††thanks: First Author and Second Author contribute equally to this work.
††thanks: First Author and Second Author contribute equally to this work.

# Multiqubit Toffoli gates and optimal geometry with Rydberg atoms

Dongmin Yu1 Han Wang1 Jin-ming Liu1 Shi-Lei Su2,‡ Jing Qian1,† Weiping
Zhang3,4,5 1State Key Laboratory of Precision Spectroscopy, Department of
Physics, School of Physics and Electronic Science, East China Normal
University, Shanghai 200062, China  2School of Physics, Zhengzhou University,
Zhengzhou 450001, China  3School of Physics and Astronomy, Tsung-Dao Lee
Institute, Shanghai Jiao Tong University, Shanghai 200240, China
4Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, China  5Shanghai Research Center for Quantum Sciences,
Shanghai 201315, China<EMAIL_ADDRESS>
###### Abstract

Due to its potential for implementing a scalable quantum computer, multiqubit
Toffoli gate lies in the heart of quantum information processing. In this
article, we demonstrate a multiqubit blockade gate with atoms arranged in a
three-dimension spheroidal array. The gate performance is greatly improved by
the method of optimizing control-qubit distributions on the spherical surface
via evolutionary algorithm, which leads to an enhanced asymmetric Rydberg
blockade. This spheroidal configuration, not only arises a well preservation
for the dipole blockade energy between arbitrary control-target pairs, which
keeps the asymmetric blockade error at a very low level; but also manifests an
unprecedented robustness to the spatial position variations, leading to a
negligible position error. Taking account of intrinsic errors and with typical
experimental parameters, we numerically show that a C6NOT Rydberg gate can be
created with a fidelity of 0.992 which is only limited by the Rydberg state
decays. Our protocol opens up a new platform of higher-dimensional atomic
arrays for achieving multiqubit neutral-atom quantum computation.

##  I Introduction

Rydberg atoms serve as a reliable platform for studying quantum computing and
quantum simulation because of their strong and tunable interactions, which can
block the excitation of surrounding atoms in the vicinity of a preexcited atom
[1, 2, 3]. Via this so-called Rydberg blockade mechanism, versatile quantum
gates can be created [4, 5, 6, 7, 8] which manifest as basic logic-calculation
units for universal quantum computation [9, 10]. Among existing Rydberg-
mediated quantum gates, a multiqubit Toffoli (CnNOT) gate is an important
family member, which can offer an efficient implementation of Grover quantum
search algorithm to speedup the searches on a programmable quantum computer
[11] or to extend into any dimensional quantum systems [12]. A conventional
three-qubit Toffoli (C2NOT) gate can be implemented in a 1D array [13, 14]
where two outer control atoms constrain the behavior of middle target atom
with strong control-target interactions. However such a linear model is
unsuited for engineering a multiqubit gate because one target atom can not be
simultaneously manipulated by two nearest-neighbor control qubits due to the
blockade mechanism [15]. Therefore previous contributions to a multiqubit
Toffoli gate often rely on the assembly of several elementary gates [16, 17,
18, 19, 20] or the parallel operation on some clusters of atoms in a 1D array
of optical tweezers [21]. Direct execution of multiqubit Toffoli gates ($n\geq
3$) remains a big challenge both in theory and experiment.

To date, several studies for multiqubit gates use the way of adiabatic
passages in which the evolution of states can be performed by obeying a
multiqubit dark eigenstate with complex optimal pulses [22, 23]. An
alternative way for this target depends on nonadiabatic holonomic quantum
computation showing a C3NOT gate with an error of 0.0018 [24]. Another
prominent idea to the realization of multiqubit Rydberg gates adopts
asymmetric blockade as proposed in [25], in which there exists a large
separation of scales between different types of Rydberg interactions [26, 27,
28, 29]. However we note that, the asymmetric interaction condition breaks
easily when the number of qubits is enlarged, especially for atoms arranged in
1D or 2D arrays where distant control-target interaction suffers from a
dramatic decrease. Recently J. Young and coworkers propose a 2D multiqubit
gate by placing many control and many target atoms at the same time, in which
the strong control-control and control-target interactions can be engineered
via extra microwave fields, leading to a perfect asymmetric blockade [29]. But
this scheme is still unsuitable for implementing individual multicontrol [30,
31, 32] or multitarget [33] gates due to the absence of strong and tunable
interactions between distant atoms.

In the present work, inspired by the development of defect-free atom arrays
from 2D to 3D platforms where arbitrary atoms can be arranged expectantly in
space [34, 35, 36, 37, 38, 39] (a recent work has reported mixed-species atom
arrays with arbitrary geometry [40]), we propose a scheme for implementing
CnNOT gates with atoms individually arranged in a 3D spheroidal atomic array.
As illustrated in Fig.1, we consider a single target atom(in green) located at
the center and $n$ control atoms(in red) on the surface. Such a 3D atomic
array can be treated as an assembling of multi-layer 2D lattices and easily
achieve single-site Rydberg addressing [41]. Compared to the existing
asymmetric-blockade-based protocols [25, 28, 29, 27, 26], our scheme benefits
from an optimal 3D configuration to maximize the asymmetry of blockade,
representing an unprecedented robustness to the 3D atomic position variations.
The synthetic interplay between interatomic Rydberg-Rydberg interactions and
the optimal geometry results in a huge asymmetric blockade, making the gate
imperfection dominated by a intrinsic decay error, and the asymmetric blockade
error can be suppressed to a negligible level. Our results show that a simple
estimate of decay errors gives rise to an acceptable fidelity of 0.9537 for a
multiqubit C12NOT gate. This 3D Rydberg quantum gates can serve as a new gate-
unit for parallel operation in 3D optical tweezers, promising for scalable
quantum computation with more flexibility.

##  II Maximizing asymmetric blockade via optimization

Figure 1: Realization of spheroidal multiqubit Toffoli gates in a 3D atomic
array. Upper panels: optimized distributions of $n$ control atoms on the
surface accompanied by the optimal geometries explicitly shown below. Main
panel: Amplification of the C6NOT gate with four control atoms $c_{1\sim 4}$
at the equatorial plane and two control atoms $c_{5\sim 6}$ at the south and
north poles. The target atom is placed at the center. (a) Distribution of
$c_{2}$ after $10^{4}$ optimizations via evolutionary algorithm. (b-c) The
atomic energy levels as well as the atom-light interactions. In the presence
of a static electric field, we fix the quantization axis along $+\hat{z}$ to
simplify the optimization, which arises an angular-dependent dipole-dipole
interaction $U_{c_{j}t}(\theta_{c_{j}t})$ for each control-target atomic pair,
while the control-control interaction
$U_{c_{j}c_{j}^{\prime}}(|\mathbb{r}_{j}-\mathbb{r}_{j^{\prime}}|)$ is of
vdWs-type which depends on the intraspecies distance
$R_{c_{j}c_{j}^{\prime}}$.

To achieve desirable asymmetric interactions we adopt optimization with
evolutionary algorithm. Since each atom contains two Rydberg states
$|p_{j,t}\rangle$ and $|s_{j,t}\rangle$ with subscript $j(t)$ for the
control(target) atom, we consider the interaction between atoms in product
states $|p_{j}s_{t}\rangle$ and $|s_{j}p_{t}\rangle$ is of resonant dipole-
dipole feature [42]

$\hat{U}_{c_{j}t}(\theta_{c_{j}t})=\frac{C_{3}^{sp}(\theta_{c_{j}t})}{R_{ct}^{3}}(|p_{j}s_{t}\rangle\langle
s_{j}p_{t}|+|s_{j}p_{t}\rangle\langle p_{j}s_{t}|),$ (1)

with $C_{3}^{sp}(\theta_{c_{j}t})=C_{3}(1-3\cos^{2}\theta_{c_{j}t})$ and
$C_{3}=|\mu_{sp}|^{2}/(8\pi\epsilon_{0})$ (see Appendix A for more details).
Note that the spheroidal structure can preserve the control-target distance
$R_{ct}$ unchanged so $U_{c_{j}t}$ only depends on the polarizing angle
$\theta_{c_{j}t}$. On the other hand there is also a vdWs interaction between
two control atoms, given by [43]

$\hat{U}_{c_{j}c_{j^{\prime}}}(|\mathbb{r}_{j}-\mathbb{r}_{j^{\prime}}|)=\frac{C_{6}}{R_{c_{j}c_{j^{\prime}}}^{6}}|p_{j}p_{j^{\prime}}\rangle\langle
p_{j}p_{j^{\prime}}|,$ (2)

with $R_{c_{j}c_{j^{\prime}}}=|\mathbb{r}_{j}-\mathbb{r}_{j^{\prime}}|$ the
control-control distance, which is a vdWs energy shift of the pair state
$|p_{j}p_{j^{\prime}}\rangle$ considered arising from a second-order
approximation of the nonresonant dipole-dipole interaction.

Below we focus on how to achieve best asymmetric blockade using 87Rb Rydberg
states: $|s_{t,j}\rangle=|(m+1)S_{1/2},m_{j}=\frac{1}{2}\rangle$,
$|p_{t,j}\rangle=|mP_{3/2},m_{j}=\frac{3}{2}\rangle$ as in [26]. The $C_{3}$
coefficient for $m=60$ is about $C_{3}/2\pi=4.194$ GHz$\cdot\mu$m3 and
$C_{6}/2\pi=-12.0$ GHz$\cdot\mu$m6 calculated by the ARC open source library
[44]. Since $U_{c_{j}t}\propto(1-3\cos^{2}\theta_{c_{j}t})$ and
$U_{c_{j}c_{j^{\prime}}}\propto 1/R_{c_{j}c_{j^{\prime}}}^{6}$, the condition
of strongly asymmetric interactions $U_{c_{j}t}\gg U_{c_{j}c_{j^{\prime}}}$
i.e. any control atom can block the excitation of the target atom without
blocking other control atoms, can be readily met if $R_{c_{j}c_{j}^{\prime}}$
is appropriate. In Appendix B we verify the establishment of strong asymmetric
interactions by calculating the leakage error due to nonresonant Rydberg
couplings nearby. We also note that the asymmetry increases for small
principal quantum number because the coefficient $C_{3}$($C_{6}$) scales as
$\sim m^{4}$($\sim m^{11}$) [25]. Lowering $m$ can realize a 3D quantum gate
with more control qubits (details in Sec. V).

