and $|D_{+}\rangle$ and $|00\rangle$ are excited to $|DD\rangle$ in the Raman
transition. As a result, only the state $|D_{-}\rangle$ is dark, and
$|D_{+}\rangle$, $|00\rangle$, and $|DD\rangle$ actually decay through the
rapidly decaying state $|p\rangle$. For this reason, only $|D_{-}\rangle$
survives in the dissipation. This method depends on that the decay rate of
$|r\rangle$ shall be much smaller than that of $|p\rangle$.

Later on, the authors of Ref. Bhaktavatsala Rao and Mø lmer (2013) extended
their scheme to prepare entanglement in $N$ Rydberg atoms Rao and Mølmer
(2014). Consider the transition chain
$|1\rangle\xleftrightarrow{\Omega_{{1r_{1}}}}|r_{1}\rangle\xleftrightarrow{\Omega_{{r_{1}r_{2}}}}|r_{2}\rangle$
involving two Rydberg states $|r_{1(2)}\rangle$, one can show that when the
interactions of the states $|r_{i}r_{j}\rangle$ for any two atoms with
$i,j\in\\{1,2\\}$ are large compared to $\Omega_{{1r_{1}}}$ and
$\Omega_{{r_{1}r_{2}}}$, the initial state $|11\cdots 1\rangle$ will only be
excited to the state $\sum_{k=1}^{N}|1r_{1}^{(k)}\cdots 1\rangle/\sqrt{N}$,
which is further excited to $|R\rangle\equiv\sum_{k=1}^{N}|1r_{2}^{(k)}\cdots
1\rangle/\sqrt{N}$, where the superscript $(k)$ labels the atom in the
ensemble. Thus, a three-level system is formed so that there is a many-body
dark state $|D\rangle\propto\Omega_{{r_{1}r_{2}}}|11\cdots
1\rangle-\sqrt{N}\Omega_{{1r_{1}}}|R\rangle$. The bright states have
population in $|r_{1}\rangle$, which can be coupled to a fast-decaying low-
lying state so as to obtain a short lifetime. Then, starting from any initial
state, the dissipation through $|r_{1}\rangle$ will leave the population of
the many-body state in the dark state $|D\rangle$. Like that in Ref.
Bhaktavatsala Rao and Mø lmer (2013), this method also depends on long
lifetimes of a Rydberg state (which is $|r_{2}\rangle$ in Rao and Mølmer
(2014)).

By using dissipation and Rydberg blockade, Ref. Zhu  _et al._ (2020) showed
that it is possible to prepare entanglement between two atoms of the form
$\sum_{j=1}^{3}|a_{j}b_{j}\rangle/\sqrt{3}$, where $|a(b)_{j}\rangle$ denotes
a ground or Rydberg state of the first (second) atom. For the case of a highly
dissipative cavity, entanglement can also be generated between ground and
Rydberg-state atoms via dissipation Shao  _et al._ (2017b).

The dissipation-based schemes are in principle robust against the fluctuation
of the interactions, but it is an open question whether in the presence of
atomic motion the Doppler dephasing of the ground-Rydberg transition will
degrade their fidelities since long times are required to establish the
desired target states.

###  III.8 Rydberg gates and entanglement with pulse shaping

Pulse shaping has been frequently used in theoretical study on Rydberg gates.
There are many Rydberg gate protocols based on adiabatic rapid passage (ARP)
and stimulated Raman adiabatic passage (STIRAP) shown in, for example, Refs.
Beterov  _et al._ (2020); Saffman  _et al._ (2020) and references therein.
Also, there are many gate protocols based on the optimal control theory Goerz
_et al._ (2011); Müller  _et al._ (2011); Goerz  _et al._ (2014); Levine  _et
al._ (2019). ARP usually denotes population transfer between two states by
time-dependent control fields, while STIRAP refers to population transfer
between two states via a fast-decaying intermediate state Bergmann  _et al._
(1998); Král  _et al._ (2007); Vitanov  _et al._ (2017). To achieve better
gate performance which involves more parameters than the usual study on ARP or
STIRAP, optimal control by numerical optimization can be used to find specific
time dependence in the external control fields. As experimentally demonstrated
in Refs. Omran  _et al._ (2019); Levine  _et al._ (2019), optimal control is a
useful tool in the study of Rydberg atom quantum science.

####  III.8.1 Two-qubit phase gates derived from Ref. Jaksch  _et al._ (2000)

In the very first proposal on quantum computing with neutral Rydberg atoms,
Ref. Jaksch  _et al._ (2000) not only presented quantum entangling gates with
rectangular pulses, but also with adiabatic pulses. By imposing time
dependence in the field amplitude and detuning, Ref. Jaksch  _et al._ (2000)
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
$\langle\alpha\beta|e^{-i\mathcal{T}\int\hat{H}(t)dt/\hbar}|\alpha\beta\rangle$
for the input state $|\alpha\beta\rangle$ is tunable by choosing appropriate
fields, where $\alpha,\beta=0$ or $1$. When only the state $|1\rangle$ is
Rydberg excited for both qubits with the same duration, the dynamical phases
$\\{\varphi_{1},\varphi_{1},\varphi_{2}\\}$ for
$\\{|10\rangle,|01\rangle,|11\rangle\\}$ can satisfy the condition
$\varphi_{2}-2\varphi_{1}=\pi$, 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 Ref. Jaksch  _et al._
(2000) 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 Ref. Müller  _et al._ (2014).

The original adiabatic version of the blockade gate in Ref. Jaksch  _et al._
(2000) 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, Refs. Keating  _et al._ (2013, 2015); Mitra  _et al._ (2020) proposed
to use dynamical phases in the blockade regime to implement entangling gates,
where quantum annealing was studied in Keating  _et al._ (2013). 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 Keating  _et al._ (2013, 2015); Mitra  _et al._ (2020) 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 Ref. Shen  _et al._ (2019) a Rydberg gate based on shortcut to adiabaticity
was proposed which leads to a fast gate operation 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.

####  III.8.2 Two-qubit CNOT gates with electromagnetically induced
transparency

A CNOT gate was proposed in Ref. Müller  _et al._ (2009) based on
electromagnetically induced transparency (EIT). The CNOT gate in Ref. Müller
_et al._ (2009) only needs three laser fields for the target qubit with the
following sequence. (i) Apply a $\pi$ pulse to the control qubit for the
transition $|1\rangle\rightarrow|r\rangle$. (ii) Apply an adiabatic pulse to
induce a state transfer $|0\rangle\leftrightarrow|1\rangle$ in the target
qubit conditioned on that there is a blockade effect from Rydberg
interactions. (iii) Apply a $\pi$ pulse to the control qubit for the
transition $|r\rangle\rightarrow|1\rangle$. The adiabatic pulse in step (ii)
is as follows: the states $|0\rangle$ and $|1\rangle$ are coupled to a low-
lying state $|p\rangle$ with a large detuning, and the state $|p\rangle$ is
coupled with $|r\rangle$ with a matched detuning so that the transition
$|r\rangle\leftrightarrow|0(1)\rangle$ is resonant. The field for the
transition $|r\rangle\leftrightarrow|p\rangle$ is constant, but that for
$|p\rangle\leftrightarrow|0(1)\rangle$ is time-dependent. The time-dependence
is designed so that when the two-qubit input state is $|0\alpha\rangle$, where
$\alpha=0$ or $1$, the state $|0\alpha\rangle$ follows adiabatically from
$|0\alpha\rangle$ 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 $|1\alpha\rangle$, the Rydberg interaction shifts away the two-photon
resonance $|r\rangle\leftrightarrow|0(1)\rangle$, and as a consequence it
allows the transition $|0\rangle\leftrightarrow|1\rangle$ to occur via the
intermediate state $|p\rangle$ in the target qubit.

It is useful to compare the above method with another CNOT method with three
pulses proposed in Ref. Shi (2018b). The method in Ref. Shi (2018b) needs to
couple the two qubit states $|0\rangle$ and $|1\rangle$ with the same Rydberg
state $|r\rangle$ for the target qubit. For an $s$ or $d-$orbital Rydberg
state $|r\rangle$, this requires four laser fields for Rydberg excitation in
the target qubit, two for $|0\rangle\leftrightarrow|r\rangle$, and two for
$|1\rangle\leftrightarrow|r\rangle$. In comparison, the CNOT gate in Ref.
Müller  _et al._ (2009) only needs three laser fields for the target qubit.

The scheme in Ref. Müller  _et al._ (2009) is specifically useful if the
target qubit is replaced by an ensemble of qubits so as to design a C-NOTN
gate, i.e., a CNOT between a mesoscopic ensemble of $N$ atoms and a single
control atom. As long as the Rabi frequency for
$|r\rangle\leftrightarrow|p\rangle$ is much larger than the peak value of the
Rabi frequency of $|p\rangle\leftrightarrow|0(1)\rangle$, 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 $|r\rangle\leftrightarrow|p\rangle$ is large enough, the error
can be largely suppressed.

In Ref. Müller  _et al._ (2009), the adiabatic scheme is employed so as to
induce the controlled swap process, and it is subjected to the the usual
blockade error since it depends on shifting away the resonance condition 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.

####  III.8.3 Two-qubit phase gates with analytical derivative removal by
adiabatic gate

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 $\Omega$ should be large. This can lead to large population leakage
to states that are near the optically-targeted Rydberg state. To tackle this
issue, Ref. Theis  _et al._ (2016) showed that by using the method of
analytical derivative removal by adiabatic gate (DRAG) Motzoi  _et al._
(2009); Motzoi and Wilhelm (2013), the population leakage can be well
suppressed when shaped but analytic Rydberg pulses are used for Rydberg
excitation. The analysis in Theis  _et al._ (2016) 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 Ref. Theis  _et al._ (2016) 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 Theis  _et al._ (2016) 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
Sec. II.3.2, and it is an open question whether the theory in Theis  _et al._
(2016) can be extended to protocols where strong enough dipole-dipole
interaction (which is not in the form of a pure energy shift) is considered.

####  III.8.4 Two-qubit phase gates by geometrical phases with adiabatic
excitation

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

It was proposed in Ref. Møller  _et al._ (2008) 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
$|0\rangle,~{}|1\rangle,~{}|2\rangle$ and a Rydberg state $|r\rangle$, where
$|0\rangle$ and $|1\rangle$ are qubit states. For one qubit upon which the
transitions $|1\rangle\leftrightarrow|2\rangle$ and
$|2\rangle\leftrightarrow|r\rangle$ are driven by external fields with Rabi
frequencies $\Omega_{1}$ and $e^{-i\phi_{r}}\Omega_{r}$, respectively, the
dark state of the three-level system is
$|D_{1}\rangle=\cos\theta|1\rangle-e^{i\phi_{r}}\theta|r\rangle$, where
$\tan\theta\equiv\Omega_{1}/\Omega_{r}$ Shi (2020c). With one adiabatic cycle
where the wavefunction evolves starting from $|1\rangle$ and ending at
$|1\rangle$, the state acquires a Berry phase $\varphi_{1}=-\int\sin^{2}\theta
d\phi_{r}$. What is interesting is that when the same pulse is applied to two
nearby atoms, although the single-atom dark state $|D_{1}\rangle$ no longer
exists, Ref. Møller  _et al._ (2008) pointed out that for large enough
blockade interaction there is another dark state $|D_{2}\rangle$, and there is
a time-dependent component in $|D_{2}\rangle$ with a phase term

$\displaystyle\mathscr{B}\equiv\langle
D_{2}|(|1r\rangle+|r1\rangle)=\frac{-\sin\theta\cos\theta
e^{-i\phi_{r}}}{\sqrt{\cos^{4}\theta+2\sin^{4}\theta}}.$ (80)

Then, by appropriately choosing a closed loop for the varying phase
$\phi_{r}$, one can simultaneously map $\\{|01\rangle,~{}|10\rangle\\}$ back
to themselves with a phase accumulation $\varphi_{1}$, and map $|11\rangle$ to
itself with another phase accumulation
$\varphi_{2}=-\int|\mathscr{B}|^{2}d\phi_{r}$. Here, one shall bare in mind
that $\varphi_{1}$ arises when only one atom responds to the excitation field,
while $\varphi_{2}$ arises when both qubits are optically excited, i.e., if
the input state is $|11\rangle$. When single-qubit rotations are applied, a
controlled-phase gate diag$\\{1,1,1,e^{\varphi_{2}-2\varphi_{1}}\\}$ is
realized with $\varphi_{2}-2\varphi_{1}$ tunable in the interval
$(0,~{}2\pi]$. By this method, Fig. 2(b) of Ref. Møller  _et al._ (2008)
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 $|D_{2}\rangle$ is only valid in the limit of large blockade
shift in $|rr\rangle$. The novelty of the scheme in Ref. Møller  _et al._
(2008) lies in that when it is applied to a multiqubit system, it can prepare
large-scale GHZ states. For more details, see Ref. Møller  _et al._ (2008);
besides, Ref. Wu  _et al._ (2017) did a more detailed analysis about the
scheme in Ref. Møller  _et al._ (2008) and studied some of its extensions.

Berry phases are particularly useful for designing quantum gates in ensemble
qubits Beterov  _et al._ (2013) which will be reviewed in Sec. III.8.7.

####  III.8.5 Two and three-qubit gates by geometrical phases with
nonadiabatic excitation

In Sec. III.5 we have reviewed Rydberg gates by geometric phases without using
pulse shaping. Although the gate in Zhao  _et al._ (2017, 2018) is implemented
with constant Rabi frequencies, cyclic state evolution occurs where geometric
phases are accumulated. In Ref. Kang  _et al._ (2018), another nonadiabatic
holonomic Rydberg quantum gate was studied based on defining qubit states by
superposition states of two hyperfine ground states. The gate in Ref. Kang
_et al._ (2018) is a C${}_{\text{Z}}$ 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 Ref. Kang  _et al._ (2018) considered using auxiliary atoms to
realize heralded nonadiabatic holonomic Rydberg gates Kang  _et al._ (2020).

