i.e. will induce a π phase change. If the blockade interaction of the state |rr⟩ is large enough, the three pulses lead to

$$\begin{align}\hfill \vert 00\rangle & \to \vert 00\rangle \to \vert 00\rangle \to \vert 00\rangle ,\hfill \\ \hfill \vert 01\rangle & \to \vert 01\rangle \to -\vert 01\rangle \to -\vert 01\rangle ,\hfill \\ \hfill \vert 10\rangle & \to -i\vert r0\rangle \to -i\vert r0\rangle \to -\vert 10\rangle ,\hfill \\ \hfill \vert 11\rangle & \to -i\vert r1\rangle \to -i\vert r1\rangle \to -\vert 11\rangle .\hfill \end{align} \tag{ 17 }$$

But the actual blockade interaction is finite, so that the second pulse will induce a marginal change as an error to the input state |11⟩ which is −i|r1⟩ at the beginning of the second pulse. Because the transition −i|r1⟩ → −|rr⟩ is no longer resonant in the presence of dipole–dipole interaction of the state |rr⟩, −i|r1⟩ will transit to −|rr⟩ with a very small probability. The result is that there will be a small population leakage out of the input state, as well as a phase error to it, which is called the blockade error as analyzed in reference [220]. However, most published papers only treat the population error as the blockade error because the phase error is in principle removable by a certain change of the phase in the Rydberg lasers [221]. The blockade error sets a fundamental limit to the achievable gate fidelity of about 0.998 in the blockade gate [221].

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The blockade error in the second pulse is sensitive to the interaction V of the state |rr⟩. This can be understood by studying the Hamiltonian

$$\begin{equation}\hat{H}=\left(\frac{\hslash {\Omega}}{2}\vert u\rangle \langle l\vert +\text{H.c.}\right)+D\vert u\rangle \langle u\vert ,\end{equation} \tag{ 18 }$$

which can be diagonalized with the following eigenvalues and eigenvectors

$$\begin{equation}\begin{aligned}{{\epsilon}}_{\pm }& =[D\pm \sqrt{{(\hslash {\Omega})}^{2}+{D}^{2}}]/2,\\ \vert {v}_{\pm }\rangle & =\left(\frac{\hslash {\Omega}}{2}\vert l\rangle +{{\epsilon}}_{\pm }\vert u\rangle \right)/{N}_{\pm }\end{aligned},\end{equation} \tag{ 19 }$$

where ${N}_{\pm }=\sqrt{{(\hslash {\Omega})}^{2}/4+{{\epsilon}}_{\pm }^{2}}$, the subscript u and l denote the 'upper' and 'lower' states, respectively, and D denotes an energy shift in the state |u⟩. Here, we have {|l⟩, |u⟩} = {|r1⟩, |rr⟩} and D is the Rydberg interaction V.

For an initial state of |ψ(t = 0)⟩ = −i|r1⟩ at the beginning of the second pulse, the state evolves as [219]

$$\begin{equation}\vert \psi (t)\rangle =\frac{-2\mathrm{i}}{\hslash {\Omega}}\frac{{{\epsilon}}_{-}{N}_{+}\,{\mathrm{e}}^{-\mathrm{i}t{{\epsilon}}_{+}/\hslash }\vert {v}_{+}\rangle -{{\epsilon}}_{+}{N}_{-}\,{\mathrm{e}}^{-\mathrm{i}t{{\epsilon}}_{-}/\hslash }\vert {v}_{-}\rangle }{{{\epsilon}}_{-}-{{\epsilon}}_{+}}.\end{equation} \tag{ 20 }$$

At the end of the second pulse, i.e. when Ωt = 2π, the state becomes

$$\begin{align}\hfill \vert \psi (t)\rangle & =\frac{-2\mathrm{i}}{\hslash {\Omega}}\frac{{{\epsilon}}_{-}{N}_{+}{\text{e}}^{-\text{i}\frac{2\pi {{\epsilon}}_{+}}{\hslash {\Omega}}}\vert {v}_{+}\rangle -{{\epsilon}}_{+}{N}_{-}{\text{e}}^{-\text{i}\frac{2\pi {{\epsilon}}_{-}}{\hslash {\Omega}}}\vert {v}_{-}\rangle }{{{\epsilon}}_{-}-{{\epsilon}}_{+}},\hfill \end{align} \tag{ 21 }$$

then the blockade error, i.e. the population in |rr⟩, is = |⟨rr|ψ(t)⟩|², which is

$$\begin{equation}{\epsilon}=\frac{{{\Omega}}^{2}}{{(V/\hslash )}^{2}+{{\Omega}}^{2}}\,{\mathrm{sin}}^{2}\,\frac{\pi \sqrt{{(V/\hslash )}^{2}+{{\Omega}}^{2}}}{{\Omega}}.\end{equation} \tag{ 22 }$$

Because deep in the blockade regime we expect V ≫ ℏΩ, we can approximate the above error as ${\epsilon}\approx \frac{{\hslash }^{2}{{\Omega}}^{2}}{{V}^{2}}\,{\mathrm{sin}}^{2}\,\frac{\pi V}{\hslash {\Omega}}$. In a practical implementation, the fluctuation in V is severe. Thus one can average by taking an integration ${\mathrm{sin}}^{2}\,\frac{\pi V}{\hslash {\Omega}}$ in an interval when $\frac{2\pi V}{\hslash {\Omega}}$ changes from 2kπ to 2(k + 1)π where k is a very large integer, leading to $\bar{{\epsilon}}\approx \frac{{\hslash }^{2}{{\Omega}}^{2}}{2{V}^{2}}$. In order to avoid crosstalk and to measure qubit states with a high fidelity, the qubit spacing L is usually set large enough, so that the dipole–dipole interaction, which drops off as 1/L³, is small. When the interaction is small compared to the energy gaps between nearby two-atom Rydberg states coupled via the dipole–dipole interaction, the interaction can be usually described with the second-order perturbation theory. This leads to finiteness of the Rydberg interaction V and, thus, the blockade error $\bar{{\epsilon}}$ sets a strict limit to the gate fidelity [221].

The blockade mechanism can be used to realize three-qubit gates. For example, reference [222] analyzed in detail the fidelities for multiqubit C_k NOT gates, and showed that a fidelity as high as 99.97% is achievable for k = 3. Based on the method in reference [99], reference [223] proposed a protocol to realize the Deutsch gate [20], which is a three-qubit gate that by itself constitutes a universal set in the circuit model of quantum computation. Due to the peculiar feature that one type of gate can constitute a universal set, there have been proposals to realize the Deutsch gate in silicon [224] or superconducting systems [225].

Up until now, most studies on quantum gates and entanglement based on Rydberg interactions focus on alkali-metal atoms. There was one experiment of entanglement between the clock and Rydberg states of alkaline-earth strontium atoms in reference [67]. To use the Rydberg blockade for quantum entanglement between nuclear spins in the ground states of divalent AEL atoms, reference [226] proposed a method of Rydberg excitation of ytterbium or strontium atoms with moderate magnetic fields and found that fidelities about 99.8% are possible for realizing a C_Z gate.