Here, in order to maximize asymmetric blockade, we have to optimize the
spatial positions of all control atoms accompanied by a dipole-angle
optimization. For arbitrary control atom $c_{j}$ a factor characterizing
asymmetry is defined as

$\chi_{j}=U_{c_{j}t}/U_{c_{j}c_{j^{\prime}}},$ (3)

which must be maximized. Intuitively, as increasing $n$ the vdWs interaction
is enhanced so as to easily break the asymmetry. To determine maximal
$n_{max}$ permitted for a chosen radius $R_{ct}$(the scale of 3D array), we
perform a global optimization to the atomic positions via evolutionary
algorithm [45]. A detailed description of optimization algorithm can be found
in Appendix C. For achieving strong asymmetric blockade, we set $\chi_{j}>100$
which means the minimal value of $\chi_{j}$ should satisfy
$\min(\chi_{j})>100$ for any $c_{j}$. This limitation leads to $n_{max}=8$
when $R_{ct}=5.0$ $\mu$m and $m=60$. Several optimal geometries are shown in
Fig. 1(upper panels) where the positions of control atoms denoted as red dots,
are precisely obtained by sufficient optimization. This optimal structure does
not depend on the coefficients $C_{3}$ or $C_{6}$ chosen and is stably
existing. For an even $n$ value, the structure looks more regular. Physics
behind these optimal geometries can be understood by seeking for a maximal
asymmetry between dipole-dipole interaction and vdWs interaction, where the
potential energy of system reaches its global minimum, corresponding to a
maximal magnitude of dipole-dipole interaction. This specific geometry is
formed by a competition between attractive vdWs interactions and inhomogeneous
dipole-dipole interactions which is discussed in Appendix D. Fig.1(a)
represents an amplified position distribution of $c_{2}$ under $10^{4}$
optimization. They are extremely condensed in space, confirming the accuracy
of algorithm.

##  III Gate performance and decay error

As examples we investigate the gate performance of an optimal 3D C6NOT gate.
The effective non-Hermitian Hamiltonian including the dissipative dynamics, is
expressed as

$\hat{\mathcal{H}}_{eff}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{I}-\frac{i}{2}\sum_{k}\hat{\mathcal{L}}_{k}^{\dagger}\hat{\mathcal{L}}_{k},$
(4)

with $k$ the indices of Rydberg levels $|p_{j,j^{\prime},t}\rangle$ and
$|s_{j,t}\rangle$, and the Hamiltonians

$\displaystyle\hat{\mathcal{H}}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\\{\Omega_{c}\sum_{j}^{n}(|0_{j}\rangle\langle
p_{j}|+|p_{j}\rangle\langle 0_{j}|)$ $\displaystyle+$
$\displaystyle\Omega_{t}[|1_{t}\rangle\langle s_{t}|+|s_{t}\rangle\langle
1_{t}|+(|0_{t}\rangle\langle s_{t}|+H.c.)]\\},$
$\displaystyle\hat{\mathcal{H}}_{I}$ $\displaystyle=$
$\displaystyle\sum_{j,j^{\prime}>j}U_{c_{j}c_{j^{\prime}}}+\sum_{j}U_{c_{j}t}$
(6)

represent the atom-light couplings and the atom-atom Rydberg interactions.
$\Omega_{c(t)}$ is the Rabi frequency for the control(target) atoms. To
characterize the gate performance we calculate the average gate fidelity

$\displaystyle\bar{\mathcal{F}}_{n}=\frac{1}{2^{n+1}}\text{Tr}\\{[\sqrt{\rho_{et}}|\bar{\Psi}_{out}\rangle\langle\bar{\Psi}_{out}|\sqrt{\rho_{et}}]^{1/2}\\}$
(7)

by solving the stochastic Schrödinger equation subject to arbitrary
computational basis $|\Psi\rangle$ [46]:

$\partial_{t}|\Psi\rangle=-i\hat{\mathcal{H}}_{eff}|\Psi\rangle.$ (8)

During each time interval $\delta t$, one generates a random number $\delta p$
and compares it with the instantaneous population on Rydberg states. If
$\delta p$ is larger, the system will evolve by obeying the Schrödinger
equation (8); otherwise one generates a random Rydberg excitation via a
quantum jump [47]. The total random number is $t_{det}/\delta t$ where
$t_{det}=2\pi/\Omega_{c}+3\pi/\Omega_{t}$ is the gate duration. By
initializing $2^{n+1}$ input states, $\bar{\Psi}_{out}$ denotes the average
output at $t=t_{det}$ after 500 stochastic evolutions and $\rho_{et}$ is an
etalon matrix. In addition, the operator

$\hat{\mathcal{L}}_{k}=\sqrt{\Gamma_{k}}(|1\rangle\langle k|+|0\rangle\langle
k|)$ (9)

indicates the spontaneous population decay of Rydberg levels, in which the
decay rates are $\Gamma_{k}=\Gamma_{p}$ for $k=p_{j,j^{\prime},t}$ and
$\Gamma_{k}=\Gamma_{s}$ for $k=s_{j,t}$.

Figure 2: Average gate fidelity $\bar{\mathcal{F}}_{n}$ as a function of the
control atomic number $n$, estimated by the numerical Monte Carlo method(red
triangles) and the theoretical expression(blue stars).

By performing further calculations, for $n=6$ we find
$(\min(U_{c_{jt}}),\max(U_{c_{j}c_{j}^{\prime}}))/2\pi=(33.552,0.096)$ MHz,
leading to the asymmetry: $\min(\chi_{j})=349.5>100$. Such a huge asymmetry
can keep the intrinsic asymmetric error originating from imperfect control-
target(control) (anti)blockade at a very low level $<10^{-5}$. In turn we
extend this asymmetric condition to a more generalized form, as

$\min(U_{c_{j}t}),\Omega_{c}\gg\Omega_{t}\gg\max(U_{c_{j}c_{j^{\prime}}}),$
(10)

which is also related to relevant pulse strengths [25]. Based on Eq.(10) we
assume

$\Omega_{t}=\min(U_{c_{j}t})/20,\Omega_{c}=5\min(U_{c_{j}t})$ (11)

throughout the paper. The decay rate is $\Gamma_{s}=5.0$ kHz and
$\Gamma_{p}=3.4$ kHz in a cryogenic environment [48], we find a gate fidelity
of $\bar{\mathcal{F}}_{6}=0.9920$ which is mainly constrained by the decay
error from Rydberg levels(the decay error is about $8\times 10^{-3}$ estimated
by Eq.(12)). The overall gate time is $t_{det}\approx 3\pi/\Omega_{t}=894$ ns.
Detailed description of the gate operation can be found in Appendix E. In
Fig.2 we show that the average gate fidelity $\bar{\mathcal{F}}_{n}$(red
triangles and texts, estimated by MC) decreases with the control atom number
$n$. For comparison, it is instructive to recall the decay-error expression
[26]

$\mathcal{E}_{n,se}=\frac{1}{2^{n}}\frac{\pi\Gamma_{s}}{\Omega_{t}}+n\frac{3\pi\Gamma_{p}}{2\Omega_{t}}+n\frac{\pi\Gamma_{p}}{2\Omega_{c}},$
(12)

by which the gate fidelity can be analytically obtained according to
$\bar{\mathcal{F}}_{n}\approx 1-\mathcal{E}_{n,se}$(blue stars and texts). A
good agreement is observable between the theoretical and numerical predictions
which confirms that other intrinsic asymmetric errors including blockade error
and antiblockade error, are both negligible due to the huge asymmetry in our
scheme.

##  IV Resilience to position variations

Owing to the finite temperature which leads to atomic position variations in
the optical trap, the interatomic interaction strength is slightly different
for each measurement. This so-called position error could catastrophically
break the implementation of Rydberg antiblockade(RAB)-based gates which depend
on a severely modified RAB condition [49, 50, 51, 52]. Although the excitation
annihilation as reviewed in [53] or transition slow-down effect [54] makes
blockade gates benefited from a robustness against interaction fluctuations,
most current achievements are still constrained to fewer-qubit gates [55, 56]
because the blockade strength decreases significantly for two distant atoms.
Here we express the control(target) atom position as

$\mathbb{r}_{j(t)}=\mathbb{r}_{0,j(t)}+\delta\mathbb{r}_{j(t)},$ (13)

where $\mathbb{r}_{0,j}=(R_{ct},\theta_{c_{j}t},\phi_{j})$ is obtained by
optimization and $\mathbb{r}_{0,t}=(0,0,0)$. The displacements
$\delta\mathbb{r}_{j(t)}$ originating from thermal motion of atoms, can be
modeled as a 3D Gaussian function with widths
$\sigma_{x,y,(z)}=\sqrt{k_{B}T_{a}/mw_{x,y,(z)}^{2}}$ for radial(axial)
localizations. Inspired by the experimental data in Ref. [57] we consider two
cases: $\sigma_{x}\in[0,2.0]$ $\mu$m, $\sigma_{y,z}=0.27$ $\mu$m and
$\sigma_{z}\in[0,2.0]$ $\mu$m, $\sigma_{x,y}=0.27$ $\mu$m. For Rb atoms held
at a low temperature $T_{a}=10$ $\mu$K, the optical trap with frequencies
$2\pi\times 18.22$ kHz and $2\pi\times 2.46$ kHz arise a position uncertainty
of 0.27 $\mu$m and $2.0$ $\mu$m, respectively. To estimate the errors from 3D
position variations we also use the way of stochastic Schrödinger equation and
obtain the numerical solution by averaging over 500 independent trajectories.