Quantum gates can be efficiently created between a single atom and an ensemble
of atoms, shown in Ref. Guo  _et al._ (2020), which proposed a three-pulse
nonadiabatic non-Abelian geometric Rydberg controlled-$U(\theta,\phi)$ gate,
where
$U(\theta,\phi)=\sigma_{z}\cos\theta+(\sigma_{x}\cos\phi+\sigma_{y}\sin\phi)\cos\theta$
and the angles $(\theta,\phi)$ are determined by laser fields and dipole
transition matrix elements. The scheme in Ref. Guo  _et al._ (2020) is
particularly robust against the fluctuation of the field intensity in the
lasers. The gate in Ref. Guo  _et al._ (2020) can be easily tuned to various
forms including C${}_{\text{Z}}$ and CNOT by adjustment of parameters of laser
fields. In particular, Ref. Guo  _et al._ (2020) 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, Ref. Li  _et al._
(2021) extended this scheme and studied an interesting multiple-qubit CkU gate
based on inverse engineering, shortcut-to-adiabaticity, and optimal control.
In particular, Ref. Li  _et al._ (2021) showed that the operation U in the CkU
gate can be an arbitrary single-qubit operation in the target qubit, and
fidelities about 99.8% (at 0 K; see Sec. III D therein) are achievable for k
up to 4 when U is either the NOT or the Hadamard operation.

####  III.8.6 Multiqubit entanglement and phase gates by optimal control

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
$-F(a,~{}b,~{}c,~{}\cdots)$, 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 $a,~{}b,~{}c,~{}\cdots$). 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., Refs. Goerz  _et al._ (2011); Saffman  _et al._ (2020).

It was shown in Ref. Müller  _et al._ (2011) 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 Ref. Müller  _et al._
(2011)). Later on, Ref. Goerz  _et al._ (2014) showed that optimal control can
lead to high-fidelity two-qubit entangling gates between two optically trapped
neutral atoms. Importantly, the nonanalytical pulses in Ref. Goerz  _et al._
(2014) 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
Ref. Goerz  _et al._ (2014) and Ref. Müller  _et al._ (2011), 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 Ref. Goerz  _et al._ (2014)
can work equally well as the method in Ref. Theis  _et al._ (2016) 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, Ref. Omran  _et al._ (2019)
reported the creation of “Schrödinger cat” states of the GHZ type with qubit
numbers up to 20 using pulses found by optimal control. In Ref. Omran  _et
al._ (2019), 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 Ref. Omran  _et al._ (2019), 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 Ref. Omran  _et al._
(2019) rigorously showed that neutral atoms are competitive to superconducting
systems Song  _et al._ (2019); Wei  _et al._ (2020) regarding the capability
to create large-scale many-particle entanglement. Besides, the same groups
experimentally demonstrated Levine  _et al._ (2019), 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 Ref. Levine  _et al._ (2019), 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, Ref. Larrouy  _et al._ (2020) 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${}_{\text{Z}}$ gate was only recently proposed in Ref. Saffman  _et
al._ (2020). It showed that with experimentally feasible parameters, a
C${}_{\text{Z}}$ 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, Ref. Saffman
_et al._ (2020) 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 Ref.
Shi (2019) and experimentally realized in Ref. Levine  _et al._ (2019), the
C${}_{\text{Z}}$ gate protocol in Ref. Saffman  _et al._ (2020) 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.

####  III.8.7 Two-qubit gates and entanglement with two atomic ensembles

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 Xia  _et al._ (2015); Wang  _et
al._ (2015, 2016); Bernien  _et al._ (2017). If one would like to use this
type of atomic arrays for storing quantum information where each site has at
least one atom, Sec. II.1.2 shows that we can treat both single-atom states
$|0(1)\rangle$ as qubit states, and ensemble states

$\displaystyle|\overline{0}\rangle$ $\displaystyle=$ $\displaystyle|000\cdots
0\cdots 000\rangle,$ $\displaystyle|\overline{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}|000\cdots 1_{j}\cdots
000\rangle,$ (81)

as qubit states as well, where $N$ is the number of atoms in the ensemble. The
integer $N$ in Eq. (81) 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 $V_{\text{min}}$. If a rectangular Rydberg laser pulse is
applied for $|1\rangle\leftrightarrow|r\rangle$ with Rabi frequency $\Omega$
for one atom, the actual Rabi frequency for the transition from
$|\overline{1}\rangle$ to the ensemble state

$\displaystyle|\overline{r}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}|000\cdots r_{j}\cdots
000\rangle,$ (82)

is enhanced by $\sqrt{N}$, where $j=1,2,~{}\cdots,N$. 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 $\sqrt{N}\hbar\Omega\ll
V_{\text{min}}$ since $\hbar\Omega\ll V_{\text{min}}$. The issue is, how can
one excite the state of the atomic ensemble from $|\overline{1}\rangle$ to
$|\overline{r}\rangle$ by a given type (and duration) of pulse when $N$ is
random? The answer was given in Ref. Beterov  _et al._ (2011), which shows
that the excitation from $|\overline{1}\rangle$ to $|\overline{r}\rangle$ 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

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\left[\hbar\frac{\Omega(t)}{2}|\overline{r}\rangle\langle\overline{1}|+\text{H.c.}\right]+\hbar\frac{\delta(t)}{2}\left(|\overline{r}\rangle\langle\overline{r}|-|\overline{1}\rangle\langle\overline{1}|\right],$
(83)

which has two eigenstates

$\displaystyle|v_{+}\rangle$ $\displaystyle=$
$\displaystyle\left[\Omega(t)|\overline{r}\rangle+(\sqrt{\Omega^{2}(t)+\delta^{2}(t)}-\delta(t))|\overline{1}\rangle\right]/\mathcal{N}_{+},$
$\displaystyle|v_{-}\rangle$ $\displaystyle=$
$\displaystyle\left[\Omega(t)|\overline{r}\rangle-(\sqrt{\Omega^{2}(t)+\delta^{2}(t)}+\delta(t))|\overline{1}\rangle\right]/\mathcal{N}_{-},$
(84)

with eigenvalues $\epsilon_{\pm}=\pm\sqrt{\Omega^{2}(t)+\delta^{2}(t)}/2$ and
normalization factors
$\mathcal{N}_{\pm}=\sqrt{\Omega^{2}(t)+[\sqrt{\Omega^{2}(t)+\delta^{2}(t)}\mp\delta(t)]^{2}}$.
If initially $\Omega(0)$ is small compared to $\delta(0)$, $|v_{+}\rangle$ is
approximately $|\overline{r}\rangle$, and $|v_{-}\rangle$ is approximately
$|\overline{1}\rangle$. If both $\Omega(t)$ and $\delta(t)$ change smoothly
until at the end of the pulse when $\Omega(t)=\Omega(0)$ and
$\delta(t)=-\delta(0)$, $|v_{-}\rangle$ is approximately
$|\overline{r}\rangle$. This means that if we start from the state
$|\overline{1}\rangle$, the atom will be excited to the state
$|\overline{r}\rangle$. However, it does require smooth enough $\Omega(t)$ and
$\delta(t)$ with the condition
$|d\Omega(t)/dt|/\sqrt{\Omega^{2}(t)+\delta^{2}(t)}\ll 1$ and
$|d\delta(t)/dt|/\sqrt{\Omega^{2}(t)+\delta^{2}(t)}\ll 1$. 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. Ref.
Beterov  _et al._ (2011) showed that when the detuning at the intermediate
state is zero, the standard three-level STIRAP can not excite the ground state
$|\overline{1}\rangle$ to the Rydberg state $|\overline{r}\rangle$ because
there is no way to define a dark eigenstate like $(\Omega_{p\rightarrow
r}|\overline{1}\rangle-\Omega_{1\rightarrow
p}|\overline{r}\rangle)/\sqrt{\Omega_{p\rightarrow r}^{2}+\Omega_{1\rightarrow
p}^{2}}$, where $\Omega_{p\rightarrow r}$ and $\Omega_{1\rightarrow p}$ are
the Rabi frequencies for the upper and lower transitions. However, Ref.
Beterov  _et al._ (2011) 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 $|\overline{1}\rangle$ to
$|\overline{r}\rangle$ when the counterintuitive STIRAP pulse is applied;
compared to the two-state case in Eq. (83), 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
$|\overline{r}\rangle$ with $s(d)$ and $p$-orbital Rydberg states can be found
in Ref. Beterov  _et al._ (2014) and the review Beterov  _et al._ (2020).

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 Petrosyan and Mølmer (2013). Ref. Beterov  _et al._ (2013) found
that for different atom numbers, the phases of the Rydberg state
$|\overline{r}\rangle$ 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 solved by shifting the sign of the Rabi
frequencies and detuning used for the deexcitation of the Rydberg state.
Numerical simulation in Beterov  _et al._ (2013) 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 Ref. Beterov  _et al._ (2016a) 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 Ref. Beterov  _et al._ (2020).

Beside the above mentioned protocol of quantum entangling gates in two atomic
ensembles, another method was shown in Ref. Zhao  _et al._ (2018) depending on
nonadiabatic geometric phases, and Ref. Ji  _et al._ (2020) showed
entanglement generation between two atom ensembles assisted by pulse shaping
and by coupling cavity modes with the atoms. The method of Ref. Zhao  _et al._
(2018) depends on coupling atoms to cavity modes which is challenging, but it
does not depend on pulse shaping since one $\pi$ pulse is sufficient. The gate
in Ref. Zhao  _et al._ (2018) is a two-qubit entangling gate that has an
entangling power less than the maximal Williams (2011) as in a
C${}_{\text{Z}}$ or CNOT. But any two-qubit entangling gate can form a
universal set with several single-qubit gates, so the method in Ref. Zhao  _et
al._ (2018) could be useful if the required coupling between the Rydberg atoms
and cavity modes can be achieved. Moreover, Ref. Guo  _et al._ (2020) proposed
an optimized geometric quantum computation with a mesoscopic ensemble of
Rydberg atoms which was reviewed in Sec. III.8.5. However, different from the
methods in Beterov  _et al._ (2011, 2013, 2014, 2016a, 2020), the methods in
Refs. Zhao  _et al._ (2018); Guo  _et al._ (2020) assume that $N$ is a fixed
number.

Requirement | Theoretical fidelity; type of operation; where it’s taken | References
---|---|---
Interaction-induced phase shift (Sec. IV.1) | 99.3%; C${}_{\text{Z}}$ gate; Fig. 12(b) on page 15 of Ref. Shi and Kennedy (2017) a | Ref. Shi and Kennedy (2017)
$99.4\%$; Barenco gate; middle left of page 6 of Ref. Shi (2018c) a | Ref. Shi (2018c)
Interaction-induced motional change |  |  $99.98\%$; 2-qubit gate; Table 1 on page 383 of Ref. Cozzini  _et al._ (2006)
---
(Error due to Rydberg-state decay not included here)
Ref. Cozzini  _et al._ (2006)
Rabi-like rotations via Förster resonance (with pulse shaping except of Ref. Beterov  _et al._ (2018b)) (Sec. IV.3) |  $99.4\%$; CNOT gate; Fig. 6(h) on page 6 of Ref. Beterov  _et al._ (2016b) | Ref. Beterov  _et al._ (2016b)
$98.6\%$; Bell state; left column on page 8 of Ref. Beterov  _et al._ (2016b) | Ref. Beterov  _et al._ (2016b)
$97.9\%$; Bell state; Fig. 7 on page 7 of Ref. Beterov  _et al._ (2018a) | Ref. Beterov  _et al._ (2018a)
$99.8\%$; C${}_{\text{Z}}$ gate; upper right of page 4803 of Ref. Liao  _et al._ (2019) | Ref. Liao  _et al._ (2019)
$99\%$; C${}_{\text{Z}}$ gate; Fig. 8 on page 8 of Ref. Sun  _et al._ (2020) | Ref. Sun  _et al._ (2020)
$98.3\%$; Toffoli gate; text below Eq. (2) on page 6 of Ref. Beterov  _et al._ (2018b) | Ref. Beterov  _et al._ (2018b)
Dark states formed via Förster resonance (with pulse shaping) (Sec. IV.4) |  99.995%; C${}_{\text{Z}}$ gate; text below Fig. 3 on page 4 of Ref. Petrosyan  _et al._ (2017) | Ref. Petrosyan  _et al._ (2017)
$99.96\%$; 2-qubit gate;b abstract of Ref. Yu  _et al._ (2019) | Ref. Yu  _et al._ (2019)
${\color[rgb]{0,0,1}99.9}\%$; Toffoli gate; Fig. 6(c) on page 9 of Ref. Khazali and Molmer (2020) c | Ref. Khazali and Molmer (2020)
$98\%$; 21-qubit C20-NOT gate; Fig. 6(c) on page 9 of Ref. Khazali and Molmer (2020) c | Ref. Khazali and Molmer (2020)

Table 3:  Fidelities of quantum gates or entanglement via multiqubit Rydberg
excitation. Most numbers of fidelities quoted were calculated with some decay
rates around $2\pi\times 1$ kHz for the Rydberg state in the corresponding
references unless otherwise specified. The blue color highlights the largest
fidelities for gates and entanglement with two or more than two qubits.

  * a

The quoted data assume that the qubits are cooled near to the vibrational
ground states of trapped atoms.