The characterization of a quantum entangling operation includes fidelity, purity, quantum degree, and entangling capability or entangling power as studied in [227–229]. In quantum information processing the concept of fidelity is frequently used to compare the different degrees of accuracy in different realizations of the same quantum process. In Rydberg atom quantum science, the fidelity for a quantum gate measures how accurately the realized gate operation matches the ideal one, or, more precisely, how faithfully the gate operation can transform an input state to the output state according to the presumed unitary operation. An experimental assessment of the fidelity requires characterization of the density matrix of the qubit system [228] that involves O(d²) measurements [230] or over. On the other hand, the theoretical study of the fidelity for a quantum operation has been a difficult subject because different formula can satisfy commonly accepted criteria for a reasonable definition. For the example of a two-qubit quantum gate, the input state can be an arbitrary superposition of the four states {|00⟩, |01⟩, |10⟩, |11⟩}, where |0⟩ and |1⟩ are the two qubit states and the first (second) digit in the ket represents the input state for the control (target) qubit. The gate fidelity is usually defined as an average over the fidelities of operations over all the possible input states. Although there are many definitions of fidelity in, e.g. references [231–236], we review formulas that were frequently employed for characterizing entanglement by Rydberg interactions.

Many experiments with Rydberg atoms created Bell states [53, 67, 101, 102, 109–116]. To characterize the quality of the Bell state, the fidelity can be defined as

$$\begin{equation}\mathcal{F}={\mathrm{Tr}}^{2}\sqrt{\sqrt{{\rho }_{\text{id}}}\rho \sqrt{{\rho }_{\text{id}}}},\end{equation} \tag{ 23 }$$

where ρ and ρ_id are the density matrices for the realized and ideal states, respectively. ρ and ρ_id can be exchanged in the above formula which is a basic property for a valid similarity measurement between two density matrices. Equation (23) is equivalent to the fidelity defined in [228]. When ρ_id = |Ψ⟩⟨Ψ| for a Bell state |Ψ⟩, the above equation simplifies to $\mathcal{F}=\langle {\Psi}\vert \rho \vert {\Psi}\rangle$ [231, 234, 235], which is a formula frequently used for quantifying how accurately the Bell states were prepared in experiments [53, 101, 102, 109–116]. The experimental measurement of the Bell-state fidelity involves measurement of the off-diagonal matrix elements which can be done by parity assessment after applying extra pulses to rotate the states. For theoretical investigation about the average accuracy of an entangling gate by equation (23), one can sample uniformly over the state space spanned by the four input states for the case of two-qubit gates.

Beside the above formula frequently used in experiments to characterize the quality of Bell state generation, one can compare ρ and ρ_id by defining a 'distance' between them, $\mathcal{D}(\rho ,{\rho }_{\text{id}})\equiv$ Tr|ρ − ρ_id|/2 [234], where $\vert A\vert \equiv \sqrt{{A}^{{\dagger}}A}$ for an arbitrary matrix A. $\mathcal{D}(\rho ,{\rho }_{\text{id}})$ is zero when ρ = ρ_id, which means that their 'distance' is zero since they coincide with each other. In this case, the fidelity is given by

$$\begin{equation}\mathcal{F}=1-\mathcal{D}(\rho ,{\rho }_{\text{id}}).\end{equation} \tag{ 24 }$$

One can find [221] that the fidelity defined in equation (24) is much more sensitive to dephasing errors compared to that in equation (23), and, thus, can be useful to quantify how accurately a state is created [125].

For quantum information processing, a quantum gate is required, and it is useful to measure how accurately a quantum gate is realized. For the controlled-NOT (CNOT), the probability truth table fidelity was frequently employed [53, 102, 109, 112]

$$\begin{equation}\mathcal{F}=\mathrm{Tr}({U}^{{\dagger}}\mathcal{U}),\end{equation} \tag{ 25 }$$

where U is the probability truth table for the ideal gate. For a CNOT in the basis {|00⟩, |01⟩, |10⟩, |11⟩}, U is

$$\begin{equation}U=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right),\end{equation} \tag{ 26 }$$

and $\mathcal{U}$ is the probability truth table when the gate is actually realized with some errors, i.e. $\mathcal{U}$ will deviate from the ideal one in equation (26) because of various error sources (see section 2.4). In experiments, the matrix elements in $\mathcal{U}$ can be measured with multiple gate cycles and different input states. Because the CNOT truth table fidelity does not fully take into account of dephasing errors, it can be larger than the fidelity defined in other ways as studied in [221].

A formula of gate fidelity used in references [134, 237–240] adopts the definition given in reference [232],

$$\begin{equation}\mathcal{F}=\frac{1}{\mathcal{N}+1}+\frac{1}{{\mathcal{N}}^{2}(\mathcal{N}+1)}\sum\limits _{k}\,\mathrm{Tr}(U{X}_{k}^{{\dagger}}{U}^{{\dagger}}\mathcal{E}({X}_{k})),\end{equation} \tag{ 27 }$$

where $\mathcal{N}={2}^{d}$ for a d-qubit gate or entanglement operation, U is the matrix of the quantum gate (such as the one in equation (26)), and $\mathcal{E}$ is the quantum gate or entanglement operation we would like to characterize. If we take d = 2 as an example, the summation over k means to sum over X_k ∈ {I, σ_x , σ_y , σ_z } ⊗ {I, σ_x , σ_y , σ_z }, where I is the 2 × 2 identity matrix and σ_x,y,z are the Pauli matrices. To evaluate the fidelity in a quantum gate or entanglement operation, we can calculate the sum over k by evaluating $\mathcal{E}({X}_{k})$ one by one using the quantum gate or entanglement protocol.

Another formula of gate fidelity which takes into account both population loss as well as phase errors is [236]

$$\begin{equation}\mathcal{F}=\frac{1}{\mathcal{N}(\mathcal{N}+1)}\left[\vert \mathrm{Tr}({U}^{{\dagger}}\mathcal{U}){\vert }^{2}+\mathrm{Tr}({U}^{{\dagger}}\mathcal{U}{\mathcal{U}}^{{\dagger}}U)\right],\end{equation} \tag{ 28 }$$

where $\mathcal{N}$ is similarly defined as in equation (27), which is 4 for a two-qubit controlled-Z (C_Z) or CNOT gate, and U is given in equation (26) for the CNOT gate. The fidelity defined in equation (28) can capture the phase errors because the term $\vert \mathrm{Tr}({U}^{{\dagger}}\mathcal{U})\vert$ is sensitive to the imaginary parts of the output wavefunction of the gate. For this reason, it can be used to characterize the C_Z gate [120, 139, 152, 168] where correct phases of the output states are important. Only when the phases in the nonzero elements of $\mathcal{U}$ of C_Z are correct can one transform it to an accurate CNOT [110], and large phase errors can lead to low fidelities.

Throughout this review, when a fidelity is quoted from the cited reference as shown in tables 2–4, the largest, explicitly stated fidelity with Rydberg-state decay at room temperature (or approximately at room temperature) will be shown unless otherwise specified; when the fidelity quoted is with Rydberg-state lifetimes in cryogenic environments or does not consider the Rydberg-state decay, it will be specified.