Figure 3: Imperfection of the gate fidelity based on two different geometries
vs the position variations along (a-b) $\hat{x}$ and (c-d) $\hat{z}$
directions. The standard deviations are $\sigma_{y,z}=0.27$ $\mu$m in (a-b)
and $\sigma_{x,y}=0.27$ $\mu$m in (c-d). Each point denotes an average of 500
measurements. (a,c)(or (b,d)) are obtained from an optimal 3D C6NOT gate(a 2D
honeycomb-type (6+1) CNOT gate). The shadings indicate a maximal position
error during the calculation.

The numerical solutions in Fig.3(a) and (c) indicate that the 3D gate protocol
can show an unprecedented robustness to the fluctuated interactions in all
directions. Because in a 3D optimal configuration the position variations of
atoms can be partially overcome keeping the infidelity at a small level of
$10^{-4}$. In contrast, arranging $(6+1)$ atoms in a 2D honeycomb lattice will
lead to a clear enhancement of the infidelity as shown in Fig.3(b) and (d).
Especially for the radial fluctuation the imperfection dramatically increases
with $\sigma_{x}$, agreeing with previous results [57, 58, 59]. Note that a 1D
chain model can not preserve the asymmetric blockade condition so as to be
unable to engineer a multiqubit quantum gate. Other technical imperfections
such as the sensitivity to motional dephasing, laser intensity noise and laser
phase noise would be discussed in Appendix F. A specific discussion for the
leakage error due to off-resonantly coupled Rydberg pair states, will be given
in Appendix B.

##  V Large-scale multiqubit gate

Figure 4: Estimates of maximal control-atom number $n_{max}$(top, color
blocks) together with the average gate fidelity $\bar{\mathcal{F}}_{n}$(black
texts, estimated by $1-\mathcal{E}_{n,se}$) in the space of $(R_{ct},m)$. For
different principal quantum number $m=(45,50,55,60,65,70,75,80)$ and at a
temperature of 77 K, the calculated coefficients are
$C_{6}/2\pi=(-1,-2,-6,-12,-24,-57,-137,-288)$ GHz$\cdot\mu$m6,
$C_{3}/2\pi=(1.370,1.950,2.912,4.194,5.859,7.976,10.620,13.873)$
GHz$\cdot\mu$m3, the Rydberg decay rates are
$\Gamma_{s}=(11.70,8.55,6.53,5.00,4.09,3.33,2.76,2.33)$ kHz and
$\Gamma_{p}=(6.85,5.50,4.26,3.40,2.73,2.25,1.88,1.60)$ kHz. All parameters are
taken from the ARC open source library [44].

This multiqubit Toffoli gates can be treated as a new calculation unit for
large-scale quantum information processor [60]. Compared to traditional fewer-
qubit gates [61, 62, 63, 64, 65] our protocol benefits from an optimal 3D
geometry with arbitrary $n$($<n_{max}$) control atoms. To fully determine
maximal number $n_{max}$ we graphically study it by tuning the principal
quantum number $m$ and the spherical radius $R_{ct}$ simultaneously. A shown
in Fig.4 it is clear that $n_{max}$ increases by lowering $m$ since the
asymmetry of interactions increases then, yet at the expense of the gate
fidelity. Because the decay rates $\Gamma_{s(p)}$ grow at the same time,
leading to a larger decay error. On the other hand, we find $n_{max}$ has a
dramatic increase with the radius $R_{ct}$. Because for a same $m$ the
absolute values $\Omega_{t}$ and $\Omega_{c}$ which depend on the control-
target interaction $U_{c_{j}t}$[Eq. (11)], would strongly decrease if $R_{ct}$
is enhanced. A smaller laser Rabi frequency will elongate the gate operation
time, making the decay error dominant. So for $R_{ct}=7$ $\mu$m a C16NOT gate
suffers from a very low fidelity of 0.2821 when $m=45$. A high-fidelity
multicontrol Rydberg-blockade gate can be accomplished by a dual consideration
of both the asymmetry and the Rydberg-state decay. Our theoretical estimation
shows that a C12NOT(12 control qubits and 1 target qubit) gate with a fidelity
of 0.9537 is possible when $R_{ct}=7$ $\mu$m and $m=60$.

##  VI Conclusion and outlook

We have studied a protocol of multicontrol-qubit Toffoli gates in which all
control atoms are precisely arranged on a 3D spherical surface via
optimization, which ensures a best asymmetric Rydberg blockade. These optimal
geometries(see Fig. 1) are obtained by performing sufficient optimization
based on evolutionary algorithm. Such a spheroidal gate has many advantages.
First, it allows for a perfect preservation of strong Rydberg blockade between
any control-target atom pairs, avoiding the effect of dramatic reduction in
blockade strength due to distant control-target atoms as in 1D or 2D arrays.
Second, an efficient optimization can ensure best asymmetric Rydberg blockade
leading to a negligible asymmetric blockade error. Finally and most
importantly, an unprecedented insensitivity $\sim 10^{-4}$ to the position
variations can be observed within the 3D gate due to the compensation of
three-dimensional spacial fluctuations. In comparison, Ref.[57] reports a
position error of $2.5\times 10^{-3}$ for a radius deviation of 0.16 $\mu$m
via excitation annihilation mechanism. Our work shows a minimal error of
0.0463 when 12 control atoms monitor one target atom.

The scheme for an arbitrary $(n+1)$-qubit Toffoli gate can offer a direct
route to multiqubit quantum computation. Upon the basis of one-step
implementation to fewer-qubit quantum gates by our group recently [56], this
3D blockade-gate scheme can be used to reduce the number of fewer-qubit gates,
greatly lowering the complexity of quantum device design [19, 20]. Other
straightforward applications with multiqubit gates refer to the production of
Rydberg-mediated entanglement between two atom qubits [57] or within a
mesoscopic ensemble of atoms [66], and of fast quantum computation with
neutral Rydberg qubits [67] .

###### Acknowledgements.

This work is supported by the National Key Research and Development Program of
China under Grant No. 2016YFA0302001; by the National Natural Science
Foundation of China under Grants Nos. 12174106, 11474094, 11104076, 11804308,
91950112, 11174081; the Science and Technology Commission of Shanghai
Municipality under Grant No. 18ZR1412800, and the ECNU Academic Innovation
Promotion Program for Excellent Doctoral Students under Grant No.
YBNLTS2019-023.

## Appendix A: Asymmetric interactions

Taking account of the scheme feasibility we present details on the Rydberg
pair states and their interactions in order to show the establishment of
asymmetric interactions. We assume the two Rydberg states of each atom which
are $|s_{t,j}\rangle=|61S_{1/2},\frac{1}{2}\rangle$,
$|p_{t,j}\rangle=|60P_{3/2},\frac{3}{2}\rangle$. In the presence of a static
electric field, when two atoms(control and target) are prepared in two
different dipole-coupled Rydberg states such as $|p_{j}\rangle$ and
$|s_{t}\rangle$, the pair state $|p_{j}s_{t}\rangle$ is directly coupled to
the same-energy state $|s_{j}p_{t}\rangle$ by a resonant dipole-dipole
exchange interaction. Typically this dipole-dipole interaction $\mathcal{B}$
between a pair of Rydberg atoms can be given by [68]

$\mathcal{B}=\frac{1}{4\pi\varepsilon_{0}}[\frac{\bm{\mu}_{1}\cdot\bm{\mu}_{2}}{R^{3}}-3\frac{(\bm{\mu}_{1}\cdot\bm{R})(\bm{\mu}_{2}\cdot\bm{R})}{R^{5}}],$
(14)

where $\bm{\mu}_{1,2}$ stands for the electric dipole transition operators and
$\varepsilon_{0}$ the permittivity of a vacuum. $\bm{R}$ is the internuclear
distance and $R=|\bm{R}|$. Moreover an external static electric field
$\vec{E}$ defines the quantization axis $\hat{z}$ which controls the
orientation of the dipole moments relative to the separation vector $\bm{R}$,
yielding an anisotropic dipole-dipole interaction,

$\begin{split}&\mathcal{B}(\theta_{c_{j}t})=\\\
&\frac{1}{4\pi\varepsilon_{0}R^{3}}\\{\mu_{1+}\mu_{2-}+\mu_{1-}\mu_{2+}+\mu_{1z}\mu_{2z}(1-3\cos^{2}\theta_{c_{j}t})\\\
&-\frac{3\sin^{2}\theta_{c_{j}t}}{2}(\mu_{1+}\mu_{2+}+\mu_{1+}\mu_{2-}+\mu_{1-}\mu_{2+}+\mu_{1-}\mu_{2-})\\\
&-\frac{3\sin\theta_{c_{j}t}\cos\theta_{c_{j}t}}{\sqrt{2}}(\mu_{1+}\mu_{2z}+\mu_{1-}\mu_{2z}+\mu_{1z}\mu_{2+}+\mu_{1z}\mu_{2-})\\},\end{split}$
(15)

with $\theta_{c_{j}t}$ the polarizing angle between the internuclear axis and
the quantization axis $\hat{z}$. $\mu_{v,(x,y,z)}$ denotes the projections of
dipole matrix element $\bm{\mu}_{v}$ onto axis $\hat{x},\hat{y},\hat{z}$ and
$\mu_{v,\pm}=\mu_{v,x}\pm i\mu_{v,y}$ with $v\in(1,2)$. Accounting for the use
of $\sigma$-polarized transition between $|60P_{3/2},m_{j}=3/2\rangle$ and
$|61S_{1/2},m_{j}=1/2\rangle$ with respect to $\Delta m_{j}=\pm 1$, we can
ignore the term $\mu_{1z}\mu_{2z}$ which requires $\Delta m_{j}=0$. Then
Eq.(15) can be reorganized as