  * b

The lower part of page 23085 of Ref. Yu  _et al._ (2019) indicates that the
decay rate of the Rydberg-state decay used in the estimate is $2\pi\times 0.2$
kHz; the gate in Ref. Yu  _et al._ (2019) is not an entangling gate.

  * c

Fig. 6(c) on page 9 of Ref. Khazali and Molmer (2020) showed fidelities with
Ck-NOT gates (which was termed Toffoli gate therein) with k from 1 to 20,
where the Toffoli gate with $k=3$ has a fidelity about $99.9\%$. Ref. Khazali
and Molmer (2020) also studied the fan-out gates C-NOTk which can have high
fidelities similar to those of the Ck-NOT gates.

##  IV Rydberg gates by exciting multiqubits to Rydberg states

There are Rydberg gates based on populating multi-atom Rydberg states, which
is in contrast to the protocols in Sec. III where multi-atom Rydberg states
should not be populated. In this section, a full population of multi-atom
Rydberg state is required in the methods in Sec. IV.1, while a partial
population is involved in the methods of Secs. IV.2, IV.3, and IV.4. Besides
the gate methods reviewed below, there are Rydberg gates based on partially
populating two-atom Rydberg states via the detuned Rabi oscillations Shi
(2017, 2019) that have been reviewed in Sec. III.2 because of their close
relevance to the other methods therein. Table 3 lists the published
theoretical fidelities with the places denoted about where to find those
numbers; in many references more than one type of quantum gates or entangling
operations were studied, and a relative larger fidelity is quoted in the
table.

###  IV.1 Two-qubit phase gates with dynamical phases from van der Waals
interactions

When two atoms are both in Rydberg states, the interaction between them can
induce phase shifts, which can be used to entangle two atoms shown in Fig.
5(b). The original proposal Jaksch  _et al._ (2000) needs to excite both
qubits to Rydberg states if they are initialized in $|11\rangle$, then wait
for some time $\pi\hbar/V$, and finally deexcite them back to ground states.
For a frozen $V$, this leads to a $\pi$ phase shift to the input state
$|11\rangle$. Because only the qubit state $|1\rangle$ is excited, the states
$|01\rangle$ and $|10\rangle$ also pick up a $\pi$ phase, leading to a
controlled-Z gate. A variation of this method was experimentally demonstrated
in Ref. Jo  _et al._ (2020). Section II.3.2 shows that when the two qubits are
placed along the quantization axis and excited to the same Rydberg state, a
van der Waals interaction appears as an energy shift from a second-order
perturbation theory. The energy shift is well suited for this protocol, but it
also means that the interaction is limited because the two-atom separation
should be sufficiently large. The entanglement fidelity $59\%$ reported in the
experiments of Ref. Jo  _et al._ (2020), however, pointed to the possibility
to employ interaction-induced dynamical phase for quantum information
processing in future if the error due to the fluctuation of qubit spacing can
be suppressed.

Ref. Shi and Kennedy (2017) showed that it is possible to realize a fast
Rydberg gate by the method of Jaksch  _et al._ (2000). The basic idea is to
place two qubits not too far away in the van der Waals regime, and excite both
qubits to a certain superposition of Rydberg states where the interaction is
annulled by state mixing via microwave fields. Then, a GHz-rate excitation by
microwave fields is used to excite the Rydberg state to other states where a
strong interaction arises, and a short wait duration induces a $\pi$ phase
shift. However, only by sufficiently cooling and using deep enough trap can
the error from the distance fluctuation of qubits be suppressed. With a trap
depth $1.5~{}$mK Kaufman  _et al._ (2012) and qubits cooled to the motional
ground states in the traps, Ref. Shi and Kennedy (2017) analyzed that a
fidelity about $99\%$ could be achieved for a C${}_{\text{Z}}$ gate. Though
this fidelity is smaller than most fidelities shown in Table 3, the gate has a
fast speed. It is an open question whether some robust protocols like those in
Beterov  _et al._ (2016b, 2018a) can be combined with the exotic annulled van
der Waals interaction in Shi and Kennedy (2017) for fast and robust
entanglement.

Ref. Shi (2018c) showed that by appropriately superposing different Rydberg
states, one can design the Barenco gate by using the dynamical phase from the
Rydberg interaction. Each Barenco gate is characterized by two specific angles
and can form a universal set by itself, i.e., based on the Barenco gate alone,
the universal quantum computation can be achieved Barenco (1995). The fidelity
of the Barenco gate is mainly limited by the Rydberg-state decay and
fluctuation of qubit spacings, which depend on the gate durations and further
on the two angles possessed in each Barence gate. By assuming cooling qubits
to the motional ground states in the optical traps, the fidelity of a Barenco
gate can approach $99.7\%$ shown in Fig. 6(a) of Ref. Shi (2018c) if qubits
are hosted in chambers at 4.2 K. The gate fidelities shown in Table 3 for the
Barenco gate are data with Rydberg-state decay at room temperature.

Compared to other methods, the Rydberg gates by phase shifts from van der
Waals interactions are quite sensitive to the fluctuation of interatomic
distances. Implementation of these protocols needs sufficiently cooled qubits
if a high fidelity is desired.

###  IV.2 Two-qubit phase gates via change of motional states by Rydberg
interactions

There is one gate proposal using vibrational states of atoms in dipole traps
for Rydberg-mediated entanglement Cozzini  _et al._ (2006) which is based on a
mechanism used for entangling trapped ions García-Ripoll  _et al._ (2003).
Instead of using Rydberg interactions to induce a phase shift by a blocked
Rabi transition or by a phase shift $\sim Vt/\hbar$, Ref. Cozzini  _et al._
(2006) analyzed a method to use the Rydberg interaction between two atoms
trapped in two traps to modify their vibrational states in their traps. By
choosing appropriate parameters and assuming ground-state vibration, it is
possible to induce a phase shift to the state $|11\rangle$ by letting the
atoms visit the two-atom Rydberg state for a little while, while the other
three input states will not pick up any extra phase because they do not
experience Rydberg interaction and thus their motional states are not altered.
The method has a fidelity limited by the Rydberg state decay, blockade error,
as well as an extra error from the thermal effect of the qubits unless ground-
state cooling is achieved, and Table 1 of Ref. Cozzini  _et al._ (2006) showed
that a fidelity $99.98\%$ (which does not include Rydberg-state decay) is
achievable with available experimental techniques. However, it is unclear
whether the gate can still obtain a large fidelity in the presence of the
fluctuation of $V$ for qubits whose motional states are far from the motional
ground states.

###  IV.3 Two and three-qubit phase gates by Förster-resonance-induced Rabi
rotations

For two atoms in a Rydberg state $|r_{1}r_{2}\rangle$, the electric dipole-
dipole interaction can flip $|r_{1}r_{2}\rangle$ to the state
$|r_{3}r_{4}\rangle$, where the parity of $|r_{1}\rangle$ shall be different
from that of $|r_{3}\rangle$, and the parity of $|r_{2}\rangle$ shall be
different from that of $|r_{4}\rangle$. However, there can be some energy
difference between the two states $|r_{1}r_{2}\rangle$ and
$|r_{3}r_{4}\rangle$, and $|r_{1}r_{2}\rangle$ can be flipped to a lot of
states other than $|r_{3}r_{4}\rangle$ which is the intrinsic nature of the
dipole-dipole interaction. An important special case is the Förster resonance
when two dipole-coupled states $|r_{1}r_{2}\rangle$ and $|r_{3}r_{4}\rangle$
have exactly the same energy, and other dipole-coupled states are largely
detuned. Then, one can isolate the dipole interaction
$|r_{1}r_{2}\rangle\leftrightarrow|r_{3}r_{4}\rangle$ from the manifold of
dipole-coupled states. This can be experimentally achieved by dc Stark shifts
Safinya  _et al._ (1981); Anderson  _et al._ (1998); Mourachko  _et al._
(1998); Westermann S.  _et al._ (2006); Nipper  _et al._ (2012); Richards and
Jones (2016); Kondo  _et al._ (2016) or by radio-frequency control Tretyakov
_et al._ (2014). This Stark-tuned dipole-dipole interaction between two
individual Rydberg atoms was first observed and reported in Ryabtsev  _et al._
(2010). Later on, three-body Förster resonances Faoro  _et al._ (2015);
Tretyakov  _et al._ (2017); Cheinet  _et al._ (2020) and even four-body
Förster resonances were reported Liu  _et al._ (2020a).

The Förster resonance can be used to generate entangling gates between two
Beterov  _et al._ (2016b, 2018a) or three qubits Beterov  _et al._ (2018b).
For two-qubit gates, the method consists of four steps which can be understood
by looking at the input state $|11\rangle$ since the other three input states
do not involve Rydberg interactions. (i) First, excite the state $|11\rangle$
to the Rydberg state $-|r_{1}r_{2}\rangle$ whose interaction is small compared
to the Rabi frequency of the Rydberg excitation. (ii) Second, tune the energy
difference between $|r_{1}r_{2}\rangle$ and $|r_{3}r_{4}\rangle$
adiabatically, from some large value to zero, and then to some large value
again. The pulse is designed so that the transition
$-|r_{1}r_{2}\rangle\rightarrow i|r_{3}r_{4}\rangle$ is realized (we assume
zero phase in the dipole-coupling Rabi frequency) as in a $\pi$-pulse Rabi
transition. (iii) Third, repeat the last step so that $i|r_{3}r_{4}\rangle$
transitions back to $|r_{1}r_{2}\rangle$. (iv) Fourth, repeat the first step
so that the input state $|r_{1}r_{2}\rangle$ returns to $-|11\rangle$. Because
the adiabatic following of the Förster resonance does not occur for the other
three input states, a C${}_{\text{Z}}$ gate map
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}\rightarrow-\\{-|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$
can be realized. Compared to the method of directly using the $\pi$ phase
shift in the Rabi-like dipole-dipole flip
$|r_{1}r_{2}\rangle\leftrightarrow|r_{3}r_{4}\rangle$, the adiabatic tuning of
the Förster resonance has a much smaller error from the fluctuation of the
interatomic distance, as numerically demonstrated in Refs. Beterov  _et al._
(2016b, 2018a) which showed that high-fidelity CNOT gates can be prepared;
note, however, that these numbers for the fidelity assume constant dipole-
dipole interactions. As shown in Fig. 8 on page 7 of Ref. Beterov  _et al._
(2018a), there will be extra errors in the gate fidelity from the fluctuation
of the interatomic distance which leads to fluctuation of the the dipole-
dipole interactions.

In Ref. Beterov  _et al._ (2018b) a fast three-qubit Toffoli gate was proposed
by using phase shift from resonant Borromean three-body interactions. The data
below Eq. (2) of Ref. Beterov  _et al._ (2018b) shows that an intrinsic gate
fidelity $98.3\%$ is realizable with realistic parameters for the dipole-
dipole interactions, where the gate fidelity is limited mainly by the Rydberg-
state decay. However, it is an open question whether there can be an adiabatic
three-qubit gate protocol like the two-qubit gates in Beterov  _et al._
(2016b, 2018a) where the error due to the fluctuation of the interatomic
distance can be partially suppressed.

It is also possible to combine Förster resonance with pulse shaping for
Rydberg gates, as shown in Refs. Liao  _et al._ (2019); Sun  _et al._ (2020).
The difference between Ref. Liao  _et al._ (2019) and Sun  _et al._ (2020) is
that adiabatic following of a dark state was used in Sun  _et al._ (2020), but
the scheme in Ref. Liao  _et al._ (2019) can obtain geometric phases for
certain input states with either adiabatic or nonadiabatic excitations.

Although the Förster-resonance method has a fidelity limited by the finiteness
of Rydberg interactions, Rydberg-state decay, and Doppler dephasing, it has
several special attractions. A special benefit of using Förster-resonant Rabi-
like rotations for Rydberg quantum gates is that the distance $L$ between two
qubits can be quite large so as to easily excite two or three qubits to
Rydberg states. This leads to three positive consequences. First, the
fluctuation of the dipole-dipole coupling strength $C_{3}/L^{3}$ will only
have a small fluctuation ratio $3\delta L/L$ when the distance fluctuation
$\delta L$ is much smaller than $L$; this scaling compares favorably to gates
by a van der Waals interaction $C_{6}/L^{6}$ where a distance fluctuation
leads to a change of interaction $6\delta L/L$. Second, the large distance
drastically reduces crosstalk between qubits when single-qubit Rydberg
addressing is used. Third, it can be used to couple two qubits far away in a
dense qubit array. For example, the lattice constant in the optical lattice in
the experiment of Graham  _et al._ (2019) was $3.1~{}\mu$m , and the
parameters used therein are unlikely able to entangle two qubits separated by
more than three lattice periods; however, it was shown in Beterov  _et al._
(2016b, 2018a) that CNOT gates in two qubits separated by $L=25~{}\mu$m can
have a fidelity $99.4\%$ by the Förster-resonance method (see Fig. 6(h) on
page 6 of Ref. Beterov  _et al._ (2016b)). Thus, the Förster-resonance method
can be useful for large-scale quantum computing where entanglement of two
well-separated qubits is required. The main challenge is to accurately isolate
the desired Förster-resonant processes.