In the very first proposal on QC with neutral Rydberg atoms, reference [99] not only presented quantum entangling gates with rectangular pulses, but also with adiabatic pulses. By imposing time dependence in the field amplitude and detuning, reference [99] showed that with slowly varying fields the atomic state will adiabatically follow the eigenstate of the system Hamiltonian, from the input state to some other states and then to itself finally, and the dynamical phase $\langle \alpha \beta \vert {\text{e}}^{-\text{i}\mathcal{T}\int \hat{H}(t)\mathrm{d}t/\hslash }\vert \alpha \beta \rangle$ for the input state |αβ⟩ is tunable by choosing appropriate fields, where α, β = 0 or 1. When only the state |1⟩ is Rydberg excited for both qubits with the same duration, the dynamical phases {φ₁, φ₁, φ₂} for {|10⟩, |01⟩, |11⟩} can satisfy the condition φ₂ − 2φ₁ = π, which corresponds to an entangling gate that can be transformed to a CNOT gate with several single-qubit rotations. The original adiabatic version of the blockade gate in reference [99] does not require single-qubit addressing. Being free from single-site addressing can simplify experimental implementation which later on motivated a more detailed analysis of the method in reference [123].

The original adiabatic version of the blockade gate in reference [99] is not robust to noise in the laser fields and fluctuation of interactions. This is because the dynamical phase is a function of Rydberg interaction and the amplitude and detuning of the Rydberg lasers. The fluctuation of Rydberg interaction can be a major source of error since it is very sensitive a function of the spacing between the qubits. To tackle this issue, references [124, 137, 275] proposed to use dynamical phases in the blockade regime to implement entangling gates, where quantum annealing was studied in [275]. In the blockade regime, the two-qubit Rydberg state is not coupled, and the dynamical phase is dependent on detuning and amplitude of the external control fields. Thus, it is possible to implement adiabatic entangling gates by ramping the frequency and amplitude of Rydberg lasers in the multi-atom system. The protocols in [124, 137, 275] are robust against the fluctuation of the Rabi frequencies which results from their strategy to adiabatically evolve a state which is a superposition of ground and Rydberg states. These methods have a fidelity limited by the Rydberg blockade error and the Doppler dephasing; compared to other methods, their fidelity is more limited by the Rydberg-state decay because the speed of the gate shall be slow enough to establish adiabaticity.

In reference [132] a Rydberg gate based on shortcut to adiabaticity was proposed which leads to a fast gate since adiabaticity is no longer required. In this case, a nontrivial phase can arise with both the geometrical and dynamical characters. This method can also create multiqubit gates.

A CNOT gate was proposed in reference [121] based on electromagnetically induced transparency (EIT). The CNOT gate in reference [121] only needs three laser fields for the target qubit with the following sequence. (i) Apply a π pulse to the control qubit for the transition |1⟩ → |r⟩. (ii) Apply an adiabatic pulse to induce a state transfer |0⟩ ↔ |1⟩ in the target qubit conditioned on that there is a blockade effect from Rydberg interactions. (iii) Apply a π pulse to the control qubit for the transition |r⟩ → |1⟩. The adiabatic pulse in step (ii) is as follows: the states |0⟩ and |1⟩ are coupled to a low-lying state |p⟩ with a large detuning, and the state |p⟩ is coupled with |r⟩ with a matched detuning so that the transition |r⟩ ↔ |0(1)⟩ is resonant. The field for the transition |r⟩ ↔ |p⟩ is constant, but that for |p⟩ ↔ |0(1)⟩ is time-dependent. The time-dependence is designed so that when the two-qubit input state is |0α⟩, where α = 0 or 1, the state |0α⟩ follows adiabatically from |0α⟩ to a time-dependent combination of two dark states, and back to itself at the end of the pulse (as in EIT); when the two-qubit input state is |1α⟩, the Rydberg interaction shifts away the two-photon resonance |r⟩ ↔ |0(1)⟩, and as a consequence it allows the transition |0⟩ ↔ |1⟩ to occur via the intermediate state |p⟩ in the target qubit.

It is useful to compare the above method with another CNOT method with three pulses proposed in reference [223]. The method in reference [223] needs to couple the two qubit states |0⟩ and |1⟩ with the same Rydberg state |r⟩ for the target qubit. For an s or d-orbital Rydberg state |r⟩, this requires four laser fields for Rydberg excitation in the target qubit, two for |0⟩ ↔ |r⟩, and two for |1⟩ ↔ |r⟩. In comparison, the CNOT gate in reference [121] only needs three laser fields for the target qubit.

The scheme in reference [121] is specifically useful if the target qubit is replaced by an ensemble of qubits so as to design a C-NOT^N gate, i.e. a CNOT between a mesoscopic ensemble of N atoms and a single control atom. As long as the Rabi frequency for |r⟩ ↔ |p⟩ is much larger than the peak value of the Rabi frequency of |p⟩ ↔ |0(1)⟩, the eigenstate of the two-qubit system will adiabatically follow after a superposition of the dark states of the system Hamiltonian. There will be an error due to trying to populate the state where all the atoms in the target ensemble are in the Rydberg state since there will be strong interactions when all the atoms in the target ensemble are in Rydberg states. However, as long as the Rabi frequency for the transition |r⟩ ↔ |p⟩ is large enough, the error can be largely suppressed.

In reference [121], the adiabatic scheme is employed so as to induce the controlled swap process, and it is subjected to the usual blockade error since it depends on shifting away the resonance by Rydberg interaction. When the blockade error is suppressed, the gate fidelity by this scheme can be large when fine tuning of the laser fields is achieved.

The intrinsic errors in Rydberg gates based on the blockade mechanism is limited by the Rydberg-state decay and the blockade error, where the former can be made smaller by using faster Rydberg excitations. But to shrink the blockade error, larger Rydberg interactions are desirable, which can arise for high-lying Rydberg states. The energy spacing between two nearby Rydberg states becomes smaller when the principal quantum number is larger. For fast gate operations in order to suppress the Rydberg-state decay, the Rydberg Rabi frequency Ω should be large. This can lead to large population leakage to states that are near the optically-targeted Rydberg state. To tackle this issue, reference [126] showed that by using the method of analytical derivative removal by adiabatic gate (DRAG) [276, 277], the population leakage can be well suppressed when shaped but analytic Rydberg pulses are used for Rydberg excitation. The analysis in [126] showed that gate fidelities over 99.99% are realizable at room temperature; if qubits are hosted in cryogenic chambers at 4.2 K, their gate can have fidelities over 99.999% thanks to the exceedingly short gate durations of several tens of nanoseconds. By comparing the gate fidelity of reference [126] with those from other methods in table 2, one can see that DRAG can lead to the largest fidelities for Rydberg gates via the blockade mechanism. However, the analyses in [126] were based on a GHz-scale Rydberg interaction in the form of a pure energy shift which is challenging to be found as discussed in section 2.3.2, and it is an open question whether the theory in [126] can be extended to protocols where strong enough dipole–dipole interaction (which is not in the form of a pure energy shift) is considered.

A Berry phase or geometric phase is accumulated when a state dependent on a varying parameter ϕ_r evolves slowly around a closed loop of ϕ_r . This type of geometric phase only depends on the path along which ϕ_r evolves, and is robust against noise of various forms.