$\displaystyle\mathcal{B}$ $\displaystyle(\theta_{c_{j}t})$ $\displaystyle=$
$\displaystyle-$
$\displaystyle\frac{(1-3\cos^{2}\theta_{c_{j}t})}{8\pi\varepsilon_{0}R^{3}}(\mu_{1+}\mu_{2-}+\mu_{1-}\mu_{2+})$
$\displaystyle-$
$\displaystyle\frac{3\sin^{2}\theta_{c_{j}t}}{8\pi\varepsilon_{0}R^{3}}[\mu_{1+}\mu_{2+}+\mu_{1-}\mu_{2-}$
$\displaystyle+$
$\displaystyle\frac{\sqrt{2}\cos\theta_{c_{j}t}}{\sin\theta_{c_{j}t}}(\mu_{1+}\mu_{2z}+\mu_{1-}\mu_{2z}+\mu_{1z}\mu_{2+}+\mu_{1z}\mu_{2-})].$

Apparently, there are three types of angular dependence in Eq.(Appendix A:
Asymmetric interactions) while only the first term
$\propto(1-3\cos^{2}\theta_{c_{j}t})$ is appropriate. This corresponds to a
resonant exchange energy between states
$|s_{j}p_{t}\rangle\leftrightarrows|p_{j}s_{t}\rangle$ where $\Delta m_{j}=+1$
for one atom and $\Delta m_{j}=-1$ for the other. Other possible transitions
connecting with same combinations of $\Delta m_{j}=\pm 1$, are off-resonantly
coupled due to a big Stark shift via the electric field [68]. An estimation of
the leakage error to the gate fidelity from these nonresonant Rydberg levels
is illustrated in Appendix B.

In the main text we expert a resonant dipole-dipole interaction strength that
only varies as

$\langle
s_{j(t)}p_{t(j)}|\mathcal{B}(\theta_{c_{j}t})|p_{j(t)}s_{t(j)}\rangle=\frac{C_{3}^{sp}(\theta_{c_{j}t})}{R_{ct}^{3}},$
(17)

where $R_{ct}=R$ means the two-atom separation and the interaction coefficient
$C_{3}^{sp}$ scaling as $m^{4}$ takes a complex form of

$\begin{split}C_{3}^{sp}(\theta_{c_{j}t})&=\frac{|\mu_{sp}|^{2}(1-3\cos^{2}\theta_{c_{j}t})}{8\pi\varepsilon_{0}},\end{split}$
(18)

and the transition matrix element is

$\displaystyle\mu_{sp}=\langle
61S_{1/2},m_{j}=1/2|\mu|60P_{3/2},m_{j}=3/2\rangle.$

Finally, we can obtain the electric dipole-dipole Hamiltonian between a pair
of control and target atoms, which is

$\hat{U}_{c_{j}t}^{sp}(\theta_{c_{j}t})=\frac{C_{3}^{sp}(\theta_{c_{j}t})}{R_{ct}^{3}}(|p_{j}s_{t}\rangle\langle
s_{j}p_{t}|+|s_{j}p_{t}\rangle\langle p_{j}s_{t}|).$ (19)

On the other hand, as for two control atoms which are prepared in same Rydberg
level such as $|60P_{3/2},m_{j}=3/2\rangle$ the electric dipole-dipole
interaction $\mathcal{B}$ only plays roles at the second-order in perturbation
theory since an atomic state has a vanishing average electric dipole moment to
the first-order of perturbation [69]. As a result via $\mathcal{B}$ the pair
state $|p_{j}p_{j^{\prime}}\rangle=|60P_{3/2},60P_{3/2}\rangle$ is coupled to
other nearby pair states of opposite parity where the energy of those states
differs from that of $|60P_{3/2},60P_{3/2}\rangle$ by a big quantity. The
average effect gives rise to a second-order vdWs shift of the considered pair
state $|p_{j}p_{j^{\prime}}\rangle$ scaling as $\propto
C_{6}/R_{c_{j}c_{j}^{\prime}}^{6}$ where the coefficient $C_{6}$ roughly
scales as $m^{11}$($m$ is the principal quantum number). Details about the
influence from original nonresonant dipole-dipole coupled states would be
discussed in Appendix B. Therefore, the reduced vdWs-type interaction
Hamiltonian can be described by

$\hat{U}_{c_{j}c_{j^{\prime}}}=\frac{C_{6}}{R_{c_{j}c_{j^{\prime}}}^{6}}|p_{j}p_{j^{\prime}}\rangle\langle
p_{j}p_{j^{\prime}}|.$ (20)

From Eqs.(19) and (20) it is apparent that both the dipole-dipole and vdWs
interactions between two Rydberg atoms are separation-dependent. To reach a
huge asymmetry in the interaction i.e.
$\frac{C_{3}^{sp}(\theta_{c_{j}t})}{R_{ct}^{3}}\gg\frac{C_{6}}{R_{c_{j}c_{j^{\prime}}}^{6}}$,
we have to seek for optimal distributions of all control atoms on the
spherical surface, see more details in Sec.II.

## Appendix B: Leakage error estimation

Leakage error based on two-atom states. As illustrated in Fig.5a we consider a
resonant dipole-dipole interaction between one control atom and one target
atom for the $|p_{j}s_{t}\rangle\leftrightarrows|s_{j}p_{t}\rangle$
transition. In a real implementation these two-atom pair states might still
experience nonresonant dipole-dipole couplings to other undesired Rydberg pair
states, resulting in a leakage error to the gate fidelity. Here the
nonresonant coupling strength and the Förster energy defect are denoted as
$B_{\kappa}$ and $\delta_{\kappa}$ respectively. Our task is to find out the
influence of these nonresonant couplings to the gate fidelity estimated in our
protocol. In principle we should sum over all selection-rule permitted
transitions over a wide range of principal quantum numbers and calculate the
leakage error. Here we have checked all possible transitions from
$|p_{j}s_{t}\rangle$ and $|s_{j}p_{t}\rangle$ to other leakage states and find
that the influence of a farther state can be almost negligible due to its
weaker coupling strength $B_{\kappa}$ or a larger energy defect
$\delta_{\kappa}$. In the calculation the factor $B_{\kappa}/\delta_{\kappa}$
is used to characterize the leakage strength that is proportional to the
leakage error. If $B_{\kappa}/\delta_{\kappa}\ll 1$ the leakage from the
resonantly-coupled state can be suppressed [63].

In Fig.5a and Table I(left) we show the possible transitions with the change
of principal quantum number up to $\pm 2$ from the resonant pair states
$|p_{j}s_{t}\rangle\leftrightarrows|s_{j}p_{t}\rangle$. To estimate the gate
error due to the leakage of population from these states, we solve the
stochastic Schrödinger equation (8) with respect to the Hamiltonian

$\displaystyle\mathcal{H}_{ps}$ (21) $\displaystyle=$
$\displaystyle\frac{1}{2}\Omega_{2}(t)|p_{j}1_{t}\rangle\langle
p_{j}s_{t}|+\frac{1}{2}\Omega_{3}(t)|p_{j}s_{t}\rangle\langle p_{j}0_{t}|$
$\displaystyle+$ $\displaystyle(B_{0}|p_{j}s_{t}\rangle\langle
s_{j}p_{t}|+B_{\kappa}|p_{j}s_{t}\rangle\langle a_{\kappa j}b_{\kappa
t}|+B_{\kappa}|s_{j}p_{t}\rangle\langle b_{\kappa j}a_{\kappa t}|$
$\displaystyle+$ $\displaystyle H.c.)+\delta_{\kappa}(|b_{\kappa j}a_{\kappa
t}\rangle\langle b_{\kappa j}a_{\kappa t}|+|a_{\kappa j}b_{\kappa
t}\rangle\langle a_{\kappa j}b_{\kappa t}|),$

where we treat $|p_{j}1_{t}\rangle$ as the initial state and apply two pulses
$\Omega_{2}(t)$ and $\Omega_{3}(t)$. Ideally a pre-excitation of the control
atom would block the excitation of the target atom so the population missing
from state $|p_{j}1_{t}\rangle$ can be regarded as the leakage error. See the
last column of Table I(left), the population rotation error shows a clear
decrease from case 1 to case 4 and it is almost negligible in cases 3 and 4
due to the tiny leakage error in the range of $10^{-7}\sim 10^{-5}$. Based on
the results we choose two pairs of nonresonant coupled states:
$|a_{1j}b_{1t}\rangle$ and $|b_{1j}a_{1t}\rangle$($\kappa=1$),
$|a_{2j}b_{2t}\rangle$ and $|b_{2j}a_{2t}\rangle$($\kappa=2$) as the dominant
leakage states with respect to the resonant transition between
$|p_{j}s_{t}\rangle\leftrightarrows|s_{j}p_{t}\rangle$.