###  IV.4 Two-qubit and multiqubit gates by dark states formed with Förster
resonance

There is a special way to use adiabatic following of a dark state formed by
Förster resonance for Rydberg-mediated entanglement Petrosyan  _et al._
(2017). Consider a three pulse sequence of the original method shown in Eq.
(54) and Fig. 5(a), where the key step is the blockade that leads to no phase
shift to the input $|11\rangle$ during the $2\pi$ pulse. Now, consider the
case that the $2\pi$ pulse is replaced by an adiabatic pulse that has the same
pulse area, but can allow an adiabatic following after the dark state
$|D\rangle\propto 2V|r_{1}1\rangle-\Omega(t)|r_{3}r_{4}\rangle$ of the
relevant Hamiltonian

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\left[\frac{\hbar\Omega(t)}{2}|r_{1}r_{2}\rangle\langle
r_{1}1|+V|r_{1}r_{2}\rangle\langle r_{3}r_{4}|\right]+\text{H.c.},$ (85)

where $|r_{1}\rangle$ is the Rydberg state excited during the first pulse on
the control qubit, and $|r_{2}\rangle$ is the Rydberg state that is excited
during the adiabatic pulse on the target qubit. Even if the dipole-dipole
coupling strength $V$ fluctuates from shot to shot, the dark state persists
for each cycle of the gate where $V$ is assumed to be constant. By smoothly
varying $\Omega(t)$ from 0 to some large value, and back to 0 again, the state
$|r_{1}1\rangle$ follows after the dark state $|D\rangle$ and ends at
$|r_{1}1\rangle$ without any phase change in it. This is remarkable in that
although the two-atom Rydberg state is populated during the adiabatic pulse,
no phase accumulates because the eigenenergy of the dark state is zero in the
rotating frame. After the adiabatic pulse, another pulse is applied for
restoring the control qubit to the ground state, resulting in a
C${}_{\text{Z}}$ gate as in Eq. (54). The method in Petrosyan  _et al._ (2017)
is not only insensitive to the fluctuation of $V$ in different gate cycles,
but comes with less blockade error compared to the initial Rydberg blockade
gate proposed in Jaksch  _et al._ (2000). As a consequence, it is possible to
have an extremely large intrinsic fidelity $99.995\%$ for a C${}_{\text{Z}}$
gate with this method. The fidelity reported there is in principle robust
against the change of interactions from shot to shot, but the influence on the
adiabatic following by the change of $V$ and the Doppler dephasing due to the
free flight of qubits within one gate cycle is unclear.

Figure 7: Results of numerical simulation for the state transform from
$\\{|00\rangle,~{}|01\rangle,~{}-i|r_{1}0\rangle,~{}-i|r_{1}1\rangle\\}$ to
$\\{|00\rangle,~{}-|01\rangle,~{}-i|r_{1}0\rangle,~{}-i|r_{1}1\rangle\\}$ in
the second pulse of the gate proposed in Ref. Petrosyan  _et al._ (2017) for
different atomic temperatures $T_{a}$; different temperatures lead to
different atomic speeds $v_{\text{rms}}=\sqrt{k_{B}T_{a}m}$, and a larger
speed results in larger dephasing of the ground-Rydberg transition. The pulse
is a shifted Gaussian one of duration $T_{t}=0.29~{}\mu$s. (a) The phase
accumulated by the input state $|01\rangle$ at the end of the adiabatic pulse.
Ideally the phase is $\pi$ indicated by the horizontal dashed line. (b) The
phase acquired (scaled by $10^{4}$) by the state $-i|r_{1}0\rangle$ upon the
adiabatic pulse. Ideally it should be zero. (c) The error (scaled by $10^{4}$)
in the fidelity $f$ for the quantum operation of the adiabatic pulse averaged
over the four input states.

To investigate the robustness of the adiabatic following of Ref. Jaksch  _et
al._ (2000), we consider the input states
$\\{|00\rangle,~{}|01\rangle,~{}-i|r_{1}0\rangle,~{}-i|r_{1}1\rangle\\}$ at
the beginning of the the adiabatic pulse. The desired output states for the
adiabatic pulse are
$\\{|00\rangle,~{}-|01\rangle,~{}-i|r_{1}0\rangle,~{}-i|r_{1}1\rangle\\}$. By
using
$\Omega=\Omega_{m}[e^{-(t-T_{t}/2)^{2}/(2\sigma^{2})}-e^{-T_{t}^{2}/(8\sigma^{2})}$
and a dipole-dipole coupling coefficient $C_{3}/(2\pi)=-33$GHz$\mu m^{3}$ of
the states
$\\{|r_{1}\rangle,~{}|r_{2}\rangle\\}\equiv\\{95p_{3/2},84p_{3/2}\\}$ (see
Table I on page 5 of Ref. Petrosyan  _et al._ (2017)), we numerically study
the fidelity $f$ of the operation of the adiabatic pulse by using the formula
of Eq. (69). Because the 133Cs $p$-orbital Rydberg states are usually excited
by one-photon excitation with 319 nm lasers Hankin  _et al._ (2014), it is in
general not easy to have a large Rabi frequency. So, we consider a peak Rabi
frequency $\Omega_{m}/(2\pi)=7.643$ MHz and $T_{t}=0.29~{}\mu$s which is 10
time longer than the adiabatic pulse used in Appendix A2 of Ref. Petrosyan
_et al._ (2017). We suppose that the speed of the atom along the direction of
the Rydberg laser field is $v_{\text{rms}}=\sqrt{k_{B}T_{a}m}$, where $k_{B}$
is the Boltzmann constant, $T_{a}$ is the atomic temperature, and $m$ is the
mass of the atomic qubit. Figure 7(a) shows that the state $|01\rangle$
obtains some phase error upon the adiabatic excitation pulse; the dashed line
shows the correct phase $\pi$. For the state $-i|r_{1}1\rangle$, Fig. 7(b)
shows that the phase error is below $1.2\times 10^{-3}$ radians for $T_{a}\leq
50~{}\mu$K. The population errors in both $|01\rangle$ and $-i|r_{1}1\rangle$
are negligible. The total error for the operation of the adiabatic pulse
averaged over the four states is shown in Fig. 7(c). Note that the errors
shown in Fig. 7(c) should be smaller than the final gate error since the
operation fidelities in the first and third pulses of the gate of Ref.
Petrosyan  _et al._ (2017) are not included. The data in Fig. 7(b) rigorously
show that the adiabatic following for the input state $|11\rangle$ is robust
against the fluctuation of the interaction as well as the Doppler dephasing.
But the data in Fig. 7(c) indicate that the Doppler dephasing still hampers
the operation fidelities since the adiabatic following does not take effect
for the other two input states $|01\rangle$ and $|10\rangle$ nor in the first
and third pulses.

The method in Petrosyan  _et al._ (2017) can be extended in various ways. For
one example, Ref. Yu  _et al._ (2019) used a similar method to study a quantum
gate with the Rydberg interactions of both the direct dipole-dipole type and a
van der Waals type, and found that fidelities 99.96% are realizable with a
Rydberg-state decay rate $2\pi\times 0.2$ kHz for a certain gate with a map
diag$\\{1,~{}-1,~{}-1,~{}1\\}$ (see Fig. 2 on page 23083 of Ref. Yu  _et al._
(2019)). This gate induces no entanglement regarding the entangling power
Williams (2011) and can be easily created by using two single-qubit
$\pi$-phase gates on the qubits. However, it should be useful to extend the
method in Ref. Yu  _et al._ (2019) to design an entangling gate. For another
example, Ref. Khazali and Molmer (2020) extended the scheme in Petrosyan  _et
al._ (2017) to protocols for creating multiqubit Ck-NOT and C-NOTk gates.
Detailed analysis in Khazali and Molmer (2020) showed that for a multiqubit
system, there is still a dark state useful for entanglement generation via the
adiabatic following. Figure 6(c) in Ref. Khazali and Molmer (2020) shows that
a fidelity about $98\%$ with $k$ up to 20 is possible. Ref. Khazali and Molmer
(2020) also extended the scheme to superconducting systems because they also
possess the Ising-type interactions like the Förster resonance of Rydberg
atoms. It is worth mentioning that the Ising-like coupling shared in different
physical systems has been well recognized in the study of multiqubit quantum
gates Rasmussen  _et al._ (2020). Finally, we note that for all these gates,
the relatively long gate durations can lead to large Rydberg-state decay. For
example, the gate fidelity has a large sensitivity to the Rydberg decay rate
as analyzed in Ref. Yu  _et al._ (2019) (see Fig. 5 on page 23088 therein).

##  V Rydberg gates and entanglement by antiblockade

It was found in Ates  _et al._ (2007); Amthor  _et al._ (2010) that when the
Rydberg interaction $V$ is exactly equal to the detuning of a Rydberg Rabi
frequency, it appears as a resonance in the rotating frame so that the two-
atom Rydberg state can be fully excited. The antiblockade-induced resonance
can also appear in more than two atoms Pohl and Berman (2009); Mazza  _et al._
(2020); Gambetta  _et al._ (2020) which can lead to experimentally observable
many-body effects Bai  _et al._ (2020). In Table 4, we list the theoretical
fidelities in the literature with the places denoted to point out where the
fidelity number appears in the corresponding reference; in many references
more than one type of quantum gates or entangling operations were studied, and
a relatively large fidelity is quoted in the table.

Figure 8: Two typical strategies to use the antiblockade effect. (a) By
sending to two qubits the same laser fields for Rydberg excitation with Rabi
frequency $\Omega$ and detuning $\Delta$, the state $|11\rangle$ can be
excited to $|rr\rangle$ with an effective Rabi frequency $\Omega^{2}/\Delta$
when $V=2\hbar\Delta$. (b) If the laser fields are resonant but periodically
varying so that the Rydberg Rabi frequency obtains a time dependence as
$\Omega\cos(\omega t)$, the state $|11\rangle$ can be excited to $|rr\rangle$
with an effective Rabi frequency $\Omega^{2}/(4\omega)$ when $V=2\hbar\omega$.
These two methods are the representative strategies, based on which there are
various antiblockade methods reviewed in Sec. V.

Briefly speaking, there are two general strategies to realize the
antiblockade, by matching the interaction with frequency detuning of the
lasers, and by matching the interaction with a modulation frequency of the
amplitude of the laser fields, as schematically shown in Fig. 8 with their
most basic protocols. The first strategy has been widely studied so that
several ways have been found to use it for entanglement. Overall, there are at
least five ways to induce entanglement by antiblockade.

###  V.1 Compensating Rydberg interactions by detunings in Rabi frequencies

####  V.1.1 Simultaneous excitation of multi-atoms with detuned laser fields

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 Ref. Zuo and Nakagawa (2010), which presented protocols for
entanglement (such as the GHZ states) between ground and Rydberg states. The
model in Ref. Zuo and Nakagawa (2010) 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, Refs.
Shao  _et al._ (2014a); Tian  _et al._ (2015); Su  _et al._ (2016); Zhu  _et
al._ (2019) 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 Shao  _et al._ (2014a) (see text
below Eq. (9) on page 831 of Ref. Shao  _et al._ (2014a)). The perturbation
used in Refs. Shao  _et al._ (2014a); Su  _et al._ (2016) led to ac Stark
shifts that can be canceled by adding additional control fields. To avoid
adding extra control fields, Ref. Su  _et al._ (2017b) 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 Ref. Su  _et al._
(2018) 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, Ref. Li and Shao (2018)
studied three-qubit gates and quantum error correction by a type of
“Unconventional Rydberg pumping”, Refs. Su  _et al._ (2020b); Zheng  _et al._
(2020a) studied a novel nondestructive parity meter in ground-state atoms,
Ref. Xing  _et al._ (2020) studied a three-qubit controlled-phase gate with a
tunable phase, Refs. Zheng  _et al._ (2020b); Haase  _et al._ (2021) showed
methods to inter-convert the GHZ and W states, and Ref. Liu  _et al._ (2020b)
studied nonadiabatic noncyclic geometric phases in a Rydberg gate based on
which Ref. Guo  _et al._ (2021) 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, Ref. Yin  _et al._ (2020) used this strategy to show that high-
fidelity three and four-qubit controlled-NOT gates can be created.

####  V.1.2 Sequential excitation of atoms with detuned laser fields

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 Refs. Shao  _et al._ (2014a); Su  _et al._ (2016,
2017b, 2018). The method to entangle two and three atoms by this fast
antiblockade strategy was analyzed in detail in Ref. Su  _et al._ (2017a).

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 Refs. Tian  _et al._
(2015); Yan  _et al._ (2021), that used pulse shaping and antiblockade for
entanglement generation, where Ref. Yan  _et al._ (2021) is not quoted in
Table 4 because it did not show achievable gate fidelities by their scheme.