It was proposed in reference [127] that an arbitrary two-qubit controlled-phase gate can be realized by using a Berry phase in two atoms, each with four atomic states, i.e. three ground states |0⟩, |1⟩, |2⟩ and a Rydberg state |r⟩, where |0⟩ and |1⟩ are qubit states. For one qubit upon which the transitions |1⟩ ↔ |2⟩ and |2⟩ ↔ |r⟩ are driven by external fields with Rabi frequencies Ω₁ and ${\text{e}}^{-\text{i}{\phi }_{r}}{{\Omega}}_{r}$, respectively, the dark state of the three-level system is $\vert {D}_{1}\rangle =\mathrm{cos}\,\theta \vert 1\rangle -{\text{e}}^{\text{i}{\phi }_{r}}\theta \vert r\rangle$, where tan θ ≡ Ω₁/Ω_r [278]. With one adiabatic cycle where the wavefunction evolves starting from |1⟩ and ending at |1⟩, the state acquires a Berry phase φ₁ = −∫sin²  θ  dϕ_r . What is interesting is that when the same pulse is applied to two nearby atoms, although the single-atom dark state |D₁⟩ no longer exists, reference [127] pointed out that for large enough blockade interaction there is another dark state |D₂⟩, and there is a time-dependent component in |D₂⟩ with a phase term

$$\begin{equation}\mathcal{B}\equiv \langle {D}_{2}\vert (\vert 1r\rangle +\vert r1\rangle )=\frac{-\mathrm{sin}\,\theta \,\mathrm{cos}\,\theta \,{\text{e}}^{-\text{i}{\phi }_{r}}}{\sqrt{{\mathrm{cos}}^{4}\,\theta +2\,{\mathrm{sin}}^{4}\,\theta }}.\end{equation} \tag{ 39 }$$

Then, by appropriately choosing a closed loop for the varying phase ϕ_r , one can simultaneously map {|01⟩, |10⟩} back to themselves with a phase accumulation φ₁, and map |11⟩ to itself with another phase accumulation ${\varphi }_{2}=-\int \vert \mathcal{B}{\vert }^{2}\mathrm{d}{\phi }_{r}$. Here, one shall bare in mind that φ₁ arises when only one atom responds to the excitation field, while φ₂ arises when both qubits are optically excited, i.e. if the input state is |11⟩. When single-qubit rotations are applied, a controlled-phase gate diag$\left\{1,1,1,{\mathrm{e}}^{{\varphi }_{2}-2{\varphi }_{1}}\right\}$ is realized with φ₂ − 2φ₁ tunable in the interval (0, 2π]. By this method, figure 2(b) of reference [127] showed that Bell states with fidelities around 99.6% can be prepared.

The above Berry-phase gate can suppress amplitude noise of the external control fields, but is still subjected to the blockade error because the two-qubit dark state |D₂⟩ is only valid in the limit of large blockade shift in |rr⟩. The novelty of the scheme in reference [127] lies in that when it is applied to a multiqubit system, it can prepare large-scale GHZ states. For more details, see reference [127]; besides, reference [128] did a more detailed analysis about the scheme in reference [127] and studied some of its extensions.

Berry phases are particularly useful for designing quantum gates in ensemble qubits [122] which will be reviewed in section 3.8.7.

In section 3.5 we have reviewed Rydberg gates by geometric phases without using pulse shaping. Although the gate in [244, 262] is implemented with constant Rabi frequencies, cyclic state evolution occurs where geometric phases are accumulated. In reference [131], another nonadiabatic holonomic Rydberg quantum gate was studied based on defining qubit states by superposition states of two hyperfine ground states. The gate in reference [131] is a C_Z gate in their qubit basis, and pulse shapes were found by shortcuts to adiabaticity. However, it is an open question whether one can efficiently manipulate (load and read) quantum information by qubit states that are not eigenstates of the external fields which should be applied for specifying the quantization axis. Later, the authors of reference [131] considered using auxiliary atoms to realize heralded nonadiabatic holonomic Rydberg gates [133].

Quantum gates can be efficiently created between a single atom and an ensemble of atoms, shown in reference [134], which proposed a three-pulse nonadiabatic non-abelian geometric Rydberg controlled-U(θ, ϕ) gate, where U(θ, ϕ) = σ_z  cos θ + (σ_x  cos ϕ + σ_y  sin ϕ)cos θ and the angles (θ, ϕ) are determined by laser fields and dipole transition matrix elements. The scheme in reference [134] is particularly robust against the fluctuation of the field intensity in the lasers. The gate in reference [134] can be easily tuned to various forms including C_Z and CNOT by adjustment of parameters of laser fields. In particular, reference [134] showed that high-fidelity (see table 2) three-qubit gates are realizable with pulses found by invariant-based inverse engineering and optimal control. Later on, reference [251] extended this scheme and studied an interesting multiple-qubit C_k U gate based on inverse engineering, shortcut-to-adiabaticity, and optimal control. In particular, reference [251] showed that the operation U in the C_k U gate can be an arbitrary single-qubit operation in the target qubit, and fidelities about 99.8% (at 0 K; see section III B therein) are achievable for k up to 4 when U is either the NOT or the Hadamard operation.

Numerical optimal control can be used to find time-dependent external fields for implementing high-fidelity Rydberg gates. By optimal control, one uses numerical search to locate a time-dependence pattern for the amplitude and frequency of external fields so as to find an optimal fidelity. For the simplest case, one can use the optimal control by minimizing −F(a, b, c, ...), where F is the gate fidelity which depends on system parameters such as the decay rates of atomic states, energy gaps between nearby atomic states, detuning of Rydberg lasers, and so on (denoted by a, b, c, ...). In practice, one shall bare in mind that an efficient gate should not require a long pulse to work, and one can add extra terms such as the total pulse area required in the functional as a constraint. For a detailed implementation of Rydberg gates by numerical optimization, see, e.g. references [135, 273].

It was shown in reference [151] that optimal control can be used to design fast two-qubit entangling gates with neutral atoms trapped on atom chips, where a two-qubit fidelity 99.7% was shown to be possible with a gate duration as short as 30 ns (see table 2 of reference [151]). Later on, reference [152] showed that optimal control can lead to high-fidelity two-qubit entangling gates between two optically trapped neutral atoms. Importantly, the nonanalytical pulses in reference [152] can lead to accurate gates that are robust to noise in the laser pulses, and it found that gate fidelities 99.99% are reachable. In both reference [152] and reference [151], the more practical two-photon excitation of Rydberg states were used for the excitation of even-parity Rydberg states without assuming effective two-photon Rabi frequencies derived by the adiabatic elimination of the intermediate state. As can be seen from table 2, the optimal control in reference [152] can work equally well as the method in reference [126] since both can lead to the largest gate fidelity compared to other schemes shown in table 2.

Optimal control has been experimentally proven to be a powerful tool in Rydberg atom quantum science. For example, reference [274] reported the creation of 'Schrödinger cat' states of the GHZ type with qubit numbers up to 20 using pulses found by optimal control. In reference [274], the initial multi-atom state adiabatically evolves to a superposition state of two ground states of a many-body Hamiltonian, leading to a 20-qubit GHZ state with a fidelity 54.2%. Because the energy gap between the ground state and excited states shrinks quickly when the number of atoms increases with the method of reference [274], longer times are required to guarantee adiabaticity for a multi-atom system with more atoms, and thus optimal control was used to minimize the pulse duration and maximize the fidelity for the GHZ state. The results in reference [274] rigorously showed that neutral atoms are competitive to superconducting systems [279, 280] regarding the capability to create large-scale many-particle entanglement. Besides, the same groups experimentally demonstrated [53], for the first time, a three-qubit neutral-atom Toffoli gate with Rydberg laser pulses found by numerical optimal control softwares. Although the intrinsic fidelity of the Toffoli gate by optimal control was about 97%, and the experimental truth-table fidelity was 87% in reference [53], their experiment gave a proof-of-principle demonstration of optimal control in quantum information processing with Rydberg atoms. Optimal control can also be used for manipulating states of one atom when many energy levels are involved. For example, reference [281] experimentally demonstrated that by using pulse sequences found from optimal control theory, one can accurately generate a nonclassical superposition state that cannot be prepared with reasonable fidelities using standard techniques.

