#  Optimal model for fewer-qubit CNOT gates with Rydberg atoms

Rui Li1, Shurui Li1, Dongmin Yu1, Jing Qian1,∗ and Weiping Zhang2,3,4 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 and Astronomy, and Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, Shanghai, 200240, China  3Shanghai Research
Center for Quantum Science, Shanghai, 201315, China  4Collaborative Innovation
Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Fewer-qubit quantum logic gate, serving as a basic unit for constructing
universal multiqubit gates, has been widely applied in quantum computing and
quantum information. However, traditional constructions for fewer-qubit gates
often utilize a multi-pulse protocol which inevitably suffers from serious
intrinsic errors during the gate execution. In this article, we report an
optimal model about universal two- and three-qubit CNOT gates mediated by
excitation to Rydberg states with easily-accessible van der Waals
interactions. This gate depends on a global optimization to implement
amplitude and phase modulated pulses via genetic algorithm, which can
facilitate the gate operation with fewer optical pulses. Compared to
conventional multi-pulse piecewise schemes, our gate can be realized by
simultaneous excitation of atoms to the Rydberg states, saving the time for
multi-pulse switching at different spatial locations. Our numerical
simulations show that a single-pulse two(three)-qubit CNOT gate is possibly
achieved with a fidelity of 99.23$\%$(90.39$\%$) for two qubits separated by
7.10 $\mu$m when the fluctuation of Rydberg interactions is excluded. Our work
is promising for achieving fast and convenient multiqubit quantum computing in
the study of neutral-atom quantum technology.

††preprint: APS/123-QED

##  I Introduction

Neutral Rydberg atoms with high-lying and long-range strongly-interacting
properties, have long-time been considered as an ideal platform for large-
scale quantum computing (see reviews e.g. refs.[1, 2, 3]). The excitation
blockade mechanism associated with Rydberg atoms can be used to prevent
transitions of more than one Rydberg excitations within a whole atomic
ensemble, enabling extensive applications in the researches of multiqubit
quantum gates [4, 5], multiparticle entanglement [6, 7, 8, 9, 10, 11] and
quantum sensing [12, 13]. So far, efforts in multiqubit quantum gates mainly
focus on the construction of fewer-qubit gates which are basic logic
calculation units for universal quantum computation. Yet most earlier
achievements adopted the way of a piecewise pulse comprising a set of
sequential $\pi$ or $2\pi$ pulses [14, 15, 16]. This method relies on a rapid
switching of the pulses at different spatial locations, as well as strong
Rydberg blockade. Any imperfection in the blockade strength can induce a
residual blockade error scaling as $\sim\Omega^{2}/V^{2}$ where $\Omega$ is
the coupling strength and $V$ is the dipole-dipole interaction [17]. Also
these pioneering schemes are found to be vulnerable to the accumulation of
decay error which increases with the number of atoms [5]. To date, whether
there exists another means for achieving fast and convenient multiqubit
quantum gates other than by excitation blockade in a strong dipole-dipole
blockade regime, remains an open question.

To our knowledge, adiabatic gates making use of a Rydberg dark eigenstate with
fewer shaped pulses offer new opportunities, although the requirement for
adiabaticity renders the gate slow and susceptible to the decay error [18, 19,
20, 21, 22, 23, 24]. The way of shortcuts to adiabaticity can accelerate the
gate operation to some extent [25]. Here we adopt another acceleration
approach based on pulse optimization. As an important example, we focus on the
implementation of fewer-qubit CNOT gates with fewer optimally-shaped optical
pulses. The importance of the CNOT gate partly stems from its ability to
generate entangled states [15]. Besides, it can basically comprise any desired
gates together with single-qubit operations [26]. Ever since the first
experimental demonstration [27], many proposals developed for constructing
CNOT gates depend on excitation blockade together with a piecewise pulse
sequence [28]. Such as in the pioneer work [27], authors used seven pulses for
the AS-CNOT gate and five pulses for the $H$-CZ-CNOT gate. Subsequent schemes
for AS-CNOT gates could work with three laser pulses [29], even in the weak
interaction regime without any rotation errors [30, 31]. Other achievements
towards $H$-CZ-CNOT gates used an “one-step implementation” to form a fast
$C_{Z}$ gate [32, 33, 34, 35], but requiring extra single-qubit Hadamard
operations which render the total gate duration elongated [36]. More recently,
the transition slow-down effect can lead to an accurate AS-CNOT gate with a
sub-microsecond duration by two pulses while still working in the strong
blockade regime [37].