Requirement | Fidelity;a type of operation; where it’s taken | References
---|---|---
Higher-order $\Omega$ via matching the detuning in laser fields with Rydberg interactions (Sec. V.1.1) | 99.68%; Bell state (via pulse shaping); lower left of page 4 of Ref. Tian  _et al._ (2015) | Ref. Tian  _et al._ (2015)
$99.95\%$; Bell state; Fig. 10(a) of page 7 of Ref. Su  _et al._ (2017b) | Ref. Su  _et al._ (2017b)
$99.73\%$; 2-qubit entanglement; Fig. 9 of page 1943 of Ref. Zhu  _et al._ (2019) | Ref. Zhu  _et al._ (2019)
$98\%$; 2-qubit entanglement; Fig. 5(b) of page 5 of Ref. Liu  _et al._ (2020b) | Ref. Liu  _et al._ (2020b)
$96.54\%$; 3-qubit gate; lower right of page 829 of Ref. Shao  _et al._ (2014a) | Ref. Shao  _et al._ (2014a)
$96.79\%$; 3-qubit gate; text below Fig. 6 of page 6 of Ref. Su  _et al._ (2018) | Ref. Su  _et al._ (2018)
$97.34\%$; 3-qubit entanglement; upper right of page 8 of Ref. Li and Shao (2018) | Ref. Li and Shao (2018)
$98.87\%$; Toffoli gate; middle right of page 5 of Ref. Xing  _et al._ (2020) | Ref. Xing  _et al._ (2020)
${\color[rgb]{0,0,1}99.65}\%$; Toffoli gate; Table 1 on page 35583 of Ref. Yin  _et al._ (2020) | Ref. Yin  _et al._ (2020)
Sequential excitation |  $99.9\%$; 2-qubit state; lower right of page 8 of Ref. Su  _et al._ (2017a) (fast)b | Ref. Su  _et al._ (2017a)
Ground-state blockade (Sec. V.2) |  $99.32\%$; Bell state (via pulse shaping); text below Fig. 4 on page 4 of Ref. Shao  _et al._ (2017a) | Ref. Shao  _et al._ (2017a)
$99.89\%$; Bell state (via cavity); text below Fig. 5 on page 4 of Ref. Shao  _et al._ (2017c) | Ref. Shao  _et al._ (2017c)
$99.66\%$ (at $0$ K); C${}_{\text{Z}}$ gate; text below Fig. 6 on page 6 of Ref. Shao (2020) (fast)c | Ref. Shao (2020)
Dissipation with asymmetric interactions (Sec. V.3.1) |  $99.88\%$; Bell state; upper right of page 3 of Ref. Carr and Saffman (2013) | Ref. Carr and Saffman (2013)
$99.91\%$; 2-qubit entanglement; text below Fig. 2 on page 4 of Ref. Shao  _et al._ (2014b) | Ref. Shao  _et al._ (2014b)
$99.47\%$; 3-qubit entanglement; text below Fig. 4 on page 4 of Ref. Chen  _et al._ (2017) | Ref. Chen  _et al._ (2017)
$99.09\%$; 3-qubit entanglement (via cavity); lower left of page 1642 of Ref. Li  _et al._ (2018a) | Ref. Li  _et al._ (2018a)
Dissipation (Sec. V.3.2) |  $99.9\%$; Bell state; Fig. 2 on page 3 of Ref. Su  _et al._ (2015) | Ref. Su  _et al._ (2015)
$99\%$; Bell state (via cavity); text above Fig. 7 on page 5 of Ref. Chen  _et al._ (2018) | Ref. Chen  _et al._ (2018)
$99.7\%$; 2-qubit entanglement; end of Sec. 4 on page 2300 of Ref. Li  _et al._ (2018b) | Ref. Li  _et al._ (2018b)
${\color[rgb]{0,0,1}99.98}\%$; 2-qubit entanglement; text above Fig. 3 on page 10124 of Ref. Jin  _et al._ (2021) | Ref. Jin  _et al._ (2021)
$98.24\%$; 3-qubit entanglement (via cavity); lower left of page 5 of Ref. Shao  _et al._ (2017d) | Ref. Shao  _et al._ (2017d)
$99\%$; 6-qubit entanglement; upper right of page 4 of Ref. Wintermantel  _et al._ (2020) | Ref. Wintermantel  _et al._ (2020)
$99.24\%$; 3-qubit entanglement; abstract of Ref. Yang  _et al._ (2021) | Ref. Yang  _et al._ (2021)
$98\%$; 3-qubit entanglement; abstract of Ref. Li and Yang (2021) | Ref. Li and Yang (2021)
Compensating Rydberg interactions by using oscillating $\Omega$ (Sec. V.4) |  $99.35\%$; CNOT gate; middle left on page 4 of Ref. Wu  _et al._ (2020a) | Ref. Wu  _et al._ (2020a)
$99.6\%$; C${}_{\text{Z}}$ gate; text below Eq. (10) on page 1203 of Ref. Wu  _et al._ (2020b) | Ref. Wu  _et al._ (2020b)
$98\%$; Bell state; Fig. 9(c) on page 8 of Ref. Li  _et al._ (2020) | Ref. Li  _et al._ (2020)
$99\%$; C${}_{\text{Z}}$ gate; lower left of page 7 of Ref. Wu  _et al._ (2021a) (fast) b | Ref. Wu  _et al._ (2021a)
$99.1\%$; SWAP gate; texts below Fig. 3 and Fig. 4(a) on pp. 816 and 817 of Ref. Wu  _et al._ (2021c) | Ref. Wu  _et al._ (2021c)

Table 4:  Fidelities of quantum gates and entanglement with antiblockade. The
order of the list follows mainly after the order of the discussions in the
text. Two-qubit operations are first shown, then multiqubit ones according to
their publication dates. Here, “2-qubit” and “3-qubit” are labeled for phase
gates that can be transformed to C${}_{\text{Z}}$, CNOT, or Toffoli gates via
single-qubit gates. Most numbers of fidelities quoted were calculated with
some decay rates around $2\pi\times 1$ kHz for the Rydberg state in the
corresponding references unless otherwise specified. The blue color highlights
the largest fidelities for gates and entanglement with two or more than two
qubits.

  * a

Most references show more than one type of gates or entanglement, where an
operation with larger fidelity is quoted.

  * b

Here, “fast” only means that they do not depend on Rydberg excitation with a
Rabi frequency derived with a perturbation theory via the antiblockade, while
a practical experimental implementation has a speed limited by experimentally
feasible parameters.

  * c

In Ref. Shao (2020) a relative slow gate was shown when taking realistic data
though the theory does not involve high-order Rydberg Rabi frequencies derived
by the antiblockade conditions. The text below Fig. 4 on page 5 of Ref. Shao
(2020) shows that state decay at 0 K was considered.

###  V.2 Blocking transitions from certain two-atom ground states by
antiblockade

The second method of quantum gates with antiblockade is to use Rydberg
antiblockade to design an effective ground-state blockade, as proposed in Ref.
Shao  _et al._ (2017a) with van der Waals interaction and Ref Shao (2020) with
isolated dipole-dipole interactions.

####  V.2.1 Ground-state blockade via Rydberg antiblockade with van der Waals
interactions

Consider the Raman transition with an effective Rabi frequency
$\Omega_{\text{hyper}}$ between $|0\rangle$ and $|1\rangle$ via a largely
detuned low-lying $p$-orbital state, there will be the following transition
chain

$\displaystyle|00\rangle\leftrightarrow(|01\rangle+|10\rangle)/\sqrt{2}\leftrightarrow|11\rangle.$
(86)

However, if one excites $|11\rangle$ to a two-atom Rydberg state $|rr\rangle$
via the Rydberg antiblockade mechanism Shao  _et al._ (2014a); Su  _et al._
(2016, 2017b, 2018) so that $|11\rangle\leftrightarrow|rr\rangle$ is realized
with a Rabi frequency $\Omega_{\text{Ryd}}$ and detuning $\Delta$, there will
be Stark shifts $(\Delta\pm\sqrt{\Delta^{2}+\Omega_{\text{Ryd}}^{2}})/2$ for
$|11\rangle$, which can be large compared to $\Omega_{\text{hyper}}$ so as to
block the transition from $(|01\rangle+|10\rangle)/\sqrt{2}$ to $|11\rangle$.
This means that there will be an isolated transition from $|00\rangle$ to the
Bell state $(|01\rangle+|10\rangle)/\sqrt{2}$, so that entanglement can be
generated rapidly. This idea is similar to the dressing method reviewed in
Sec. III.3. The ground-state blockade mechanism can also entangle three atoms
Shao  _et al._ (2017a) and can be used to generate entanglement by dissipation
in the context of cavity quantum electrodynamics Shao  _et al._ (2017c).

####  V.2.2 Ground-state blockade via Rydberg antiblockade with dipole-dipole
interactions

Another ground-state blockade mechanism with isolated dipole-dipole
interactions similar to the one in Shao  _et al._ (2017a) was proposed in Ref.
Shao (2020). The dipole-dipole interactions employed in Ref. Shao (2020)
couple three two-atom Rydberg states, and by coupling each of two nearby atoms
for the transition $|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$
with appropriate Rabi frequencies and detunings (see Fig. 1 of Ref. Shao
(2020)), it is possible to create a transition between $|11\rangle$ and
$(|1r\rangle+|r1\rangle)/\sqrt{2}$ while the other states
$|00\rangle,~{}|01\rangle$, and $|10\rangle$ 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 $|11\rangle$ and $(|1r\rangle+|r1\rangle)/\sqrt{2}$, a
C${}_{\text{Z}}$ gate can be created. However, the antiblockade 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.

###  V.3 Quantum gates and entanglement by dissipation

####  V.3.1 Entanglement with dissipation and asymmetries in Rydberg
interactions

The third method of quantum gates by antiblockade is to use asymmetric Rydberg
interactions and dissipation, as proposed in Ref. Carr and Saffman (2013). The
asymmetric Rydberg interaction can be used in the excitation blockade regime
as reviewed in Sec. III.6. To use it in the antiblockade regime, Ref. Carr and
Saffman (2013) assumed that the interactions $V_{\text{ss}},~{}V_{\text{sp}}$
and $V_{\text{pp}}$ for the two-atom Rydberg states
$|ss\rangle,~{}|sp(ps)\rangle$ and $|pp\rangle$ satisfy the condition
$V_{\text{pp}}=V_{\text{ss}}\neq V_{\text{sp}}$, which can be realized by
choosing different Zeeman substates $|s\rangle$ and $|p\rangle$ within one
Rydberg level. Then, by using Rydberg excitation fields with a detuning that
exactly compensates the Rydberg interaction $V_{\text{pp}}=V_{\text{ss}}$, one
can populate the states $|ss\rangle$ and $|pp\rangle$ respectively from
$|00\rangle$ and $|11\rangle$ via the excitation
$|0\rangle\xrightarrow[]{\Omega_{\text{eff}}}-i|s\rangle$ and
$|1\rangle\xrightarrow[]{\Omega_{\text{eff}}}-i|p\rangle$, where
$\Omega_{\text{eff}}=2\hbar\Omega^{2}/V_{\text{ss}}$ is a Rabi frequency from
the second-order perturbation theory. The decay of the states $|ss\rangle$ and
$|pp\rangle$ will populate the states $|00\rangle,~{}|01\rangle,~{}|10\rangle$
and $|11\rangle$ with an equal probability, but among which only $|00\rangle$
and $|11\rangle$ will be reexcited to the two-atom Rydberg states which
enables recycling. Then, it is possible to create an entangled state
$(|01\rangle+e^{i\alpha}|10\rangle)/\sqrt{2}$ with an undetermined phase. By
adding a Raman transition between $|0\rangle$ and $|1\rangle$, Ref. Carr and
Saffman (2013) showed that the maximally entangled state with $\alpha=\pi$ 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 Carr and Saffman (2013) can also generate anti-ferromagnetic state
where a purity of $96\%$ was found in a 4-atom system. Later on, Ref. Shao
_et al._ (2014b) extended the scheme to prepare an entangled state in the form
of, in the two-atom case, $\sum_{k=1}^{3}|g_{k}g_{k}\rangle/\sqrt{3}$, i.e., a
maximally entangled state between three ground states $|g_{k}\rangle$ with
$k=1,2,3$. Besides depending on dissipation and antiblockade, Ref. Shao  _et
al._ (2014b) requires the interactions $V_{ij}$ of the Rydberg states
$|r_{i}r_{j}\rangle$ to satisfy the condition $V_{ii}=V_{jj}\neq V_{ij}$,
where $i,j\in\\{1,2,3\\}$ and $i\neq j$. When the Rydberg interactions between
three atoms satisfy specific relations, more complicated entanglement can
arise. For example, Ref. Chen  _et al._ (2017) showed that a three-qubit
singlet state can be created, and Ref. Li  _et al._ (2018a) showed that a
three-qubit $W$ state can be created by coupling atoms with cavity modes.

####  V.3.2 Entanglement with dissipation

The methods reviewed in Sec. V.3.1 depend on both dissipation and asymmetries
in the Rydberg interactions. Soon after the introduction of the methods of
Ref. Carr and Saffman (2013), Ref. Su  _et al._ (2015) put forward a concise
scheme using dissipation and antiblockade to achieve the goal of Refs. Carr
and Saffman (2013); Shao  _et al._ (2014b). Ref. Su  _et al._ (2015) showed
that two transitions are sufficient, namely, a Rydberg excitation
$|0\rangle\xrightarrow[]{}-i|r\rangle$ (detuned by $\Delta$) and a resonant
Raman transition $|0\rangle\xrightarrow[]{}-i|1\rangle$. An appropriate choice
of $\Delta$ can block the transition from
$|D\rangle\equiv(|01\rangle-|10\rangle)/\sqrt{2}$ to Rydberg states, but any
two-atom states not identical to $|D\rangle$ can be excited to Rydberg states
that will then decay to $|D\rangle$. Then, $|D\rangle$ 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 Refs. Li  _et al._
(2018b); Wintermantel  _et al._ (2020); Yang  _et al._ (2021); Li and Yang
(2021). Dissipation can also be used to generate entanglement with dipole-
dipole flip-flop processes as shown in Ref. Jin  _et al._ (2021). Unlike that
in Sec. V.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 Sec. V.3.1, the scheme based on dissipation and
antiblockade could also be used in the context of cavity quantum
electrodynamics Shao  _et al._ (2017d); Chen  _et al._ (2018).