Although optimal control was frequently used in the study of Rydberg atom quantum technology, an optimal-control based, fast, high-fidelity, and single-step C_Z gate was only recently proposed in reference [135]. It showed that with experimentally feasible parameters, a C_Z gate can be rapidly realized by sending global external field to the two qubits. The optimized pulse was found by dividing the external control field to M equal-length segments where M is a moderate integer so as to search an optimal pattern for the amplitude and frequency of the laser fields in each segment. With the spontaneous emission and blackbody radiation from the intermediate p-orbital and Rydberg states included, reference [135] showed that a fidelity over 0.997 is achievable in the frequently adopted two-photon excitation of s or d-orbital Rydberg states. Compared to previous protocols based on quantum interference proposed in reference [120] and experimentally realized in reference [53], the C_Z gate protocol in reference [135] can substantially reduce the motion-induced Doppler dephasing because there is no gap time during which a Rydberg atom drifts freely in free space. Thus, if fine tuning of the control lasers can be achieved, optimal control can be of particular help in the near-term Rydberg atom quantum science where noise from the lasers and atomic motion dominate the gate infidelity based on current technology.

There are two main challenges for designing quantum gates by Rydberg blockade in atomic ensembles. First, it is difficult to deterministically excite Rydberg states when the atom number is random, and second, the phase change of the atomic states in response to Rydberg excitation depends on the number of atoms in the ensemble which makes it difficult to deterministically generate quantum entanglement.

Excitation of Rydberg states.—When we try to load one atom in each optical dipole trap in a large array, there is a possibility to fail, as well as some chance to load more than one atoms in one trap [45, 46, 47, 282]. If one would like to use this type of atomic arrays for storing quantum information where each site has at least one atom, section 2.1.2 shows that we can treat both single-atom states |0(1)⟩ as qubit states, and ensemble states

$$\begin{equation}\begin{aligned}\hfill \vert \bar{0}\rangle & =\vert 000\dots 0\dots 000\rangle ,\hfill \\ \hfill \vert \bar{1}\rangle & =\frac{1}{\sqrt{N}}\sum\limits _{j=1}^{N}\vert 000\dots {1}_{j}\dots 000\rangle ,\hfill \end{aligned}\end{equation} \tag{ 40 }$$

as qubit states as well, where N is the number of atoms in the ensemble. The integer N in equation (40) can fluctuate from site to site. Consider the case when all the atoms in one site are near enough, i.e. the trap is deep enough, the Rydberg interaction between any pair of atoms in the trap can be larger than a certain value V_min. If a rectangular Rydberg laser pulse is applied for |1⟩ ↔ |r⟩ with Rabi frequency Ω for one atom, the actual Rabi frequency for the transition from $\vert \bar{1}\rangle$ to the ensemble state

$$\begin{equation}\vert \bar{r}\rangle =\frac{1}{\sqrt{N}}\sum\limits _{j=1}^{N}\vert 000\dots {r}_{j}\dots 000\rangle ,\end{equation} \tag{ 41 }$$

is enhanced by $\sqrt{N}$, where j = 1, 2, ..., N. For a small N, the Rydberg blockade effect can hold with a practical value of N in the experiment, i.e. we can still have the condition $\sqrt{N}\hslash {\Omega}\ll {V}_{\mathrm{min}}$ since ℏΩ ≪ V_min. The issue is, how can one excite the state of the atomic ensemble from $\vert \bar{1}\rangle$ to $\vert \bar{r}\rangle$ by a given type (and duration) of pulse when N is random? The answer was given in reference [129], which shows that the excitation from $\vert \bar{1}\rangle$ to $\vert \bar{r}\rangle$ with N up to 7 can be accurately accomplished with practical atomic parameters and experimental resources. To show the essence of the theory, we consider the Hamiltonian of a one-photon excitation of Rydberg states

$$\begin{equation}\hat{H}=\left[\hslash \frac{{\Omega}(t)}{2}\vert \bar{r}\rangle \langle \bar{1}\vert +\text{H.c.}\right]+\hslash \frac{\delta (t)}{2}\left(\vert \bar{r}\rangle \langle \bar{r}\vert -\vert \bar{1}\rangle \langle \bar{1}\vert \right),\end{equation} \tag{ 42 }$$

which has two eigenstates

$$\begin{align}\hfill \vert {v}_{+}\rangle & =\left[{\Omega}(t)\vert \bar{r}\rangle +(\sqrt{{{\Omega}}^{2}(t)+{\delta }^{2}(t)}-\delta (t))\vert \bar{1}\rangle \right]/{\mathcal{N}}_{+},\hfill \\ \hfill \vert {v}_{-}\rangle & =\left[{\Omega}(t)\vert \bar{r}\rangle -(\sqrt{{{\Omega}}^{2}(t)+{\delta }^{2}(t)}+\delta (t))\vert \bar{1}\rangle \right]/{\mathcal{N}}_{-},\hfill \end{align} \tag{ 43 }$$

with eigenvalues ${{\epsilon}}_{\pm }=\pm \hslash \sqrt{{{\Omega}}^{2}(t)+{\delta }^{2}(t)}/2$ and normalization factors ${\mathcal{N}}_{\pm }=\sqrt{{{\Omega}}^{2}(t)+{[\sqrt{{{\Omega}}^{2}(t)+{\delta }^{2}(t)}\mp \delta (t)]}^{2}}$. If initially Ω(0) is small compared to δ(0), |v₊⟩ is approximately $\vert \bar{r}\rangle$, and |v₋⟩ is approximately $\vert \bar{1}\rangle$. If both Ω(t) and δ(t) change smoothly until at the end of the pulse when Ω(t) = Ω(0) and δ(t) = −δ(0), |v₋⟩ is approximately $\vert \bar{r}\rangle$. This means that if we start from the state $\vert \bar{1}\rangle$, the atom will be excited to the state $\vert \bar{r}\rangle$. However, it does require smooth enough Ω(t) and δ(t) with the condition $\vert \mathrm{d}{\Omega}(t)/\mathrm{d}t\vert /\sqrt{{{\Omega}}^{2}(t)+{\delta }^{2}(t)}\ll 1$ and $\vert \mathrm{d}\delta (t)/\mathrm{d}t\vert /\sqrt{{{\Omega}}^{2}(t)+{\delta }^{2}(t)}\ll 1$. This adiabatic method does not require the pulse area to be equal to a certain value, and it is applicable to excite an atomic ensemble with unknown but small enough N.

The above method is useful for exciting a p-orbital Rydberg state. For s- or d-orbital Rydberg state, an intermediate state shall be used. Reference [129] showed that when the detuning at the intermediate state is zero, the standard three-level STIRAP can not excite the ground state $\vert \bar{1}\rangle$ to the Rydberg state $\vert \bar{r}\rangle$ because there is no way to define a dark eigenstate like $({{\Omega}}_{p\to r}\vert \bar{1}\rangle -{{\Omega}}_{1\to p}\vert \bar{r}\rangle )/\sqrt{{{\Omega}}_{p\to r}^{2}+{{\Omega}}_{1\to p}^{2}}$, where Ω_p→r and Ω_1→p are the Rabi frequencies for the upper and lower transitions. However, reference [129] found that by adding a large, constant detuning to the intermediate state, there is an eigenstate of the many-body Hamiltonian which adiabatically changes from $\vert \bar{1}\rangle$ to $\vert \bar{r}\rangle$ when the counterintuitive STIRAP pulse is applied; compared to the two-state case in equation (42), the STIRAP excitation of an ensemble Rydberg state only needs to tune the Rabi frequency without tuning the detuning. More details about the high-fidelity excitation of $\vert \bar{r}\rangle$ with s(d) and p-orbital Rydberg states can be found in reference [130] and the review [136].