Figure 5: (a) For one control and one target atoms, some dominant leakage
levels with respect to the resonant dipole-dipole interaction between
$|p_{j}s_{t}\rangle$ and $|s_{j}p_{t}\rangle$, where
$|p_{j,t}\rangle=|60P_{3/2},3/2\rangle$ and
$|s_{j,t}\rangle=|61S_{1/2},1/2\rangle$. Other selection-rule permitted pair
states are off-resonantly coupled by obeying
$|p_{j}s_{t}\rangle\leftrightarrows|a_{\kappa j}b_{\kappa t}\rangle$(forward)
and $|s_{j}p_{t}\rangle\leftrightarrows|b_{\kappa j}a_{\kappa
t}\rangle$(backward) with strength $B_{\kappa}$ and Förster energy defect
$\delta_{\kappa}$ and $\kappa\in(1,2,3,4)$. (b) Some leakage levels related to
the detuned dipole-dipole interaction between two control atoms where
$|p_{1}p_{2}\rangle=|60P_{3/2},3/2;60P_{3/2},3/2\rangle$. $B_{\kappa}$ and
$\delta_{\kappa}$ represent the coupling strength and the corresponding energy
defect between $|p_{1}p_{2}\rangle$ and other undesired nonresonant Rydberg
pair states $|Pr_{\kappa}\rangle=\\{|a_{c1}b_{c1}\rangle$,
$|a_{c1}b_{c2}\rangle$, $|a_{c2}b_{c1}\rangle$, $|a_{c2}b_{c2}\rangle\\}$,
where $|a_{c1}\rangle=|60S_{1/2},1/2\rangle$,
$|a_{c2}\rangle=|58D_{5/2},5/2\rangle$,
$|b_{c1}\rangle=|61S_{1/2},1/2\rangle$,
$|b_{c2}\rangle=|59D_{5/2},5/2\rangle$.

$\kappa$ | $|a_{\kappa j}\rangle=|a_{\kappa t}\rangle$ | $|b_{\kappa j}\rangle=|b_{\kappa t}\rangle$ | $C_{3}^{sp(\kappa)}(0)$ | $B_{\kappa}/\delta_{\kappa}$ | $1-P_{p_{j}1_{t}}$ | $|Pr_{\kappa}\rangle$ | $C_{3}^{(\kappa)}$ | $\epsilon_{1}$ | $\epsilon_{2}$
---|---|---|---|---|---|---|---|---|---
1 | $|60S_{1/2},\frac{1}{2}\rangle$ | $|61P_{3/2},\frac{3}{2}\rangle$ | $-9.134$ | $8.3\times 10^{-2}$ | $9.11\times 10^{-4}$ | $|60S_{1/2},\frac{1}{2};61S_{1/2},\frac{1}{2}\rangle$ | $4.301$ | $4.27\times 10^{-5}$ | $3.21\times 10^{-2}$
2 | $|59D_{5/2},\frac{5}{2}\rangle$ | $|60P_{1/2},-\frac{1}{2}\rangle$ | $-9.254$ | $9.5\times 10^{-3}$ | $1.32\times 10^{-4}$ | $|60S_{1/2},\frac{1}{2};59D_{5/2},\frac{5}{2}\rangle$ | $5.919$ | $2.69\times 10^{-8}$ | $8.09\times 10^{-6}$
3 | $|58D_{5/2}\frac{5}{2}\rangle$ | $|61P_{3/2},-\frac{1}{2}\rangle$ | $-3.926$ | $3.6\times 10^{-3}$ | $3.93\times 10^{-5}$ | $|58D_{5/2},\frac{5}{2};61S_{1/2},\frac{1}{2}\rangle$ | $3.203$ | $2.15\times 10^{-8}$ | $8.76\times 10^{-6}$
4 | $|59S_{1/2},\frac{1}{2}\rangle$ | $|62P_{3/2},\frac{3}{2}\rangle$ | $-0.14$ | $2.1\times 10^{-4}$ | $3.57\times 10^{-7}$ | $|58D_{5/2},\frac{5}{2};59D_{5/2},\frac{5}{2}\rangle$ | $4.408$ | $1.43\times 10^{-8}$ | $8.75\times 10^{-6}$

Table 1: Leakage error due to the presence of several nonresonant dipole-
dipole coupled states nearby. Table(left): Leakage states for the resonant
pair states $|p_{j}s_{t}\rangle$ and $|s_{j}p_{t}\rangle$ which are
$|p_{j}s_{t}\rangle\leftrightarrows|a_{\kappa j}b_{\kappa t}\rangle$(forward)
and $|s_{j}p_{t}\rangle\leftrightarrows|b_{\kappa j}a_{\kappa
t}\rangle$(backward) with strength
$B_{\kappa}=C_{3}^{sp(\kappa)}/R_{ct}^{3}$($C_{3}^{sp(\kappa)}$ is in unit of
$2\pi\times$GHz$\cdot\mu$m3) and energy defect
$\delta_{\kappa}/2\pi=(0.8771,7.8032,8.6142,5.3452)$ GHz. Here
$\theta_{c_{j}t}=0$ and $B_{0}=C_{3}^{sp}(0)/R_{ct}^{3}$ with
$C_{3}^{sp}(0)/2\pi=-8.388$ GHz$\cdot\mu$m3. $R_{ct}=5$ $\mu$m treats as the
radius of sphere and the leakage error $1-P_{p_{j}1_{t}}$ is estimated by
calculating the population missing from state $|p_{j}1_{t}\rangle$.
Table(right): Off-resonance pair states $|Pr_{\kappa}\rangle$ with respect to
$|p_{1}p_{2}\rangle$ i.e.
$|p_{1}p_{2}\rangle\leftrightarrows|Pr_{\kappa}\rangle$ enabled by a
nonresonant dipole-dipole interaction
$B_{\kappa}=C_{3}^{(\kappa)}/R_{c_{1}c_{2}}^{3}$($C_{3}^{(\kappa)}$ is in unit
of $2\pi\times$GHz$\cdot\mu$m3). Here the energy defect is
$\delta_{\kappa}/2\pi=(0.2784,7.0614,7.4587,14.8012)$ GHz and $\epsilon_{1,2}$
stand for the leakage error from state $|0_{1}0_{2}\rangle$ under different
interatomic distances $R_{c_{1}c_{2}}=(2R_{ct},R_{ct})$ $\mu$m. All parameters
are taken from the ARC open-source library [44].

In addition we study the leakage error due to the nonresonant dipole-dipole
couplings with respect to the identical Rydberg pair state
$|p_{1}p_{2}\rangle$ for two control atoms. Here we apply four pairs of
dipole-allowed off-resonantly coupled states $|Pr_{\kappa}\rangle$ with
strength $B_{\kappa}=C_{3}^{(\kappa)}/R_{c_{1}c_{2}}^{3}$ and Förster energy
defect $\delta_{\kappa}$, see Table I(right). In principle we have to sum up
all possible nonresonant states at the same time which leads to a second-order
level shift of $|p_{1}p_{2}\rangle$ represented by a coefficient
$C_{6}\approx\sum_{\kappa}(C_{3}^{(\kappa)})^{2}/\delta_{\kappa}$.

In the calculation we again calculate the dynamics of a two-atom state
$|0_{1}0_{2}\rangle$ by following the Hamiltonian of

$\mathcal{H}_{pp}=\sum_{j}^{2}\mathcal{H}_{j}+(B_{\kappa}|p_{1}p_{2}\rangle\langle
Pr_{\kappa}|+H.c.)+\delta_{\kappa}|Pr_{\kappa}\rangle\langle Pr_{\kappa}|$
(22)

where $\mathcal{H}_{j}=\frac{1}{2}(\Omega_{1}(t)|p_{j}\rangle\langle
0_{j}|+\Omega_{5}(t)|0_{j}\rangle\langle p_{j}|)$. Starting from the initial
state $|0_{1}0_{2}\rangle$ we numerically estimate the population missing from
$|0_{1}0_{2}\rangle$($\epsilon_{1(2)}=1-P_{0_{1}0_{2}}$) by applying two
separated $\pi$ pulses $\Omega_{1}(t)$ and $\Omega_{5}(t)$. Note that the
nonresonant coupling strength $B_{\kappa}$ is inversely proportional to the
interatomic distance $R_{c_{1}c_{2}}$ so we use different values
$R_{c_{1}c_{2}}=(2R_{ct},R_{ct})$ $\mu$m. See the last two columns in Table I,
for a larger distance the leakage error $\epsilon_{1}$ is always below
$10^{-4}$ in all cases which means these off-resonantly coupled levels play a
negligible effect and it is reasonable to assume a pure vdWs shift represented
by $C_{6}/R_{c_{1}c_{2}}^{6}$ for the $|p_{1}p_{2}\rangle$ state. However if
the two-atom distance is too small e.g. $R_{c_{1}c_{2}}=R_{ct}$, it is
inappropriate to assume a vdWs interaction because the pair state
$|Pr_{1}\rangle$ can cause a big leakage error $\sim 0.0321$ which is even
larger than the intrinsic decay error. In all cases we find the effect of
other nonresonant states $|Pr_{2\sim 4}\rangle$ can be ignored as compared
with $|Pr_{1}\rangle$. Based on the analysis above we treat $|Pr_{1}\rangle$
as the dominant leakage state with respect to $|p_{1}p_{2}\rangle$ and will
include it in evaluating the leakage error of a realistic multiqubit quantum
gate.