###  V.4 Compensating Rydberg interactions by periodically changing Rydberg
Rabi frequencies

The fourth method of quantum entanglement via antiblockade is to use
periodically changing Rydberg Rabi frequencies to compensate the Rydberg
interactions. In Ref. Wu  _et al._ (2020a), it was shown that instead of using
the mismatch between the laser frequency and atomic transition frequency to
compensate the Rydberg interactions as in the above methods, laser fields with
amplitudes that are periodically alternating can effectively compensate the
Rydberg interactions. Because the laser fields used in the Rydberg excitation
have wavelengths of hundreds of nanometers, one can still use rotating-wave
approximation to derive a Hamiltonian for the atom-laser interaction,

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\frac{\hbar\Omega}{2}|r\rangle\langle 1|+\text{H.c.},$ (87)

where $|1\rangle$ and $|r\rangle$ are the ground and Rydberg states,
respectively, and $\Omega=\Omega_{m}\cos(\omega t)$, where the engineered
modulation has an angular frequency $\omega$ that is orders of magnitude
smaller than the central frequency of the lasers. Then, using the same
strategy for deriving Eq. (87) and the rotating-wave approximation conditioned
on $\omega\ll\Omega_{m}$, one can again use a rotating-frame transform to
derive

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\left(\frac{\hbar\Omega_{m}}{4}|r\rangle\langle
1|+\text{H.c.}\right)+\hbar\omega|r\rangle\langle r|,$ (88)

where the last term above can also be $-\hbar\omega|r\rangle\langle r|$ if
another rotating frame is employed. The last term in Eq. (88), then, becomes
an effective detuning that can be used to compensate the Rydberg interaction
involved in the two-atom state $|rr\rangle$. Based on this formalism, Ref. Wu
_et al._ (2020a) showed C${}_{\text{Z}}$ and CNOT gate protocols by using the
time-dependent Rabi oscillations with two atoms. The scheme in Ref. Wu  _et
al._ (2020a) leads to an effective Rabi frequency via the second-order
perturbation theory, where a detailed analysis showed that high-fidelity
C${}_{\text{Z}}$ gates can be realized Wu  _et al._ (2020b). To realize a
faster gate, Ref. Wu  _et al._ (2021a) proposed to excite one qubit via
constant Rydberg Rabi frequency, but to excite the other qubit via a
periodically alternating laser fields with a Hamiltonian of Eq. (88). Then,
resonant Rydberg excitations can be realized based on this method with
fidelities over $99\%$ for the two-qubit C${}_{\text{Z}}$ gate Wu  _et al._
(2021a). This exotic antiblockade strategy can also be used together with
dissipation for entanglement generation Li  _et al._ (2020).

The antiblockade by periodical driving can be used to construct certain
quantum gates that were rarely studied by other means of Rydberg-mediated gate
methods. For example, Ref. Wu  _et al._ (2021c) used the periodical driving
and a modified antiblockade scheme to study a controlled-SWAP gate in three
qubits, where the state of one control qubit control the SWAP operation of the
states of two target qubits. Ref. Wu  _et al._ (2021c) also studied a SWAP
gate by the same mechanism. Importantly, both the two gate protocols in Ref.
Wu  _et al._ (2021c) require only one step. These gates can obtain relatively
large fidelities; for example, the texts below Fig. 5 and Fig. 7(a) on page
818 indicate that their C-SWAP gate can have a fidelity 98.7% at room
temperature. To consider the achievable fidelity of their schemes, Ref. Wu
_et al._ (2021c) showed that with experimentally affordable cooling of atomic
qubits, motion-induced Doppler dephasing only lead to an extra error smaller
than 1%.

Periodically modulating the amplitudes of laser fields can be used for
antiblockade in ways other than compensating the interaction. For example,
Ref. Jin  _et al._ (2021) considered using the strategy for speeding up
entanglement creation via dissipation; the number quoted from Ref. Jin  _et
al._ (2021) in Table 4, however, does not involve dissipation. It is also of
interest to study many-body dynamics by periodical driving of Rydberg atoms
Mallavarapu  _et al._ (2021).

Though the antiblockade mechanism can lead to high-fidelity Rydberg gates in
theory, it is challenging to realize it in experiments because the distance
and relative angular positions between qubits in real space can fluctuate,
which leads to off-diagonal Rydberg interactions shown in Sec. II.3.2 as well
as change of the diagonal interactions. Except rare cases such as Ref. Zheng
_et al._ (2020a), most studies of Rydberg antiblockade did not explore errors
from the fluctuation of Rydberg interactions. It demands efforts to realize
the theoretical fidelities of quantum gates in the antiblockade regime by, for
example, adopting ground-state cooling of qubits or using sufficiently deep
traps.

Category | Rydberg blockade | Frozen interaction
---|---|---
Feature | Robust to the fluctuation of $V$ | Sensitive to the fluctuation of $V$ a
Type of $V$ | Dipole-dipole, or van der Waals, or in a | Van der Waals |
regime with both types of interactions |  | Dipole-dipole
Application |  | Blocking Rydberg excitation of one atom by
---
the interaction from a nearby Rydberg atom
Phase shift | Antiblockade |  | Two-atom state flip
---
Proposed in | Jaksch  _et al._ (2000) (individual atoms); Lukin  _et al._ (2001) (ensembles of atoms) | Ates  _et al._ (2007) | (Intrinsic)
| Experiments
---
(CNOT, C${}_{\text{Z}}$,
Toffoli)
| Two atoms: Isenhower  _et al._ (2010); Zhang  _et al._ (2010); Maller  _et
al._ (2015); Zeng  _et al._ (2017); Levine  _et al._ (2019); Graham  _et al._
(2019)
---
Three atoms (Toffoli): Levine  _et al._ (2019)
Two photons: Tiarks  _et al._ (2019)
|  |
| Experiments
---
(Individual-atom
Entanglement)
| Two atoms: Wilk  _et al._ (2010); Isenhower  _et al._ (2010); Zhang  _et
al._ (2010); Maller  _et al._ (2015); Zeng  _et al._ (2017); Levine  _et al._
(2018); Picken  _et al._ (2019); Levine  _et al._ (2019); Graham  _et al._
(2019); Madjarov  _et al._ (2020)
---
Twenty atoms: Omran  _et al._ (2019)
Two photons: Tiarks  _et al._ (2019)
Jo  _et al._ (2020) |  |
| Achievable gate
---
or entanglement
fidelity in theory
| 2-qubit: $99.99\%$
---
4-qubit: $99.97\%$ b
(see Table 2)
|  | 2-qubit:$99.98\%$
---
3-qubit: $99.65\%$
(see Table 4)
| 2-qubit:$99.995\%$
---
3-qubit: $99.9\%$
(see Table 3)

Table 5:  Summary of quantum entanglement and logic gates based on two-body
Rydberg interactions of neutral atoms. There are in general two categories,
the strong blockade regime and the frozen interaction regime, where the former
was widely studied because of its robustness to the fluctuation of Rydberg
interactions.

  * a

For the gates in the frozen interaction regime, the methods in Secs. IV.3 and
IV.4 can also be robust against the fluctuation of interatomic interactions.

  * b

A four-qubit gate is quoted here because Table 1 on page 763 of Ref. Isenhower
_et al._ (2011) listed data for gates CkNOT with $k\geq 3$. The fidelity of
Toffoli gate C2NOT should be larger than 99.97% according to Eq. (2) of Ref.
Isenhower  _et al._ (2011).

##  VI Challenges

###  VI.1 Fidelity of Rydberg gates

The most visited experimental method used for Rydberg-mediated entanglement is
via the blockade mechanism, as summarized in Table 5. This is because it is
robust against the fluctuation of the Rydberg interaction $V$ from shot to
shot; within each experimental cycle, $V$ will also change, and the blockade
method is also robust to it. To suppress the blockade error, only very large
$V$ can be useful because the blockade error scales as
$\sim\hbar^{2}\Omega^{2}/(2V^{2})$ shown around Eq. (59). The blockade error
can be made negligible by using very large direct dipole-dipole interaction.
Although the direct dipole-dipole interaction is large for high-lying Rydberg
states, the inter-state energy spacing between two nearby Rydberg levels is
small. This makes it challenging to suppress leakage error unless prestigious
pulse shaping is employed Theis  _et al._ (2016) if very high Rydberg states
are used. Another challenge to realize gates with a large $V$ lies in that
when placing two qubits too close, it is difficult to avoid crosstalk
concerning the Rydberg excitation of a selected atom. For this reason, the
value of $V$ is well below $h\times 100$MHz in typical experiments with
alkali-metal atoms Wilk  _et al._ (2010); Isenhower  _et al._ (2010); Zhang
_et al._ (2010); Maller  _et al._ (2015); Jau  _et al._ (2016); Zeng  _et al._
(2017); Levine  _et al._ (2018); Picken  _et al._ (2019); Levine  _et al._
(2019); Graham  _et al._ (2019) although the entanglement experiment with
strontium atoms used a much larger interaction $V/h\sim 130$ MHz Madjarov  _et
al._ (2020).

Even if the blockade error is suppressed, the motional dephasing is another
stumbling block. Motion-induced dephasing is prevailing because the atom is
not static even if it is cooled to the motional ground state in the dipole
trap. If the atoms are released during the Rydberg pulses, the motional effect
within each gate cycle is actually not random, but is dependent on a linear
effect as studied in detail in Ref. Shi (2020a). The average effect of
multiple gate cycles, however, appears as a dephasing, which is in close
analogy to the nuclear spin dephasing in solid-state systems Liu  _et al._
(2010). By using the linear effect of the phase accumulation due to the atomic
drift, it was shown that either by using extra laser fields for phase
compensation during the gap times, or by using a “V”-type dual-rail Rydberg
excitation method, one can almost completely suppress the motional dephasing
Shi (2020a).

The current stage in experiments is still about removing technical
imperfections Levine  _et al._ (2018, 2019); Graham  _et al._ (2019). Only
after these issues are tackled will the suppression of the blockade error and
the dephasing error be interesting. However, given the high fidelity reported
in Madjarov  _et al._ (2020) with rapid control thanks to the large Rabi
frequencies realized, there is possibility to realize ground-state
entanglement by Rydberg interactions with a fidelity limited only by the two
factors mentioned above. With all the strengths and weaknesses for each
entangling method discussed, however, there are hope as well as extreme
challenges toward a realistic large-scale neutral-atom quantum computer.

####  VI.1.1 Suppressing blockade error by spin echo

One method to suppress the blockade error is by spin echo as shown in Ref. Shi
(2018a). To understand the spin-echo method, one should first understand that
the fluctuation of $V$ in Sec. III.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 $V_{0}$, it
does not change much during the gate sequence. Thus, one can start to
understand the spin echo by first assuming a frozen interaction $V=V_{0}$, and
then analyze fluctuation of the interaction around $V_{0}$ in different gate
cycles.

The essence of the spin-echo Rydberg gate is to break the second $2\pi$ pulse
in Fig. 5(a) into three pulses. First, a $\pi$ pulse is used to induce the
transition $|1\rangle\leftrightarrow|r\rangle$ for the target qubit with a
Rabi frequency $\Omega$. For the input state $|11\rangle$, the relevant
Hamiltonian is given in Eq. (55) with
$\\{|u\rangle,~{}|l\rangle\\}=\\{|rr\rangle,~{}|r1\rangle\\}$ and $D=V_{0}$.
Second, a microwave $\pi$ pulse is used to transfer the Rydberg state
$|r\rangle$ to another Rydberg state $|r^{\prime}\rangle$ with a Rabi
frequency $i\Omega_{\mu}$, where $\hbar\Omega_{\mu}\gg|V|$. The microwave
fields induce a Rydberg state flip $|r\rangle\rightarrow|r^{\prime}\rangle$
for both the control and target qubits. The interactions of the states
$|rr\rangle$ and $|r^{\prime}r^{\prime}\rangle$ are $V_{0}$ and
$V_{0}^{\prime}$, respectively, where $V_{0}$ and $V_{0}^{\prime}$ have
opposite signs, i.e., one is attractive and the other is repulsive. Third, a
$\pi$ pulse is used to induce the transition
$|r^{\prime}\rangle\rightarrow|1\rangle$ with a Rabi frequency
$\Omega^{\prime}$, where $\Omega^{\prime}=\Omega V_{0}^{\prime}/V_{0}$. The
Hamiltonian for the third pulse here is

$\displaystyle\hat{H}^{\prime}$ $\displaystyle=$
$\displaystyle(\frac{\hbar\Omega^{\prime}}{2}|r^{\prime}r^{\prime}\rangle\langle
r^{\prime}1|+\text{H.c.})+V_{0}^{\prime}|r^{\prime}r^{\prime}\rangle\langle
r^{\prime}r^{\prime}|.$ (89)

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

The above discussion about the spin echo has ignored the duration of the
microwave pulses, and no phase appears in the qubit states. To induce a
C${}_{\text{Z}}$ 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 Ref. Shi (2020c), where the fluctuation of
$V$ around $V_{0}$ was studied. One can see that for the spin echo to work,
the key step is to know the value of $V_{0}^{\prime}/V_{0}$. 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
$V_{0}^{\prime}/V_{0}$ is simply the ratio of the two $C_{6}$ coefficients of
the two states $|r^{\prime}r^{\prime}\rangle$ and $|rr\rangle$. 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 $V^{\prime}$ and $V$ in the third and the first pulses are still
given by the ratio of the $C_{6}$ 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 Ref.
Shi (2018a) 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 Ref. Shi
(2018a)).

Compared to the methods in Sec. III, 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 Ref. Shi
(2018a) 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 Sec. III 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.

####  VI.1.2 Suppressing blockade error by pulse shaping

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 derivative
removal by adiabatic gate reviewed in Sec. III.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 III.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 Sec. III.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
Refs. Beterov  _et al._ (2016b); Petrosyan  _et al._ (2017) as reviewed in
Secs. IV.3 and IV.4 can remove the blockade error. These two strategies are
robust against the fluctuation of the interatomic interactions as shown in
Fig. 7 for a numerical simulation with the example of Ref. Petrosyan  _et al._
(2017). 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 simultaneous 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${}_{\text{Z}}$ and CNOT) Williams (2011). With the many pulse-shaping
methods as reviewed in Sec. III.8, we hope that certain pulse-shaping methods
may lead to suppression of both the blockade error and the dephasing error.