Compensation of random phases.—The adiabatic excitation of an atomic ensemble to a Rydberg state is much more involved compared to the case of single-atom excitation [283]. Reference [122] found that for different atom numbers, the phases of the Rydberg state $\vert \bar{r}\rangle$ differ; consequently, when deexcited to the ground state, the phase of the state wavefunction depends on the number of atoms. The phase is accumulated because for N > 1 in the case of STIRAP, the adiabatic following is no longer about a dark state. Instead, the adiabatic state can sometimes have a nonzero energy although most of the time it is dark during the pulse sequence, leading to a phase with both dynamical and geometrical characters. This issue can be resolved by shifting the sign of the Rabi frequencies and detuning used for the deexcitation of the Rydberg state. Numerical simulation in [122] showed that the random phase can be exactly compensated, and thus the standard gate protocols based on the Rydberg blockade are applicable when atomic ensembles are used as qubits. Figure 7 on page 9 of reference [125] showed that gate fidelities can be 99.9% found by simulated quantum process tomography with N up to 4. More details about the theories on Rydberg gates with ensembles can be found in reference [136].

Beside the above mentioned protocol of quantum entangling gates in two atomic ensembles, another method was shown in reference [244] depending on nonadiabatic geometric phases, and reference [284] showed entanglement generation between two atom ensembles assisted by pulse shaping and by coupling cavity modes with the atoms. The method of reference [244] depends on coupling atoms to cavity modes which is challenging, but it does not depend on pulse shaping since one π pulse is sufficient. The gate in reference [244] is a two-qubit entangling gate that has an entangling power less than the maximal [229] as in a C_Z or CNOT. But any two-qubit entangling gate can form a universal set with several single-qubit gates, so the method in reference [244] could be useful if the required coupling between the Rydberg atoms and cavity modes can be achieved. Moreover, reference [134] proposed an optimized geometric quantum computation with a mesoscopic ensemble of Rydberg atoms which was reviewed in section 3.8.5. However, different from the methods in [122, 125, 129, 130, 136], the methods in references [134, 244] assume that N is a fixed number.

Early study on Rydberg-mediated entanglement with antiblockade usually focused on using detuning in the Rydberg Rabi oscillations to compensate the Rydberg interaction, so that a higher-order Rydberg Rabi frequency arises from antiblockade. An early investigation of antiblockade on quantum entanglement was made in reference [296], which presented protocols for entanglement (such as the GHZ states) between ground and Rydberg states. The model in reference [296] assumes only nearest-neighboring van der Waals interaction in a one-dimensional lattice, and thus it is an open question whether there is a similar model for fast multi-atom entanglement with realistic physical parameters in the antiblockade regime. Later on, references [145, 237, 297, 298] did a more detailed analysis, and showed that by setting the detuning equal to the Rydberg blockade, one can populate a two-atom Rydberg state with a Rabi frequency from the second-order perturbation theory; using the same method, by setting the effective detuning equal to the Rydberg blockade in three atoms, one can populate the three-atom Rydberg state with a Rabi frequency derived from the perturbation theory. This strategy can be repeated so more atoms can be populated in Rydberg states with a higher-order Rabi frequency, which can lead to four-atom GHZ states with a high fidelity 96.80% if Rydberg-state decay is ignored [297] (see text below equation (9) on page 831 of reference [297]). The perturbation used in references [237, 297] led to ac Stark shifts that can be canceled by adding additional control fields. To avoid adding extra control fields, reference [299] introduced a scheme to modify the detuning used in the excitation in specific ways so that two-atom Rydberg excitation can still be realized in one step, and reference [300] introduced a scheme to realize an N-atom Rydberg excitation in one step with ac Stark shifts canceled with specially modified detunings. Notice that these approaches to Rydberg antiblockade can lead to many interesting schemes. For example, by using Rydberg antiblockade, reference [301] studied three-qubit gates and quantum error correction by a type of 'unconventional Rydberg pumping', references [302, 303] studied a novel nondestructive parity meter in ground-state atoms, reference [304] studied a three-qubit controlled-phase gate with a tunable phase, references [305, 306] showed methods to inter-convert the GHZ and W states, and reference [307] studied nonadiabatic noncyclic geometric phases in a Rydberg gate based on which reference [308] designed a scheme to distinguish different types of GHZ states with fidelities over 99.9% for GHZ states up to five qubits. Because the different interactions between different pairs of atoms in a multi-atom system can be different, by matching the detunings in the lasers with specific Rydberg interactions, interesting dynamics can be identified. For example, reference [309] used this strategy to show that high-fidelity three and four-qubit controlled-NOT gates can be created.

The second way to compensate the interaction V is to sequentially and quickly excite one atom after another, with the detuning of each pulse specifically designed to compensate the Rydberg interaction. This means that it is no longer necessary to use adiabatic elimination to obtain an effective Rydberg Rabi frequency, so that the required time for Rydberg excitation is much shorter than those in references [237, 297, 299, 300]. The method to entangle two and three atoms by this fast antiblockade strategy was analyzed in detail in reference [238].

Because the requirement to compensate the interaction, there are relatively few protocols based on pulse shaping to implement quantum gates based on antiblockade. However, there are two examples, shown in references [145, 310], that used pulse shaping and antiblockade for entanglement generation, where reference [310] is not quoted in table 4 because it did not show achievable gate fidelities by their scheme.

Consider the Raman transition with an effective Rabi frequency Ω_hyper between |0⟩ and |1⟩ via a largely detuned low-lying p-orbital state, there will be the following transition chain

$$\begin{equation}\vert 00\rangle {\leftrightarrow}(\vert 01\rangle +\vert 10\rangle )/\sqrt{2}{\leftrightarrow}\vert 11\rangle .\end{equation} \tag{ 45 }$$

However, if one excites |11⟩ to a two-atom Rydberg state |rr⟩ via the Rydberg antiblockade mechanism [237, 297, 299, 300] so that |11⟩ ↔ |rr⟩ is realized with a Rabi frequency Ω_Ryd and detuning Δ, there will be Stark shifts $({\Delta}\pm \sqrt{{{\Delta}}^{2}+{{\Omega}}_{\text{Ryd}}^{2}})/2$ for |11⟩, which can be large compared to Ω_hyper so as to block the transition from $(\vert 01\rangle +\vert 10\rangle )/\sqrt{2}$ to |11⟩. This means that there will be an isolated transition from |00⟩ to the Bell state $(\vert 01\rangle +\vert 10\rangle )/\sqrt{2}$, so that entanglement can be generated rapidly. This idea is similar to the dressing method reviewed in section 3.3. The ground-state blockade mechanism can also entangle three atoms [146] and can be used to generate entanglement by dissipation in the context of cavity quantum electrodynamics [311].