Leakage error based on a multiqubit quantum gate. To correctly show the
results reasonable in the main text, we now discuss the real implementation of
a multiqubit quantum gate by taking account of off-resonant dipole-dipole
couplings from dominant leakage states. The system we simulate consists of $n$
control atoms and one target atom described by the Hamiltonian of

$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{I}.$ (23)

Here the first term $\hat{\mathcal{H}}_{0}$ as shown in Eq.(III) describes the
resonant atom-light couplings of control and target atoms. The second term
$\hat{\mathcal{H}}_{I}$[see Eq.(6)] describes the Rydberg states and their
interactions, where the resonant dipole-dipole interaction for control-target
atom pairs is(here $\kappa=1,2$)

$\displaystyle\hat{U}_{c_{j}t}$ $\displaystyle=$
$\displaystyle(B_{0}|p_{j}s_{t}\rangle\langle
s_{j}p_{t}|+B_{\kappa}|p_{j}s_{t}\rangle\langle a_{\kappa j}b_{\kappa t}|$
(24) $\displaystyle+$ $\displaystyle B_{\kappa}|s_{j}p_{t}\rangle\langle
b_{\kappa j}a_{\kappa t}|+H.c.)+\delta_{\kappa}(|b_{\kappa j}a_{\kappa
t}\rangle\langle b_{\kappa j}a_{\kappa t}|$ $\displaystyle+$
$\displaystyle|a_{\kappa j}b_{\kappa t}\rangle\langle a_{\kappa j}b_{\kappa
t}|),$

and the nonresonant dipole-dipole interaction between two identical control
atoms takes form of(here $\kappa=1$)

$\hat{U}_{c_{j}c_{j^{\prime}}}=(B_{\kappa}|p_{1}p_{2}\rangle\langle
Pr_{\kappa}|+H.c.)+\delta_{\kappa}|Pr_{\kappa}\rangle\langle Pr_{\kappa}|,$
(25)

with the strength $B_{\kappa}$ and the Förster energy defect $\delta_{\kappa}$
estimated in Table I. Note that the coefficient $C_{3}^{sp(\kappa)}$ depends
on a varying polarizing angle $\theta_{c_{j}t}$ while in Table I we set
$\theta_{c_{j}t}=0$.

In the numerical simulation we first consider $n=2$ which is a three-qubit
Toffoli gate showing a linear structure in space. We calculate the average
gate fidelity $\bar{\mathcal{F}}_{2}$[see Eq.(7)] over eight input states
$|\Psi_{in}\rangle=\\{|000\rangle,|001\rangle,|010\rangle,|011\rangle,|100\rangle,|101\rangle,|110\rangle,|111\rangle\\}$
by evolving the stochastic Schrödinger equation. The output state
$|\bar{\Psi}_{out}\rangle$ is obtained at $t=t_{det}$ after 500 stochastic
evolutions. Via taking account of the dominant leakage states we finally find
the average gate fidelity is $\bar{\mathcal{F}}_{2}=0.99793741$. As compared
with $\bar{\mathcal{F}}_{2}=0.99832354$ obtained in the case of no nonresonant
states, this value is slightly decreased by $3.9\times 10^{-4}$. The small
leakage error caused by nonresonant couplings, means that the asymmetric
blockade interactions can be well preserved in the implementation of a
$(2+1)$-qubit gate.

More remarkably, we also calculate a realistic spheroidal quantum gate with
four control atoms and one target atom since different atomic distances and
polarizing angles must be affected. After a computer-demanding calculation we
finally obtain $\bar{\mathcal{F}}_{4}=0.99360269$ which leads to a leakage
error of $7.1\times 10^{-4}$ with respect to the case that ignores the
nonresonant couplings. In Fig.2 we have shown that the average gate fidelity
for the case of $n=4$ is 0.99431526. This slight growth of the leakage error
with the number of control atoms mainly comes from a slightly weaker asymmetry
in the interaction. Because as $n$ increases, the nonresonant coupling
strength which depends on the interatomic distance $R_{c_{j}c_{j}^{\prime}}$,
becomes stronger arising a bigger leakage. However our simulations confirm
that, by considering off-resonantly coupled Rydberg states the leakage error
for a realistic 3D multiqubit quantum gate can still be kept at a negligible
level $\sim 10^{-4}$. Therefore the 3D multiquibit gate presented in the main
text can be implemented in a realistic environment which is enabled by a large
symmetry in the interaction.

## Appendix C: Evolutionary algorithm

Case | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ | $c_{5}$ | $c_{6}$
---|---|---|---|---|---|---
(i) | $(\pi/2,\pi/6)$ | $(\pi/2,\pi/2)$ | $(\pi/2,5\pi/6)$ | $(\pi/2,7\pi/6)$ | $(\pi/2,3\pi/2)$ | $(\pi/2,11\pi/6)$
| $(1.5731,3.1421)$ | $(1.5747,4.7111)$ | $(1.5732,0)$ | $(1.5705,1.5694)$ | $(0.0027,0.0007)$ | $(3.1412,0.0012)$
(ii) | $(0,0)$ | $(\pi/5,2\pi/5)$ | $(2\pi/5,4\pi/5)$ | $(3\pi/5,6\pi/5)$ | $(4\pi/5,8\pi/5)$ | $(\pi,2\pi)$
| $(1.5743,3.1433)$ | $(1.5744,4.7136)$ | $(1.5722,0)$ | $(1.5724,1.5753)$ | $(0.0003,0.0008)$ | $(3.1408,0.0056)$
(iii) | $(\pi/6,\pi/5)$ | $(\pi/3,2\pi/5)$ | $(\pi/2,3\pi/5)$ | $(2\pi/3,4\pi/5)$ | $(5\pi/6,\pi)$ | $(\pi,6\pi/5)$
| $(1.5701,3.1403)$ | $(1.5697,4.7125)$ | $(1.5711,0)$ | $(1.5707,1.5731)$ | $(0.0010,0.0011)$ | $(3.1419,0.0023)$
(iv) | $(\pi/7,\pi/5)$ | $(2\pi/7,\pi/6)$ | $(3\pi/7,\pi/7)$ | $(4\pi/7,\pi/8)$ | $(5\pi/7,\pi/9)$ | $(6\pi/7,\pi/10)$
| $(1.5705,3.1419)$ | $(1.5711,4.7119)$ | $(1.5695,0)$ | $(1.5703,1.5718)$ | $(0.0002,0.0005)$ | $(3.1416,0.0003)$
(v) | $(\pi/8,\pi)$ | $(\pi/4,\pi/2)$ | $(3\pi/8,\pi/3)$ | $(\pi/2,\pi/4)$ | $(5\pi/8,\pi/5)$ | $(3\pi/4,\pi/6)$
| $(1.5713,3.1408)$ | $(1.5716,4.7119)$ | $(1.5701,0)$ | $(1.5695,1.5705)$ | $(0.0013,0.0007)$ | $(3.1415,0.0001)$
Average results | $(1.5708,3.1416)$ | $(1.5707,4.7124)$ | $(1.5709,0)$ | $(1.5708,1.5707)$ | $(0.0005,0.0006)$ | $(3.1416,0.0007)$
Table 2:  (i-v) Five sets of parameters for random initialization
$(\theta_{j}^{(0)},\phi_{j}^{(0)})$ of atomic positions(first row) and for
optimized positions($(\theta_{j},\phi_{j})$, second row) under one
optimization run. Average results indicate the datum averaging over $10^{4}$
optimizations.

In this section we demonstrate how to get optimal geometries of control atoms
on the surface and determine the value $n_{max}$ via evolutionary algorithm.
Taking C6NOT gate as an example, first we randomly arrange the initial
positions of control atoms $c_{j}$ denoted as
$\mathbb{r}_{j}^{(0)}=(R_{ct},\theta_{j}^{(0)},\phi_{j}^{(0)})$ where the
superscript “0” means the initial step $p=0$. For a finite spherical radius
$R_{ct}$, one has $\mathbb{r}_{j}^{(0)}=(\theta_{j}^{(0)},\phi_{j}^{(0)})$ and
$j\in[1,6]$. Then we compare all asymmetry factors
$\chi_{j}^{(0)}=U_{c_{j}t}/U_{c_{j}c_{j^{\prime}}}$ with respect to atom
$c_{j}$ where the total number is C${}_{6}^{2}=15$, in order to find a minimal
value

$\chi_{0}=\min(\chi_{j}^{(0)}).$ (26)

Next we add a small perturbation to the position of control atoms, leading to

$\mathbb{r}_{j}^{(1)}=(\theta_{j}^{(0)}+\delta\theta_{j},\phi_{j}^{(0)}+\delta\phi_{j}),$
(27)

in which $\delta\theta_{j}$ or $\delta\phi_{j}$ is obtained randomly from the
range of $[-0.1\theta_{j}^{(0)},0.1\theta_{j}^{(0)}]$ or
$[-0.1\phi_{j}^{(0)},0.1\phi_{j}^{(0)}]$. With the new position
$\mathbb{r}_{j}^{(1)}$ we again compare all $\chi_{j}^{(1)}$ values and find
out a minimal value which is denoted as $\min(\chi_{j}^{(1)})$. To maximize
asymmetric blockade, $\chi_{1}$ is defined as

$\chi_{1}=\max(\min(\chi_{j}^{(0)}),\min(\chi_{j}^{(1)})),$ (28)

at step $p=1$ and the corresponding position $\mathbb{r}_{j}^{(0)}$ or
$\mathbb{r}_{j}^{(1)}$ will be ready for the next-step($p=2$) optimization.
This single-optimization process must be repeated with sufficient
iterations(typically $p>10^{5}$) until the condition
$|\chi_{p}-\chi_{p-1}|<10^{-5}$ is met where the universal maximal asymmetry
factor is given by

$\chi_{p}=\max(\min(\chi_{j}^{(0)}),\min(\chi_{j}^{(1)}),...\min(\chi_{j}^{(p)})).$
(29)

In order to avoid a local optimal solution we perform $10^{4}$ optimization
runs via evolutionary algorithm and achieve the optimal distribution of
control atoms $\mathbb{r}_{j}=(\theta_{j},\phi_{j})$. As shown in Fig.1(a) the
position of $c_{2}$ is obtained after $10^{4}$ optimizations which are very
condensed in space. To quantitatively verify this effect, in Table 2(i-v) we
exemplify five sets of parameters to show the robustness of our optimization
algorithm. Given the initial positions $(\theta_{j}^{(0)},\phi_{j}^{(0)})$
(first row of (i-v)), the results from single optimization(second row of
(i-v)) are very close to the average results after taking $10^{4}$
optimizations, which confirms the accuracy of evolutionary algorithm. For
ensuring a huge asymmetry we also set a limitation $\chi_{p}>100$ in the
optimization which arises a maximal $n_{max}$ permitted if $m$ and $R_{ct}$
are determined. E.g. in the case of $m=60$ and $R_{ct}=5$ $\mu$m, $n=8$ leads
to $\chi_{p}=102.986$, while $n=9$ leads to $\chi_{p}=64.310$ which breaks the
limitation, so $n_{max}=8$ is obtained.