In the present work, we analyze a fast and high-fidelity AS-CNOT(briefly CNOT
in the following) gate with fewer optical pulses. It works in the van der
Waals(vdWs) regime between two isolated Rydberg atoms prepared in a well-
defined quantum state. This can essentially avoid the influence of Rydberg
blockade. Compared to the Rydberg gates that work with strong dipole-dipole
interactions, the weak vdWs interaction will not cause the coupling of
adjacent two-atom Rydberg levels due to the non-resonant interactions, so as
to be less technically demanding. e.g. for $|62D_{3/2}\rangle$ state in 87Rb
atoms the interaction character of $1/R^{6}$ cannot be altered when the
interatomic distance is $R>5.5$ $\mu$m [38]. Therefore, accounting for the use
of simultaneous excitation with the least number of optical pulses, our
protocol not only removes the need for multiple switchings of external control
fields reducing the complexity of experimental implementation; but also
decreases extra atomic losses induced by the multi-laser noises [39].

In addition, the scheme has the added advantage of modulating the pulse
waveform accurately by optimization algorithm. We apply a global optimization
to construct amplitude and phase-modulated pulses which implement the two-
qubit CNOT or three-qubit Toffoli gates. The pulses are optimized through a
genetic algorithm [40]. The resulting gate fidelity is essentially constrained
by the precision of optimization algorithm and the Rydberg decay error in the
weak interaction regime. To make the scheme more accessible, other technical
errors arising from the fluctuations of interaction, the laser amplitude noise
as well as the thermal motion of atoms at finite temperature, are also
investigated. In contrast to previous multi-pulse piecewise protocols, our
gate benefiting from a simplified pulse design, is more straightforward for
experimental demonstration and deserves future studies in diverse systems such
as superconducting circuits [41], trapped ions [42], and solid-state quantum
systems [43].

##  II Theoretical strategy

Figure 1: (a) Experimental sketch for a two-qubit CNOT gate. Optical-tweezers-
trapped atoms are globally excited by a two-photon excitation with 480nm and
780nm lasers, which propagate in opposite directions with a tiny separation
angle $\theta<5^{o}$ for reducing the Doppler-induced frequency shift. Atom
measurement can be realized by collecting fluorescence signals on a cooled
EMCCD camera placed perpendicularly to the laser propagation. To carry out the
gate, several electro-optic modulator(EOM) and acoustic-optical modulator(AOM)
devices are needed in experiment. After producing a shaped pulse via an
amplitude-EOM(AEOM) one may use an AOM for generating a frequency difference
between the zeroth- and first-order diffraction optical fields with respect to
the 780nm laser. This light is used for an off-resonant excitation between the
ground state $|0\rangle$ or $|1\rangle$ and the intermediate state
$|5P_{3/2}\rangle$ (not shown). The other 480nm beam is left on for coupling
$|5P_{3/2}\rangle$ to the Rydberg state $|r\rangle=|70S_{1/2}\rangle$ [44]. A
phase-EOM(PEOM) can selectively provide another phase control between the two
pulses. (b) Relevant energy levels including
$\\{|0\rangle,|1\rangle,|r\rangle\\}$ as well as the atom-light couplings. The
vdWs interaction will lead to a pure energy shift $V$ to the two-atom state
$|70S_{1/2};70S_{1/2}\rangle$ excluding the influence of other nearby high-
lying Rydberg states. (c1-c4) Basic physical mechanism provided by a reduced
two-pulse protocol, in which the population transformation based on four
computational basis states $\\{|10\rangle,|11\rangle,|00\rangle,|01\rangle\\}$
are shown. Each atom contains $\\{|0\rangle$, $|1\rangle$, $|r\rangle\\}$
three states. The modified pulse 1 labeled by the two-photon Rabi frequency
$\Omega_{1}(t)$ (green arrow) globally couples $|0\rangle$ and $|r\rangle$ for
both atoms. The local addressing between $|1\rangle$ and $|r\rangle$ of the
target atom is provided by pulse 2 with the Rabi frequency $\Omega_{2}(t)$
(red arrow). Solid-arrow stands for a true excitation accompanied by the
population transfer, while the dashed-arrow means a null excitation despite
the presence of optical pulses.