Another ground-state blockade mechanism with isolated dipole–dipole interactions similar to the one in [146] was proposed in reference [312]. The dipole–dipole interactions employed in reference [312] couple three two-atom Rydberg states, and by coupling each of two nearby atoms for the transition |0⟩ ↔ |r⟩ ↔ |1⟩ with appropriate Rabi frequencies and detunings (see figure 1 of reference [312]), it is possible to create a transition between |11⟩ and $(\vert 1r\rangle +\vert r1\rangle )/\sqrt{2}$ while the other states |00⟩, |01⟩, and |10⟩ are decoupled because of being largely detuned effectively by the dipole–dipole interaction and detuning in the laser fields. This means that by applying the control for a full Rabi-like transition between |11⟩ and $(\vert 1r\rangle +\vert r1\rangle )/\sqrt{2}$, a C_Z gate can be created. However, the antiblockade condition can fail due to the fluctuation of interatomic distance, and it is unclear about to what extent will the breakdown of the Rydberg antiblockade due to this fluctuation hamper the ground-state blockade.

The third method of quantum gates by antiblockade is to use asymmetric Rydberg interactions and dissipation, as proposed in reference [272]. The asymmetric Rydberg interaction can be used in the excitation blockade regime as reviewed in section 3.6. To use it in the antiblockade regime, reference [272] assumed that the interactions V_ss , V_sp and V_pp for the two-atom Rydberg states |ss⟩, |sp(ps)⟩ and |pp⟩ satisfy the condition V_pp = V_ss ≠ V_sp , which can be realized by choosing different Zeeman substates |s⟩ and |p⟩ within one Rydberg level. Then, by using Rydberg excitation fields with a detuning that exactly compensates the Rydberg interaction V_pp = V_ss , one can populate the states |ss⟩ and |pp⟩ respectively from |00⟩ and |11⟩ via the excitation $\vert 0\rangle \stackrel{{{\Omega}}_{\text{eff}}}{\to }-i\vert s\rangle$ and $\vert 1\rangle \stackrel{{{\Omega}}_{\text{eff}}}{\to }-i\vert p\rangle$, where Ω_eff = 2ℏΩ²/V_ss is a Rabi frequency from the second-order perturbation theory. The decay of the states |ss⟩ and |pp⟩ will populate the states |00⟩, |01⟩, |10⟩ and |11⟩ with an equal probability, but among which only |00⟩ and |11⟩ will be reexcited to the two-atom Rydberg states which enables recycling. Then, it is possible to create an entangled state $(\vert 01\rangle +{\text{e}}^{\text{i}\alpha }\vert 10\rangle )/\sqrt{2}$ with an undetermined phase. By adding a Raman transition between |0⟩ and |1⟩, reference [272] showed that the maximally entangled state with α = π can be created with a high fidelity of similar value in the original Rydberg blockade method if the fluctuation of Rydberg interactions is ignored. The method in [272] can also generate anti-ferromagnetic state where a purity of 96% was found in a four-atom system. Later on, reference [313] extended the scheme to prepare an entangled state in the form of, in the two-atom case, ${\sum }_{k=1}^{3}\vert {g}_{k}{g}_{k}\rangle /\sqrt{3}$, i.e. a maximally entangled state between three ground states |g_k ⟩ with k = 1, 2, 3. Besides depending on dissipation and antiblockade, reference [313] requires the interactions V_ij of the Rydberg states |r_i r_j ⟩ to satisfy the condition V_ii = V_jj ≠ V_ij , where i, j ∈ {1, 2, 3} and i ≠ j. When the Rydberg interactions between three atoms satisfy specific relations, more complicated entanglement can arise. For example, reference [314] showed that a three-qubit singlet state can be created, and reference [315] showed that a three-qubit W state can be created by coupling atoms with cavity modes.

The methods reviewed in section 5.3.1 depend on both dissipation and asymmetries in the Rydberg interactions. Soon after the introduction of the methods of reference [272], reference [316] put forward a concise scheme using dissipation and antiblockade to achieve the goal of references [272, 313]. Reference [316] showed that two transitions are sufficient, namely, a Rydberg excitation $\vert 0\rangle \to -i\vert r\rangle$ (detuned by Δ) and a resonant Raman transition $\vert 0\rangle \to -i\vert 1\rangle$. An appropriate choice of Δ can block the transition from $\vert D\rangle \equiv (\vert 01\rangle -\vert 10\rangle )/\sqrt{2}$ to Rydberg states, but any two-atom states not identical to |D⟩ can be excited to Rydberg states that will then decay to |D⟩. Then, |D⟩ becomes the only stable state in the process of dissipation. Later on, it was found that by choosing appropriate detunings, interactions between the states, and dissipation, entanglement with various forms can be created, shown in references [318, 321–323]. Dissipation can also be used to generate entanglement with dipole–dipole flip-flop processes as shown in reference [319]. Unlike that in section 5.3.1, this way does not require asymmetric interactions and may be easier to be experimentally tested if the fluctuation of interactions can be sufficiently suppressed. As in section 5.3.1, the scheme based on dissipation and antiblockade could also be used in the context of cavity quantum electrodynamics [317, 320].

One method to suppress the blockade error is by spin echo as shown in reference [220]. To understand the spin-echo method, one should first understand that the fluctuation of V in section 3.1 denotes that in different runs of the gate, the fluctuation of the two-qubit spacing leads to randomness of V. In a first approximation, although the initial value of V at the beginning of each gate cycle can differ from the presumed blockade interaction V₀, it does not change much during the gate sequence. Thus, one can start to understand the spin echo by first assuming a frozen interaction V = V₀, and then analyze fluctuation of the interaction around V₀ in different gate cycles.

The essence of the spin-echo Rydberg gate is to break the second 2π pulse in figure 5(a) into three pulses. First, a π pulse is used to induce the transition |1⟩ ↔ |r⟩ for the target qubit with a Rabi frequency Ω. For the input state |11⟩, the relevant Hamiltonian is given in equation (18) with {|u⟩, |l⟩} = {|rr⟩, |r1⟩} and D = V₀. Second, a microwave π pulse is used to transfer the Rydberg state |r⟩ to another Rydberg state |r'⟩ with a Rabi frequency iΩ_μ , where ℏΩ_μ ≫ |V|. The microwave fields induce a Rydberg state flip |r⟩ → |r'⟩ for both the control and target qubits. The interactions of the states |rr⟩ and |r'r'⟩ are V₀ and V'₀, respectively, where V₀ and V'₀ have opposite signs, i.e. one is attractive and the other is repulsive. Third, a π pulse is used to induce the transition |r'⟩ → |1⟩ with a Rabi frequency Ω', where Ω' = ΩV'₀/V₀. The Hamiltonian for the third pulse here is

$$\begin{equation}{\hat{H}}^{\prime }=\left(\frac{\hslash {{\Omega}}^{\prime }}{2}\vert {r}^{\prime }{r}^{\prime }\rangle \langle {r}^{\prime }1\vert +\text{H.c.}\right)+{V}_{0}^{\prime }\vert {r}^{\prime }{r}^{\prime }\rangle \langle {r}^{\prime }{r}^{\prime }\vert .\end{equation} \tag{ 48 }$$

If we ignore the duration of the microwave pulse, the state evolution in response to the three pulses described above is as $\vert {\psi }^{\prime }\rangle ={\text{e}}^{-\text{i}{\hat{H}}^{\prime }\frac{\pi }{\hslash {{\Omega}}^{\prime }}}\hat{\mathcal{M}}\,{\text{e}}^{-\text{i}\hat{H}\frac{\pi }{\hslash {\Omega}}}\vert \psi \rangle$, where |ψ⟩ is the initial two-qubit state and $\hat{\mathcal{M}}$ represents the time evolution operator during the microwave pulse. One can easily verify that |ψ'⟩ = |ψ⟩ because ${\hat{H}}^{\prime }\frac{\pi }{\hslash {{\Omega}}^{\prime }}+\hat{H}\frac{\pi }{\hslash {\Omega}}=0$ when we replace |r'⟩ and ⟨r'| by |r⟩ and ⟨r| in ${\hat{H}}^{\prime }$.