## Appendix D: Optimal geometry

Figure 6: Collective potential eigenenergies $E_{\pm}$ and $E_{2,-}$
corresponding to (a) two and (b) four control atoms, vs the polarizing angle
$\theta$ and the polarizing and azimuthal angles ($\theta,\phi^{\prime}$),
respectively. Subscript $j$ is omitted for brevity. (b) In the case of four
control atoms we introduce another angle $\phi^{\prime}$ with respect to the
rotational coordinate of
$(\hat{x}^{\prime},\hat{y}^{\prime},\hat{z}^{\prime})$, in order to determine
the spatial position of atoms. On the right side of (b) we show three possible
distributions of control atoms in a 3D sphere corresponding to red dots, white
dots and green dots, where the absolute value of potential energy $E_{2,-}$
attains its maximum.

Case of two control atoms. To provide a physical understanding for these
optimal configurations we study the potentials of system. The basic ingredient
in optimization is to maximize the asymmetry of interaction, arising
$U_{c_{j}t}\gg U_{c_{j}c_{j}^{\prime}}$. To achieve this, we explore the
essential feature of system by concentrating on the Rydberg states
$|p_{j,t}\rangle$, $|s_{j,t}\rangle$ which connect with the interaction
Hamiltonians Eqs.(1-2). Here the index $j=1,...,n$ represents the number of
control atoms, and $t$ labels the target atom. We analyze the interaction
potential in the subspace only involving Rydberg states spanned by collective
states $|\pi_{i}\rangle$($i=1,...,n$), which means $i$ control atoms are in
$|p\rangle$ and others are in $|s\rangle$ [70]. For $n=2$, states
$|\pi_{i}\rangle$ can be written as

$\displaystyle|\pi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|s_{1}p_{2}p_{t}\rangle+|p_{1}s_{2}p_{t}\rangle),$
$\displaystyle|\pi_{2}\rangle$ $\displaystyle=$
$\displaystyle|p_{1}p_{2}s_{t}\rangle.$ (30)

Due to the exchange property of resonant dipole-dipole interaction
$|p_{j}s_{t}\rangle\leftrightarrows|s_{j}p_{t}\rangle$ this complete set of
basis is equivalent to the form of
$\\{|\pi_{0}\rangle,|\pi_{1}\rangle\\}=\\{|s_{1}s_{1}p_{t}\rangle,\frac{1}{\sqrt{2}}(|s_{1}p_{2}s_{t}\rangle+|p_{1}s_{2}s_{t}\rangle)\\}$.
In the basis of $|\pi_{1}\rangle$ and $|\pi_{2}\rangle$ as in Eq.(30), the
effective interaction Hamiltonian $\mathcal{H}_{n,int}$ is given by ($n=2$)

$\mathcal{H}_{2,int}=\left(\begin{array}[]{ccc}0&\frac{1}{\sqrt{2}}D_{2}\\\
\frac{1}{\sqrt{2}}D_{2}&B_{2}\\\ \end{array}\right),$ (31)

where $D_{n}=\sum_{j=1}^{n}{U_{c_{j}t}}$ and
$B_{n}=\sum_{j>j^{\prime}}{U_{c_{j}c_{j^{\prime}}}}$ and
$j,j^{\prime}\in[1,..,n]$. Analytical diagonalization of the Hamiltonian
matrix yields the eigenenergies, explicitly as

$E_{\pm}=\frac{B_{2}\pm\sqrt{B_{2}^{2}+2D_{2}^{2}}}{2}.$ (32)

It is apparent that, in Eq.(31) the off-diagonal resonant dipole-dipole
interaction $U_{c_{j}t}(\theta_{c_{j}t})\propto(1-3\cos^{2}(\theta_{c_{j}t}))$
is responsible for the population transfer between different $|\pi_{i}\rangle$
states, which leads to anisotropic spatial interactions. The diagonal term
$U_{c_{1}c_{2}}\propto|\mathbb{r}_{c_{1}}-\mathbb{r}_{c_{2}}|^{-6}$ also
depends on the relative distance between two control atoms $c_{1}$ and
$c_{2}$. We restrict all control atoms on the spherical surface so
$U_{c_{1}c_{2}}$ is minimized when two atoms are separated by a maximal
distance which is $2R_{ct}$(diameter), leading to
$\theta=\theta_{c_{1}t}=\pi-\theta_{c_{2}t}$. In this case $D_{2}$ and $B_{2}$
take explicit expressions as

$D_{2}=\frac{2C_{3}(1-3\cos^{2}\theta)}{R_{ct}^{3}},B_{2}=\frac{C_{6}}{(2R_{ct})^{6}},$
(33)

and the eigenenergies of two collective states $|\pi_{1,2}\rangle$ can be
independently controlled by a single polarizing angle $\theta$.

As shown in Fig. 6(a), we show the eigenenergies $E_{\pm}$ as a function of
the polarizing angle $\theta$. For $\theta=0$ or $\pi$ the minimum of
potential occurs which means the attractive dipole-dipole interaction $D_{2}$
attains its maximal(absolute) magnitude. In other word when the two control
atoms are placed at $(\theta,\phi)=(0,0)$ and $(\pi,0)$ the asymmetric
interaction could be maximized (note that $B_{2}$ is a constant). Our result
based on numerical optimization also gives to a same structure as the
theoretical prediction, see Fig.1(n=2). To this end we confirm the accuracy of
evolutionary algorithm which helps to obtain a maximal asymmetric interaction
by positioning atoms appropriately on the spherical surface.

Case of four control atoms. Situation becomes quite complex when more control
atoms are included, because the magnitude of potential energy is influenced by
multiple adjustable parameters. As examples we analyze the case of $n=4$. With
the inclusion of more Rydberg states the collective $|\pi_{i}\rangle$ states
can be rewritten as

$\displaystyle|\pi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|p_{1}s_{2}s_{3}s_{4}p_{t}\rangle+|s_{1}p_{2}s_{3}s_{4}p_{t}\rangle+|s_{1}s_{2}p_{3}s_{4}p_{t}\rangle$
$\displaystyle+$ $\displaystyle|s_{1}s_{2}s_{3}p_{4}p_{t}\rangle),$
$\displaystyle|\pi_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(|p_{1}p_{2}s_{3}s_{4}s_{t}\rangle+|p_{1}s_{2}p_{3}s_{4}s_{t}\rangle+|p_{1}s_{2}s_{3}p_{4}s_{t}\rangle$
$\displaystyle+$
$\displaystyle|s_{1}p_{2}p_{3}s_{4}s_{t}\rangle+|s_{1}p_{2}s_{3}p_{4}s_{t}\rangle+|s_{1}s_{2}p_{3}p_{4}s_{t}\rangle),$
$\displaystyle|\pi_{3}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|p_{1}p_{2}p_{3}s_{4}p_{t}\rangle+|p_{1}p_{2}s_{3}p_{4}p_{t}\rangle+|p_{1}s_{2}p_{3}p_{4}p_{t}\rangle$
$\displaystyle+$ $\displaystyle|s_{1}p_{2}p_{3}p_{4}p_{t}\rangle),$
$\displaystyle|\pi_{4}\rangle$ $\displaystyle=$
$\displaystyle|p_{1}p_{2}p_{3}p_{4}s_{t}\rangle,$ (34)

arising the effective interaction Hamiltonian $\mathcal{H}_{4,int}$ given by

$\mathcal{H}_{4,int}=\left(\begin{array}[]{cccc}0&\frac{\sqrt{3}}{2\sqrt{2}}D_{4}&0&0\\\
\frac{\sqrt{3}}{2\sqrt{2}}D_{4}&\frac{1}{6}B_{4}&0&0\\\
0&0&\frac{1}{2}B_{4}&\frac{1}{2}D_{4}\\\ 0&0&\frac{1}{2}D_{4}&B_{4}\\\
\end{array}\right),$ (35)

with its eigenenergies explicitly as

$\displaystyle E_{1,\pm}=\frac{3B_{4}\pm\sqrt{B_{4}^{2}+4D_{4}^{2}}}{4},$ (36)
$\displaystyle E_{2,\pm}=\frac{B_{4}\pm\sqrt{B_{4}^{2}+216D_{4}^{2}}}{12}.$
(37)

However due to multiple adjustable parameters from four control atoms(there
are overall eight degrees of freedom $(\theta_{1\sim 4},\phi_{1\sim 4})$ in
the bare spherical coordinate) it is difficult to find out its maximal
magnitude which corresponds to a maximal asymmetric interaction. By
considering the symmetry of potentials we guess a regular-tetrahedron
structure. As displayed in Fig. 6(b)(right inset) an optimal regular-
tetrahedron configuration is freely rotated in which the polarizing angle
$\theta$(denoted by $\theta_{1}$, $\theta_{2}$, $\theta_{3}$) with respect to
one control atom(vertex) varies from 0 to $\pi/2$. Note that the range of
$\theta\in(\pi/2,\pi)$ is equivalent due to the symmetry so we focus on
$\theta\in(0,\pi/2]$. Besides, $\phi^{\prime}$ is an azimuthal angle in the
bottom side of tetrahedron which is opposite to the vertex. With tunable
$\theta$ and $\phi^{\prime}$, we show the potential energy $E_{2,-}$ in Fig.
6(b) where its maximal(absolute) values are denoted by dots. Periodical
patterns along $\phi^{\prime}$ axis are explicitly observed due to the
isotropy of three atoms in the bottom side. However, only at
$(\theta_{1},\theta_{2},\theta_{3})=(\arccos(\sqrt{\frac{2}{3}}),\arccos(1/3),\pi/2)$,
the absolute value of $E_{2,-}$ which is $\propto|D_{4}|$ can reach its global
maximum, corresponding to the optimal configurations as displayed on the right
side of Fig. 6(b). Note that for $\theta_{3}=\pi/2$ there exists two types of
bottom-side atomic distributions due to the periodicity.