####  VI.1.3 Suppressing blockade error and motional dephasing by gates with
collectively-encoded qubit

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 Sec. II.1 and shown in Fig. 2(c). Almost all Rydberg
quantum gates reviewed in Sec. III, IV, and V use the first method for
defining qubits as reviewed in Sec. II.1 and shown in Fig. 2(a), i.e., using
hyperfine-Zeeman substates from single atoms as qubit states (or its superatom
version as in Sec. III.8.7). 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.

####  VI.1.4 Suppressing blockade error by the method of Ref. Levine  _et al._
(2019)

As reviewed in Sec. III.2, the interference gate explored in Levine  _et al._
(2019) also has a blockade error because it depends on the Rydberg blockade
mechanism. The supplemental material of Ref. Levine  _et al._ (2019) 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 $\Omega^{2}\hbar/(2V)$ by the renormalization of
$(|1r\rangle+|r1\rangle)/\sqrt{2}$ concerning its excitation to the two-atom
Rydberg state $|rr\rangle$. In other words, the value of $\Delta$ in Eqs.
(71)-(77) should be updated according to the renormalization
$(|1r\rangle+|r1\rangle)/\sqrt{2}$. This means that in theory the blockade
error can be suppressed in the gate of Levine  _et al._ (2019).

The method for suppressing the blockade error discussed in Levine  _et al._
(2019) 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 $(|1r\rangle+|r1\rangle)/\sqrt{2}$ in Eq. (70)
results in admixing of the state $|rr\rangle$ in it, which leads to a state
dynamics slightly different from that shown in Sec. III.2. But for
sufficiently large $V$, the admixing of the state $|rr\rangle$ is negligible.
This means that the possibility to suppress the blockade error for the gate in
Levine  _et al._ (2019) is intimately related with the capability to realize
large van der Waals interaction $V$ and to suppress the motion of the qubits.

####  VI.1.5 Suppressing blockade error and motional dephasing by adequate
cooling of alkali-earth-like atoms

The discussions in Secs. VI.1.1, VI.1.2, VI.1.3, and VI.1.4 focus on a
theoretical attack of the gate errors when alkali-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 Secs. IV and V 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 alkaline-earth-like (AEL) atoms.
Compared to alkali-metal atoms, divalent AEL atoms can be more easily cooled
Yamamoto  _et al._ (2016); Saskin  _et al._ (2019); Cooper  _et al._ (2018);
Covey  _et al._ (2019b); Madjarov  _et al._ (2020); Norcia  _et al._ (2018),
their nuclear spin states can be preserved during cooling Reichenbach and
Deutsch (2007), and both their ground and Rydberg states can be trapped in one
trap Wilson  _et al._ (2019). 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 quantum computing with AEL atoms
Jaksch  _et al._ (1999); Calarco  _et al._ (2000); Hayes  _et al._ (2007);
Daley  _et al._ (2008); Gorshkov  _et al._ (2009); Daley  _et al._ (2011);
Daley (2011); Pagano  _et al._ (2019); Jensen  _et al._ (2019) rarely used
Rydberg interactions; though a recent Rydberg blockade experiment Madjarov
_et al._ (2020) with AEL atoms reported a two-atom entanglement fidelity
$99.5\%$, the entanglement in Madjarov  _et al._ (2020) was between a Rydberg
state and a metastable state $(5s5p)^{3}P_{0}$ 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.

###  VI.2 Three-qubit gates

To execute a computation task with a quantum circuit, not only high-fidelity
two-qubit entangling gates are needed, but also high-fidelity error correction
schemes should be at hand, in which the three-qubit Toffoli gate Williams
(2011) plays an important role Nielsen and Chuang (2000). Rydberg Toffoli
gates have been extensively studied in theory Brion  _et al._ (2007d); Shi
(2018b); Levine  _et al._ (2019); Isenhower  _et al._ (2011); Khazali and
Molmer (2020), but a high-fidelity realization is difficult in experiment.
Though there are Toffoli or other three-qubit gates listed in Tables 3 and 4,
these methods depend on frozen interactions or well isolated dipole-dipole
processes which demand extreme efforts to realize in real experiments. Until
now, there was one experimental study of Toffoli gate reported in Ref. Levine
_et al._ (2019) where the blockade mechanism was used so as to be robust
against the fluctuation of Rydberg interactions. Although both two and three-
qubit gates were realized by global Rydberg pulses, Ref. Levine  _et al._
(2019) reported an experimental Toffoli-gate fidelity 87.0% which is about 10%
smaller than their theoretical estimate; in comparison, the realized two-qubit
gate fidelity was 97.4% in Ref. Levine  _et al._ (2019), which is less than 3%
smaller than their theoretical estimate. This contrast shows the difficulty to
carry out high-fidelity three-qubit Rydberg gates.

The challenge to realize the Toffoli gate lies in that for a three-atom system
lying on one line, there will be some residual dipole-dipole interaction
between the two outermost atoms when they are both in the Rydberg states,
which not only leads to blockade error, but also induces unwanted phase
shifts. Typical solutions to this problem include (i) to excite the three
qubits to three different Rydberg states, so that the interaction between the
two edge atoms is negligible Brion  _et al._ (2007d); (ii) to choose an
appropriate gate duration so as to effectively annul the induced phase shift
Shi (2018b); (iii) to use microwave fields to induce zero van der Waals
interaction Shi (2018b); Shi and Kennedy (2017); (iv) to use optimal control
to find appropriate time-dependent Rabi frequency and detuning in the pulse
Levine  _et al._ (2019). Theoretically, the last method has a fidelity bounded
below $98\%$ as analyzed in Levine  _et al._ (2019), and although the methods
(i-iii) in principle can lead to large fidelities, the residual blockade
between the edge atoms in method (i) and the fluctuation of $V$ in methods
(ii,iii) set an intrinsic challenge in their implementation. There is also an
adiabatic method using dark states for implementing a high-fidelity Toffoli
gate Khazali and Molmer (2020), as well as a fast Toffoli gate protocol based
on three-atom Förster resonance Beterov  _et al._ (2018b), where the fidelity
depends on how well one can isolate the chosen dipole-dipole processes.

##  VII Conclusions

There are in general three mechanisms to generate quantum entanglement between
individual atoms by Rydberg interactions, namely, the blockade mechanism, the
excitation of multi-atom Rydberg states, and the antiblockade mechanism. The
theoretical entanglement and gate fidelities using them are listed in Tables
2, 3, and 4, from which we can see that the largest theoretical fidelity for
two-qubit gates is 99.995% with the method of excitation of multi-atom Rydberg
state, and the largest fidelity for multiqubit gates is 99.97% via the
blockade mechanism. The total numbers of entanglement fidelities over or equal
to 99.9% (with qubits hosted at room temperature) are eight, two, and five in
Tables 2, 3, and 4, respectively. Though there are five fidelities over or
equal to 99.9% in Table 4, the analyses leading to them ignored the
fluctuation of interactions. Even if the fluctuation of atomic spacing is
absent, the comparison between Tables 2, 3, and 4 shows that in theory, it
seems more possible to reach high-fidelity quantum gates and entanglement via
the blockade mechanism, i.e., via the methods reviewed in Sec. III.

The fundamental errors for Rydberg-mediated entanglement via the blockade
mechanism stem from Rydberg-state decay, rotation errors due to imperfect
blockade, and motional dephasing in the ground-Rydberg rotations. The Rydberg-
state decay can be made smaller and smaller, but not removable. Whatever the
method is, there is always motional dephasing during the control of the
ground-Rydberg transition by using electromagnetic radiations. Although
fortunately there are several theories coping with the motional dephasing, the
blockade error is a real issue once large fidelity is desired. This is because
the fidelity by using Rydberg blockade effect is always hampered by the
blockade error, which is difficult to be completely suppressed because it is
challenging to accurately address the two qubits without crosstalk if they are
too close to each other; so, large enough qubit spacing is necessary which
limits the magnitude of the blockade interaction. The fidelity by using the
frozen interactions in Secs. IV and V can avoid the blockade error; but most
of them (except the ones as in Refs. Beterov  _et al._ (2016b); Petrosyan  _et
al._ (2017)) depend on negligible fluctuation of the qubit spacing. These
methods are physically interesting though it is an outstanding challenge to
suppress the position fluctuation of qubits to a level where the motion-
induced dephasing and change of interaction disappear. Given the fact that
many Rydberg-mediated gates have been proposed as reviewed in Secs. III, IV,
and V, there are enough reasons to expect new physics and new protocols on
Rydberg gates in near future that can indeed push the Rydberg-mediated quantum
computing to a practical level.

## ACKNOWLEDGMENTS

The author thanks the anonymous referees for their valuable comments and
suggestions, and thanks T. A. B. Kennedy and Yan Lu for discussions. This work
is supported by the National Natural Science Foundation of China under Grants
No. 12074300 and No. 11805146, the Natural Science Basic Research plan in
Shaanxi Province of China under Grant No. 2020JM-189, and the Fundamental
Research Funds for the Central Universities.

##  Appendix A Entanglement of photons via Rydberg interactions

Rydberg interaction is not only useful in quantum information processing, but
also in quantum optics Dudin and Kuzmich (2012); Peyronel  _et al._ (2012);
Firstenberg  _et al._ (2013); Li  _et al._ (2013); Gorniaczyk  _et al._
(2014); Baur  _et al._ (2014); Tiarks  _et al._ (2014); Li  _et al._ (2016);
Thompson  _et al._ (2017); Busche  _et al._ (2017); Paris-Mandoki  _et al._
(2017); Ripka and Pfau (2018); Liang  _et al._ (2018); Li  _et al._ (2019);
for a review, see Ref. Firstenberg  _et al._ (2016). In particular, it was
shown that by using Rydberg blockade one can achieve photonic quantum
entanglement by mapping single photons to Rydberg polaritons Paredes-Barato
and Adams (2014). In a Rydberg polariton, the Rydberg excitation is shared in
many atoms in a collective way, and the blockade interaction induces phase
shifts to the polariton, which can be transfered to the output photon when
released in the formalism of EIT Fleischhauer and Lukin (2002). The key step
for realizing this method has been experimentally tested Busche  _et al._
(2017). Later on, Ref. Khazali  _et al._ (2015) proposed an improved version
of the photon-photon gate. Although both methods depend on the dual-rail
coding Chuang and Yamamoto (1995), the difference between Paredes-Barato and
Adams (2014) and Khazali  _et al._ (2015) lies in that the qubit states
$|0(1)\rangle$ are all stored in Rydberg dark-state polaritons in Paredes-
Barato and Adams (2014), but only one of the two qubit states is stored in the
Rydberg polaritons in Khazali  _et al._ (2015) and the other is stored in
ground-state polaritons. In principle, these methods can induce high-fidelity
photon entanglement if the efficiency for loading and retrieving of single
photons is high. However, the dephasing of spin wave is a critical issue in a
practical implementation Dudin and Kuzmich (2012); Jenkins  _et al._ (2012);
Schmidt-Eberle  _et al._ (2019).

Another method for photon entanglement is to pass a single-photon pulse
through a medium where there is already a Rydberg collective excitation in the
medium, so that the Rydberg interaction between the flying Rydberg polariton
and the static Rydberg excitation induces a phase shift Friedler  _et al._
(2005); Gorshkov  _et al._ (2011). The first photon-photon quantum entangling
gate based on Rydberg interaction was experimentally realized with this
method, as reported in Ref. Tiarks  _et al._ (2019). Because there is
inevitable overlap between the static and flying Rydberg excitation, heavy
dissipation arises which puts fundamental limit to the achievable fidelity
Gorshkov  _et al._ (2011). The benefit of this method, however, lies in that
it is relatively simple to manipulate the system and there is less motional
dephasing because only one of the two qubits is stored in the method of Tiarks
_et al._ (2019).

There are also entanglement methods by coupling Rydberg atoms with cavity
photons Hao  _et al._ (2015); Das  _et al._ (2016) via the scheme proposed in
Ref. Duan and Kimble (2004), where entanglement can emerge between a stored
photon (in the form of collective spin wave) and a flying photon.

##  Appendix B Transition slow-down

The mechanism of transition slow-down (TSD) proposed in Ref. Shi (2020b)
belongs to the Rydberg blockade regime. It involves Rydberg excitation of both
qubits, and the qubits should be placed close enough so that strong dipole-
dipole interactions can appear. Because of the particular applicability of TSD
in realizing the CNOT gate, we show two examples of CNOT below via TSD. Though
these examples have gate speeds not as fast as that in Ref. Shi (2020b), they
can illustrate why a “slow-down” effect can help in designing a quantum gate.

Figure 9: A two-step CNOT realized by a 2-fold TSD in a ground-Rydberg
transition, where either the first or the second step consists of a pulse
sequence described around Eqs. (90) and (91) with $\alpha=\sqrt{15}$. In each
step, the pulse areas for the control and target qubits are $\alpha\pi$ and
$2\pi$, respectively. Here $|r\rangle$ and $|r^{\prime}\rangle$ are different
Rydberg eigenstates with different principal quantum numbers. For the target
qubit, the Rabi frequencies for
$|0(1)\rangle\leftrightarrow|r^{\prime}(r)\rangle$ are both
$\Omega_{\text{t}}$ in the first step, while those for
$|1(0)\rangle\leftrightarrow|r^{\prime}(r)\rangle$ are both
$-\Omega_{\text{t}}$ in the second step.