The above discussion about the spin echo has ignored the duration of the microwave pulse, and no phase appears in the qubit states. To induce a C_Z gate and to include the finite duration of the microwave pulses, one should correctly set the phase for the fields used in the pulses. A detailed analysis can be found in reference [278], where the fluctuation of V around V₀ was studied. One can see that for the spin echo to work, the key step is to know the value of V'₀/V₀. If the spacing and relative angular position of the qubits do not significantly change between the first and third pulses in the echo sequence, the value V'₀/V₀ is simply the ratio of the two C₆ coefficients of the two states |r'r'⟩ and |rr⟩. Even if the traps are shallow so that the shot-to-shot fluctuation of the initial locations of the qubits in their traps is large, the ratio between the qubit interactions V' and V in the third and the first pulses are still given by the ratio of the C₆ coefficients as long as the qubit flight is slow during the pulse sequence. This means that the spin echo sequence is robust to the fluctuation of the interaction. The numerical analysis in reference [220] showed that a large fidelity over 99.99% is achievable with Rydberg-state decay at 4.2 K (see the lower left on page 6 of reference [220]).

Compared to the methods in section 3, the spin-echo method can suppress the blockade error and the suppression is robust against the fluctuation of the Rydberg interaction. To our knowledge, the spin-echo Rydberg gate in reference [220] is the only method in the blockade regime that can suppress the blockade error without using pulse shaping. Even so, neither the spin-echo method nor the methods in section 3 can suppress the Doppler dephasing and it is an open question whether one can find an experimentally friendly way to suppress the blockade error as well as the motional dephasing.

One possible solution to the issue of blockade error may come from pulse shaping. There are in general two strategies to use pulse shaping for suppressing the blockade error. First, the method of analytical DRAG reviewed in section 3.8.3 can remove the blockade error in the blockade regime. Notice that the blockade error in the standard Rydberg gate is intrinsic since the Rydberg interaction V can not be too large. If very high Rydberg states are used, the energy levels are too dense which makes it challenging to selectively address the desired Rydberg state. Section 3.8.3 has reviewed a pulse-shaping method to use high enough Rydberg states for Rydberg gates, where a high fidelity 99.99% is achievable. The comparison in table 2 shows that the pulse-shaping strategy is among the most accurate gate protocols. Though promising, the theory in section 3.8.3 still depends on large enough interatomic interactions. If the atoms are placed too closely, selectively addressing (as required in information loading and reading) may be challenging.

Second, adiabatic pulses with Förster-resonant dipole–dipole interactions in references [138, 139] as reviewed in sections 4.3 and 4.4 can remove the blockade error. These two strategies are robust against the fluctuation of the interatomic interactions as shown in figure 7 for a numerical simulation with the example of reference [139]. Experiments have not exploited Rydberg-mediated entanglement via the Förster-resonant dipole–dipole interactions so far.

Even if the blockade error is suppressed either by spin echo or by pulse shaping, they require multi-pulses to work which inevitably leads to excess Doppler dephasing as in the original Rydberg blockade gate. It is an open question whether there is any method to simultaneously suppress the blockade error and the motional dephasing for designing an efficient two-qubit quantum entangling gate with the maximal entangling power (such as the C_Z and CNOT) [229]. With the many pulse-shaping methods as reviewed in section 3.8, we hope that certain pulse-shaping methods may lead to suppression of both the blockade error and the dephasing error.

Another possible solution to the above issues is to explore quantum entangling gates with qubits defined by, for example, the single-ensemble collective states as reviewed in section 2.1 and shown in figure 2(c). Almost all Rydberg quantum gates reviewed in sections 3–5 use the first method for defining qubits as reviewed in section 2.1 and shown in figure 2(a), i.e. using hyperfine-Zeeman substates from single atoms as qubit states (or its superatom version as in section 3.8). For a large-scale quantum computer, the many qubits defined by many atoms may lead to a rapid scaling of the Rydberg-state decay and motion-induced dephasing errors in the hyperfine-qubit method. But if one ensemble can carry the information of many qubits, then there can be novel ways to shrink the Rydberg-state decay and motional dephasing.

As reviewed in section 3.2, the interference gate explored in [53] also has a blockade error because it depends on the Rydberg blockade mechanism. The supplemental material of reference [53] discussed a mechanism to suppress the blockade error based on a static V throughout the gate sequence, which shows that a finite V leads to a modification of the detuning by an amount of Ω² ℏ/(2V) by the renormalization of $(\vert 1r\rangle +\vert r1\rangle )/\sqrt{2}$ concerning its excitation to the two-atom Rydberg state |rr⟩. In other words, the value of Δ in equations (30)–(36) should be updated according to the renormalization $(\vert 1r\rangle +\vert r1\rangle )/\sqrt{2}$. This means that in theory the blockade error can be suppressed in the gate of [53].

The method for suppressing the blockade error discussed in [53] is based on that V is a van der Waals interaction which does not change from shot to shot, nor does V change during the gate sequence within one gate cycle. In practical experiments, however, V does fluctuate, which makes the blockade error irremovable. Even if V is frozen, the renormalization of the state $(\vert 1r\rangle +\vert r1\rangle )/\sqrt{2}$ in equation (29) results in admixing of the state |rr⟩ in it, which leads to a state dynamics slightly different from that shown in section 3.2. But for sufficiently large V, the admixing of the state |rr⟩ is negligible. This means that the possibility to suppress the blockade error for the gate in [53] is intimately related with the capability to realize large van der Waals interaction V and to suppress the motion of the qubits.

The discussions in sections 6.1.1–6.1.4 focus on a theoretical attack of the gate errors when alkaline-metal atoms are used for qubits. If qubits show negligible position fluctuation, the blockade error or dephasing error can be absent by the methods reviewed in sections 4 and 5 which are not based on the blockade mechanism. Nevertheless, it is difficult to cool alkali-metal atoms to an extent where the position fluctuation is negligible for Rydberg quantum gates.

Another route toward neutral-atom quantum computation is by using atoms with two valence electrons and a nonzero nuclear spin, i.e. some alkaline-earth-metal or lanthanide atoms, which we call AEL atoms. Compared to alkali-metal atoms, divalent AEL atoms can be more easily cooled [67, 326–330], their nuclear spin states can be preserved during cooling [331], and both their ground and Rydberg states can be trapped in one trap [211]. These in principle mean that AEL qubits can show less position fluctuation if they are used in Rydberg-mediated entanglement generation. However, previous studies on QC with AEL atoms [103–105, 166, 332–336] rarely used Rydberg interactions; though a recent Rydberg blockade experiment [67] with AEL atoms reported a two-atom entanglement fidelity 99.1%, the entanglement in [67] was between a Rydberg state and a metastable state (5s5p)³ P₀ of the nuclear-spin-free strontium-88. It is an open question whether the versatile and accurate control over AEL atoms can significantly suppress the blockade error and motional dephasing for high-fidelity quantum entanglement between stable states.