To obtain a maximal asymmetric blockade among the cases of $\theta_{1}$,
$\theta_{2}$ and $\theta_{3}$, we analyze the ratio $\chi_{j}$ of each control
atom and find that its minimum is $\min(\chi_{j})\approx 828.44$ when
$\theta=\theta_{1}$(case 1). And this value lowers to 552.30 if
$\theta=\theta_{2}$(case 2). Case 3 with $\theta=\theta_{3}$ has an equivalent
structure as the former case 2. Relevant parameters estimated in calculation
are $C_{3}/2\pi=4.194$ GHz$\cdot\mu$m3, $C_{6}/2\pi=-12$ GHz$\cdot\mu$m6 and
$R_{ct}=5.0$ $\mu$m. These results explicitly suggest that case 1 has a best
asymmetry, which agrees with the optimal structure obtained by evolutionary
algorithm, see Fig.1(n=4).

## Appendix E: Gate Implementation

In the section we demonstrate how to realize a CnNOT gate via asymmetric
blockade. Consider ${}^{87}Rb$ atoms trapped in a 3D spheroidal array with
$R_{ct}=5.0$ $\mu$m. Qubit-state preparation depends on two hyperfine ground
states $|0_{j,t}\rangle=|5S_{1/2},F=1,m_{F}=0\rangle$ and
$|1_{j,t}\rangle=|5S_{1/2},F=2,m_{F}=0\rangle$ via an optically pumping method
[71]. In practice the control atoms are globally driven via an one-step UV
excitation from $|0_{j}\rangle$ to $|p_{j}\rangle=|60P_{3/2}\rangle$ with
wavelength 297nm; and the target atom will face a two-photon transition from
$|0_{t}\rangle$ to $|s_{t}\rangle=|61S_{1/2}\rangle$ with wavelengths 795nm
and 474nm, decided by the selection rules. For the target atom the
intermediate state e.g. $|5P_{1/2}\rangle$ has been safely discarded due to
dispersive interactions, and state $|1_{t}\rangle$ is also coupled to
$|s_{t}\rangle$ by same mechanism.

A straightforward realization of the multiqubit CnNOT gates should obey

$\displaystyle|00...0...\beta\rangle$ $\displaystyle=$ $\displaystyle
e^{i\varphi}|00...0...\beta\rangle$ $\displaystyle\text{and
}|\underbrace{11...1...}_{n}\beta\rangle$ $\displaystyle=$
$\displaystyle|\underbrace{11...1...}_{n}\bar{\beta}\rangle$ (38)

with $\beta\in[0,1]$ and $\bar{\beta}\equiv 1-\beta$. We have omitted the
subscripts t, j. The relative phase $\varphi$ is tunable by external fields
and here $\varphi=0$. To describe the gate implementation, we note that when
any control atom is initialized in state $|0_{j}\rangle$ the pre-excitation of
the control qubit(s) can induce a strong control-target exchange interaction
which blocks the subsequent excitation of the target atom. Only if all control
atoms are in idle state $|1_{j}\rangle$ that are uncoupled with the laser
fields, a complete state conversion mediated by $|s_{t}\rangle$ takes place
between $|0_{t}\rangle$ and $|1_{t}\rangle$ of the target atom. In the scheme,
a weak control-control interaction can facilitate an individual manipulation
for the target atom by each control qubit, avoiding the control-control
blockade.

As usual we apply a piecewise pulse sequence $\Omega_{1\sim 5}$ comprising
three fundamental steps [72]. (1) The incidence $\pi$ pulse with Rabi
frequency $\Omega_{1}$ is globally applied to all control atoms, which allows
a Rydberg excitation of $|0_{j}\rangle\to|p_{j}\rangle$. (2) A pair of $\pi$
pulses which include $\Omega_{2}$ and $\Omega_{3}$(or $\Omega_{3}$ and
$\Omega_{4}$), can generate a state swap for the target atom between
$|0_{t}\rangle$ and $|1_{t}\rangle$ if all control atoms are idle in
$|1_{j}\rangle$, obeying the routes of

$\displaystyle|1_{t}\rangle\xrightarrow{\Omega_{2}}|s_{t}\rangle\xrightarrow{\Omega_{3}}|0_{t}\rangle,$
$\displaystyle\text{or
}|0_{t}\rangle\xrightarrow{\Omega_{3}}|s_{t}\rangle\xrightarrow{\Omega_{4}}|1_{t}\rangle,$
(39)

which depends on its initial status $|1_{t}\rangle$(or $|0_{t}\rangle$). (3) A
$(-\pi)$-pulse $\Omega_{5}$ can finally return the Rydberg state
$|p_{j}\rangle$ to $|0_{j}\rangle$ via a de-excitation process. In the main
text we have assumed the Rabi frequencies with magnitudes
$\Omega_{c}=|\Omega_{1,5}|$ and $\Omega_{t}=|\Omega_{2,3,4}|$ throughout the
paper.

## Appendix F: Other technical errors

Figure 7:  Technical errors of a C6NOT gate caused by the imperfection of
experimental conditions. (a-b) Deviations in the laser amplitudes and phases.
(c) Motional dephasing of atoms under different atomic temperatures. Every
point represents an average over 500 measurements.

In this section we first address the errors from imperfect experimental
technique leading to unwanted laser amplitude and phase fluctuations. The
infidelity of the C6NOT gate vs fluctuations of laser field amplitude is
plotted in Fig.7(a) in which the fluctuation $\delta\Omega_{c(t)}$ with
respect to $\Omega_{c(t)}$ is estimated by a random number. i.e. The relative
variation of laser Rabi frequency $\delta\Omega_{c(t)}/\Omega_{c(t)}$ can be
randomly obtained from a range of $[-\delta_{\Omega},\delta_{\Omega}]$. By
increasing the value $\delta_{\Omega}$, from Fig.7(a) we know that the gate
infidelity has an apparent growth. For $\delta_{\Omega}=0.1$ the infidelity of
a 3D C6NOT gate is about $\sim 0.035$ after an average of 500 measurements.
Similarly we also study the robustness of gate against the variation of laser
phases. We assume that the fluctuated laser phase $\phi_{c(t)}$ obeys a
Gaussian distribution around the desired value $\phi_{c(t),0}=0$ with the
standard deviation $\sigma_{\phi}/\pi\in[0,0.1]$. During each measurement a
random laser phase $\phi_{c(t)}$ is adopted which leads to
$\Omega_{c(t)}\to\Omega_{c(t)}e^{i\phi_{c(t)}}$. From Fig.7(b) we learn that
the imperfection of gate fidelity also increases with the strength of phase
fluctuation $\sigma_{\phi}$. In fact once the laser amplitude or phase is
fluctuated it would arise imperfect excitation or deexcitation during the
process of state swap owing to the invalidity of $\pi$-pulses [see Eq.(39)]
which will cause the gate inefficient.

Another inevitable technical error resource is the motional dephasing effect
[73, 74]. Due to the finite temperature, the thermal motion of control and
target atoms will induce an inevitable Doppler dephasing to the excitation of
Rydberg states which can be estimated by a phase change to the laser Rabi
frequencies

$\Omega_{t}\to\Omega_{t}e^{i\Delta_{t}t},\Omega_{c}\to\Omega_{c}e^{i\Delta_{c}t},$
(40)

where the detuning $\Delta_{t(c)}$ seen by the atoms satisfies a Gaussian
distribution with its mean value $\bar{\Delta}_{t(c)}=0$ and the standard
deviation $\sigma_{\Delta_{t(c)}}$. Typically
$\sigma_{\Delta_{t(c)}}=\mathbb{k}_{eff}\mathbb{v}$ where
$\mathbb{k}_{eff}=\sum_{j}\mathbb{k}_{j}$ is the overall wavevector and
$\mathbb{v}=v_{rms}=\sqrt{k_{B}T_{a}/M}$ is the atomic root-mean-square
velocity with $k_{B}$, $T_{a}$ and $M$ being the Boltzmann constant, atomic
temperature, and atomic mass. To minimize the Doppler effect, for target atom
which undergoes a two-photon transition with wavelengths 795nm and 474nm the
effective wavevector is $k_{t,eff}=(k_{474}-k_{495})=5\times 10^{6}$ $m^{-1}$
by considering two counterpropagating lasers, leading to
$\sigma_{\Delta_{t}}=k_{t,eff}v_{rms}$ [75]. As for control atoms undergoing
an one-step UV excitation $k_{t,eff}=k_{295}=2\times 10^{7}$ $m^{-1}$ arising
$\sigma_{\Delta_{c}}=k_{c,eff}v_{rms}$. With these settings, in Fig. 7(c) we
numerically calculate the imperfection of gate performance with respect to the
temperature. For a given $T_{a}$ we adopt a random detuning $\Delta_{t(c)}$
from the Gaussian function to simulate the phase error on the population
evolution. By averaging over sufficient measurements we find the gate
infidelity attains 0.08 at $T_{a}=50$ $\mu$K where the phase variations caused
by atomic thermal motion are $\sigma_{\Delta_{t}}=0.346$ MHz,
$\sigma_{\Delta_{c}}=1.382$ MHz respectively.

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