###  II.1 Model and Hamiltonian

To our knowledge, earlier proposals for realizing e.g. $H$-CZ-CNOT often
constructed the entangling gates with a piecewise pulse sequence, accompanied
by a local $\frac{\pi}{2}$-rotation which can be implemented via a microwave
field [16]. However, the gate fidelity depends on the phase difference between
the microwave field and the optical field, which is usually difficult to be
locked in experiments. Besides, the combination of microwave and optical
pulses makes the addressing of individual atoms difficult during the gate
operation [45]. To solve these problems, we propose a full-optical and
straightforward scheme to implement a CNOT gate as illustrated in Fig.1(a),
which consists of two 87Rb atoms with a well-separated spacing $r_{0}$ between
the center of two traps (we also study the three-qubit Toffoli gate in Sec.
V). The simplified implementation is achieved via a combination of a global
coupling $|0\rangle\to|r\rangle$ of both atoms with the effective Rabi
frequency $\Omega_{1}(t)$(green arrow, pulse 1), and the local addressing by a
laser inducing the $|0\rangle\to|r\rangle$ transition of the target atom with
$\Omega_{2}(t)$ (red arrow, pulse 2). Atoms are pre-pumped into $|0\rangle$ by
a polarized laser propagating along $-\hat{x}$ axis and $|1\rangle$ is a
nearby hyperfine ground state. Additional $\pi$ pulses are applied to either
or both of the atoms to accomplish the state preparation [46]. Two laser
pulses differ in frequency by a magnitude of $\omega_{01}/2\pi\approx$ 6.83
GHz which agrees with the hyperfine energy separation between states
$|0\rangle$ and $|1\rangle$. In experiment an AOM can be used to spatially
split the laser into the zeroth-order (pulse 1) and first-order (pulse 2)
fields via diffraction, in which its modulation frequency equals
$\omega_{01}$. A phase modulation with the help of a PEOM is sometimes
required for the optimal pulse shapes.

The desired probability matrix for constructing a two-qubit CNOT gate should
contain a unitary transformation of the computational basis states
$\\{|00\rangle,|01\rangle,|10\rangle|11\rangle\\}$ as follows:

$U_{\text{CNOT}}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\\\ \end{array}\right)\\\ $ (1)

which is enabled by the two-atom Hamiltonian
$\hat{\mathcal{H}}=\hat{\mathcal{H}_{c}}\otimes\hat{{I}}+\hat{{I}}\otimes\hat{\mathcal{H}_{t}}+V\left|rr\right\rangle\left\langle
rr\right|$ with

$\displaystyle\hat{\mathcal{H}_{c}}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{1}(t)}{2}(\left|0\right\rangle\left\langle
r\right|+H.c.)$ $\displaystyle\hat{\mathcal{H}_{t}}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{1}(t)}{2}(\left|0\right\rangle\left\langle
r\right|+H.c.)+\frac{\Omega_{2}(t)}{2}(\left|1\right\rangle\left\langle
r\right|+H.c.)$

for describing the atom-light interactions of control
($\hat{\mathcal{H}_{c}}$) and target ($\hat{\mathcal{H}_{t}}$) atoms,
respectively. $U=V\left|rr\right\rangle\left\langle rr\right|$ denotes the
interatomic Rydberg interaction.