The first method uses a pulse sequence shown in Fig. 9, which consists of two
pulses. We study the effect of the first pulse. The pulse for the control
qubit is applied during $t\in[t_{0},~{}t_{0}+t_{\pi})$, and the pulse for the
target qubit is applied during $t\in[0,~{}2t_{\pi})$, where
$t_{0}\in[0,~{}t_{\pi})$ and $t_{\pi}=\pi/\Omega_{\text{t}}$. Starting from an
initial two-atom state $|11\rangle$, the wavefunction at $t=t_{0}$ becomes

$\displaystyle|\psi(t_{0})\rangle$ $\displaystyle=$
$\displaystyle\cos(\Omega_{\text{t}}t_{0}/2)|11\rangle-i\sin(\Omega_{\text{t}}t_{0}/2)|1r\rangle.$
(90)

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

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar[(\Omega_{\text{c}}|r1\rangle+\Omega_{\text{t}}|1r\rangle)\langle
11|/2+\text{H.c.}]+\hat{H^{\prime}},$ (91)

where $\hat{H^{\prime}}$ includes excitation from $|1r\rangle$ and
$|r1\rangle$ to $|rr\rangle$ and dipole-dipole interaction involving
$|rr\rangle$ and nearby dipole-coupled states. In the strong interaction
regime, $\hat{H^{\prime}}$ can be discarded, and
$\hat{H}=\hbar\bar{\Omega}|\mathscr{R}_{+}\rangle\langle 11|/2+\text{H.c.}$,
where $\bar{\Omega}\equiv\sqrt{\Omega_{\text{c}}^{2}+\Omega_{\text{t}}^{2}}$
and
$|\mathscr{R}_{+}\rangle\equiv(\Omega_{\text{c}}|r1\rangle+\Omega_{\text{t}}|1r\rangle)/\bar{\Omega}$.
When $\bar{\Omega}t_{\pi}=4k\pi$ is satisfied with an integer $k$, the
wavefunction at $t=t_{0}+t_{\pi}$, given by
exp$[-i\hat{H}t_{\pi}/\hbar]|\psi(t_{0})\rangle$, is equal to Eq. (90). The
state for the target qubit becomes $|1r\rangle$ at $t=2t_{\pi}$. So, a $2\pi$
pulse leads to the excitation from $|11\rangle$ to $|1r\rangle$. The smallest
$\alpha\equiv\Omega_{\text{c}}/\Omega_{\text{t}}$ satisfying the above
condition is $\alpha=\sqrt{15}$. By this, a fast and high-fidelity CNOT is
shown in Fig. 9, where two Rydberg states $|r\rangle$ and $|r^{\prime}\rangle$
are involved. As can be easily verified, the pulse sequence simply leads to
the state mapping
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}\rightarrow-\\{|00\rangle,|01\rangle,i|1r^{\prime}\rangle,i|1r\rangle\\}$
when the transitions
$\\{|1\rangle,|0\rangle\\}\leftrightarrow\\{|r\rangle,|r^{\prime}\rangle\\}$
are excited by a time $2\pi/\Omega_{\text{t}}$ for the target qubit with Rabi
frequencies $\Omega_{\text{t}}$, within which the transition
$|1\rangle\leftrightarrow|r\rangle$ is excited by a time
$\pi/\Omega_{\text{t}}=\alpha\pi/\Omega_{\text{c}}$ for the control qubit.
Subsequently, the transitions
$\\{|0\rangle,|1\rangle\\}\leftrightarrow\\{|r\rangle,|r^{\prime}\rangle\\}$
are excited by a time $2\pi/\Omega_{\text{t}}$ for the target qubit with Rabi
frequencies $-\Omega_{\text{t}}$ while the excitation for the control qubit
stays the same as in the first step. Then one can find that the input-output
map corresponds to a CNOT gate. The gate has an intrinsic fidelity bounded by
the Rydberg-state decay and the residual blockade error
$\sim(\Omega_{\text{c(t)}}/V)^{2}$, where the latter can be made negligible
when very large native dipole-dipole interaction is used. The Rydberg-state
decay is $\epsilon=T_{R}/\tau$, where $T_{R}$ is the expected time for the
atoms to stay in the Rydberg state averaged over the four input states, and
$\tau$ is the lifetime of the Rydberg state. By using $t_{0}=0$, we find that
the times for the input states $|00\rangle$ and $|10\rangle$ (or,
equivalently, $|01\rangle$ and $|11\rangle$) to be in the Rydberg state are
respectively $\pi/(2\Omega_{\text{t}})$ and $\pi/(\Omega_{\text{t}})$ if
$\alpha=\sqrt{15}$ for either of the two pulses in Fig. 9, so we have
$T_{R}=3\pi/(4\Omega_{\text{t}})$ when there is no gap time between the two
pulses for the target. For $\tau\sim 330~{}\mu$s corresponding to s-orbital
rubidium states with principal quantum numbers around $100$ at room
temperature Beterov  _et al._ (2009), the Rydberg-state decay leads to a gate
infidelity $\epsilon=T_{R}/\tau\approx 1.1\times 10^{-3}$ if
$\Omega_{\text{t}}/(2\pi)=1$ MHz, i.e., a fidelity $99.89\%$ is realizable. A
higher fidelity is possible with larger Rabi frequencies.

Figure 10: A three-state transition slow-down and a one-shot CNOT. (a) A
three-state TSD is realized by sending to each qubit one pulse with equal
duration. The pulse areas are $\beta\pi$ and $6\sqrt{2}\pi$ for the control
and target qubits, respectively. (b) Relevant transitions for the input states
$|10\rangle$ and $|11\rangle$. Transitions involving $|00\rangle$ and
$|01\rangle$ are not shown because they do not involve dipole-dipole
interactions. That the arrows point to the left or right does not mean
transitions with change of angular momentum $\pm\hbar$, but only for
clarifying the protocol. (c) If the input state is $|00\rangle$, it
transitions to $|01\rangle$ and back within a time of
$2\sqrt{2}\pi/\Omega_{\text{t}}$. Three cycles are shown with a pulse duration
$6\sqrt{2}\pi/\Omega_{\text{t}}$. (d) With the same excitation, the input
state $|10\rangle$ just goes to $|11\rangle$ at the end of the pulse. The
population error at the end of the pulse is zero in (c) and $1.8\times
10^{-3}$ in (d). The contrast between (c) and (d) shows that the transition
between $|0\rangle$ and $|1\rangle$ in the target qubit is slowed down by 6
folds. Here $\Omega_{\text{c}}=4.245739\Omega_{\text{t}}$ is used
corresponding to $\beta\approx 36.0$. Figure (b) is adopted from Phys. Rev.
Appl. 14, 054058 (2020).

Another TSD method in three states can lead to a one-shot CNOT. We study a
ground-Rydberg-ground transition chain
$|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$, i.e., a state
swap between the two qubit states $|0\rangle$ and $|1\rangle$ via a Rydberg
state $|r\rangle$. The three-state TSD is implemented by exciting the control
and target qubits with respective Rabi frequencies $\Omega_{\text{c}}$ and
$\Omega_{\text{t}}$ for the same duration, with transition
$|1\rangle\leftrightarrow|r\rangle$ for the control qubit and
$|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$ for the target
qubit. Because $|0\rangle$ is not excited for the control qubit, the
Hamiltonian is simply $\hbar\Omega_{\text{t}}(|00\rangle+|01\rangle)\langle
0r|/2+$H.c. for the input states $|00\rangle$ and $|01\rangle$. One can show
that a superposition of these two states is preserved when the excitation only
lasts for a duration $t=2\sqrt{2}k_{1}\pi/\Omega_{\text{t}}$ with an integer
$k_{1}$ Shi (2018b). For the remaining input states, we consider the ordered
basis $\\{|1r\rangle,|r1\rangle,|r0\rangle,|11\rangle,|10\rangle\\}$ and the
following Hamiltonian

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\left(\begin{array}[]{ccccc}0&0&0&\Omega_{\text{t}}&\Omega_{\text{t}}\\\
0&0&0&\Omega_{\text{c}}&0\\\ 0&0&0&0&\Omega_{\text{c}}\\\
\Omega_{\text{t}}&\Omega_{\text{c}}&0&0&0\\\
\Omega_{\text{t}}&0&\Omega_{\text{c}}&0&0\end{array}\right),$ (97)

where we ignore the two-Rydberg state $|rr\rangle$ with reasons shown below
Eq. (91). By diagonalizing Eq. (97), one can show that

$\displaystyle|10\rangle$ $\displaystyle=$
$\displaystyle(-|\mathscr{R}_{1}\rangle+|\mathscr{R}_{2}\rangle-|\mathscr{R}_{3}\rangle+|\mathscr{R}_{4}\rangle)/2,$
$\displaystyle|11\rangle$ $\displaystyle=$
$\displaystyle(|\mathscr{R}_{1}\rangle-|\mathscr{R}_{2}\rangle-|\mathscr{R}_{3}\rangle+|\mathscr{R}_{4}\rangle)/2,$
(98)

where $|\mathscr{R}_{k}\rangle$ with $k=1-4$ are four eigenvectors of Eq. (97)
with respective eigenvalues
$(\Omega_{\text{c}},~{}-\Omega_{\text{c}},~{}\overline{\Omega},-\overline{\Omega})/2$
[the fifth eigenvector does not enter Eq. (98)], where
$\overline{\Omega}\equiv\sqrt{\Omega_{\text{c}}^{2}+2\Omega_{\text{t}}^{2}}$.
Starting from the input state $|\psi(0)\rangle=|10\rangle$, the wavefunction
$|\psi(t)\rangle$ at $t$ becomes

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

which means that $|\psi(t)\rangle=-|11\rangle$ if
$\Omega_{\text{c}}t=4k_{2}\pi$ and $\overline{\Omega}t=2(2k_{3}+1)\pi$ can be
satisfied with two integers $k_{2}$ and $k_{3}$ for a pulse duration $t$.
Then, one can show that the same condition leads to that the input state
$|11\rangle$ becomes $-|10\rangle$. To summarize, if a set of integers
$(k_{1},k_{2},k_{3})$ can be found for the following conditions

$\displaystyle\Omega_{\text{t}}t=2\sqrt{2}k_{1}\pi,~{}\Omega_{\text{c}}t=4k_{2}\pi,~{}\overline{\Omega}t=2(2k_{3}+1)\pi,$
(99)

the following gate similar to the CNOT is realized

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

with a single pulse for each qubit; application of the phase gate
$|1\rangle\rightarrow-|1\rangle$ in the target qubit can recover a CNOT.
Another way to realize the standard CNOT is to add a $\pi$ phase difference
for the Rabi frequencies in the transitions
$|0\rangle\leftrightarrow|r\rangle$ and $|1\rangle\leftrightarrow|r\rangle$ so
that the standard CNOT can be realized.

Unlike the two-state TSD, it is difficult to realize a high-fidelity three-
state TSD because in order to avoid large Rabi frequencies we shall consider
small integers $k_{j}$, but unfortunately we can not find small enough
integers satisfying Eq. (99). As a compromise with $k_{j}<10$, $j=1-3$, we
find a large fidelity by using $\Omega_{\text{c}}=4.245739\Omega_{\text{t}}$
associated with $(k_{1},k_{2},k_{3})=(3,9.007,8.993)$. The fact that $k_{1}$
is integer while $k_{2}$ and $k_{3}$ are approximate integers means that
$\\{|00\rangle,|01\rangle\\}\rightarrow\\{|00\rangle,|01\rangle\\}$ is exact
in this one-shot CNOT, while
$\\{|10\rangle,|11\rangle\\}\rightarrow-\\{|11\rangle,|10\rangle\\}$ comes
with some error, which is $1.8\times 10^{-3}$ by numerics based on Eq. (97).
This means that the error of the CNOT gate fidelity is $\epsilon_{1}=9\times
10^{-4}$ if Rydberg-state decay and the residual blockade error are ignored.
The blockade error can be made negligible when large native dipole-dipole
interaction is employed, while the Rydberg-state decay remains. The numerical
simulation of the 6-fold TSD in three states based on Eq. (97) is shown in
Fig. 10. We find that the time for the atoms to be in Rydberg states is
$T_{R}=1.06\times 2\pi/\Omega_{\text{t}}$ during the three cycles of the
transition $|00\rangle\rightarrow|01\rangle\rightarrow|00\rangle$ shown in
Fig. 10(c). For the transition in Fig. 10(d), the time for the atoms to
experience Rydberg population is $2T_{R}$. So the decay error of the CNOT is
$\epsilon_{2}=3T_{R}/(2\tau)$ when averaging over the four input states, where
$\tau$ is the lifetime of the Rydberg state. This means that if
$\Omega_{\text{t}}/(2\pi)=1$ MHz, then $\epsilon_{2}=4.8\times 10^{-3}$ if we
consider $\tau\sim 330~{}\mu$s as in the two-pulse CNOT by the two-state TSD,
leading to an infidelity $5.7\times 10^{-3}$ for the one-pulse CNOT. Although
this error is about five times larger than that by the two-state TSD with
similar Rabi frequencies, the one-shot CNOT does not need extra switch on and
off of the external control field and thus can avoid errors due to the
switching of lasers to the largest extent. To decrease the Rydberg-state decay
for larger gate fidelity, one can use a cryogenic chamber to host the qubits
or use stronger laser fields for faster Rabi rotations.

Compared to other methods, TSD can lead to a CNOT gate with a minimal number
of pulse switchings which are related with Rydberg population loss in the
standard Rydberg gates Maller  _et al._ (2015). Moreover, the Doppler
dephasing can be suppressed in the one-shot TSD-based CNOT because there is no
time required to leave a Rydberg atom in free flight that can lead to strong
motional dephasing Shi (2020a). The protocol shown in Fig. 10 is the only one-
pulse CNOT gate based on the blockade mechanism; there are, of course, some
one-shot CNOT proposals based on the antiblockade effect reviewed in Sec. V.

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