To facilitate the scheme, the Rydberg interaction is assumed to be of vdWs-
feature appearing between two isolated atoms, which excludes the couplings
between $ss$ and other higher angular momentum states such as $p$-orbital
states [47]. More precisely we have summed up all non-resonant dipole-dipole
interactions among Zeeman sublevels due to the larger Förster defects, leading
to a second-order vdWs shift to the two-atom state
$|70S_{1/2};70S_{1/2}\rangle$ [48]. This differs from the Rydberg phase-shift
gates proposed in [49, 50], where the large Rydberg interaction is provided by
a static electric field to remove the degeneracy of high-lying Rydberg levels.
We will verify the existence of the vdWs interaction for appropriate two-atom
distances $r_{0}$ in Appendix A.

###  II.2 Physical realization

Before we proceed to a detailed numerical discussion we consider the
theoretical feasibility of the gate implementation. Ideally, the gate
operation can be simply understood by two optical pulses $\Omega_{1}(t)$ and
$\Omega_{2}(t)$, satisfying

$|\Omega_{1}(t)|>|\Omega_{2}(t)|.$ (2)

This assumption is given for simplifying the following discussion but is not
crucial for the realization of the gate. The systematic dynamics can be
understood by considering the behavior of the four computational basis states:

(i) If the input state is $|1\beta\rangle$ with $\beta\in\\{0,1\\}$, i.e. the
control atom is an idler, the two pulses $\Omega_{1}(t)$ and $\Omega_{2}(t)$
transfer the target atom through a two-photon Raman process, obeying

$|10\rangle\rightleftarrows|1r\rangle\rightleftarrows|11\rangle$ (3)

[see Fig.1(c1-c2)]. Since the control atom is never excited, there is no
Rydberg interaction on it leading to null rotation error. The major intrinsic
errors stem from the transient residence on $|r\rangle$ of the target atom,
i.e. the Rydberg decay error, as well as the imperfect Raman transfer due to
the accuracy of optimization. To minimize the decay error we apply the Raman
pulses within a relatively short duration $\sim 1.0$ $\mu$s (Ref.[51] showed a
similar Cz-gate duration with optimized adiabatic pulses). Also, the
insufficient pulse optimization will bring extra optimization error here.

(ii) If the control atom is initially in $|0\rangle$ the traditional multi-
pulse blockade gate relies on a pre-excitation of the control atom. Once the
control atom has been excited, it causes an inevitable energy shift to state
$|r\rangle$ of the target atom, preventing its excitation. However, we notice
that two pulses are applied simultaneously arising two competitive excitation
channels that strongly depend on the interaction strength. With a larger
interaction strength, $V>|\Omega_{1}|,|\Omega_{2}|$, the $|00\rangle$ state
tends to have a collective transition [see (c3), top]

$|00\rangle\to\frac{|0r\rangle+|r0\rangle}{\sqrt{2}}\to|00\rangle.$ (4)

This channel only involves single-atom Rydberg excitation so as to be
insensitive to the fluctuation of interactions. Oppositely, when the
interaction strength is small, $V\leq|\Omega_{1}|,|\Omega_{2}|$, it
facilitates a simultaneous two-atom excitation [see (c3), bottom]

$|00\rangle\to|rr\rangle\to|00\rangle.$ (5)

This is a which-way problem. The realistic excitation channel is interaction-
dependent, so no matter big or small the interaction strength is, our scheme
can apply targeted approaches. However, for the target atom initially in
$|1\rangle$ [see (c4)] we are interested in the regime
$|\Omega_{1}|>|\Omega_{2}|$, in which the weak coupling $\Omega_{2}(t)$ is
unable to overcome the energy shift induced by the vdWs interaction, since the
control atom is pre-excited by $\Omega_{1}(t)$. As a result, the target atom
is left on $|1\rangle$ without excitation, giving rise to an ideal transfer of

$|01\rangle\to|r1\rangle\to|01\rangle.$ (6)

Such a process may suffer from an imperfection. When the shifted energy is
small leading to an excitation of the target atom to $|r\rangle$ by
$\Omega_{2}(t)$, the subsequent coupling between $|r\rangle$ and $|0\rangle$
by $\Omega_{1}(t)$ would destroy the targeted state transformation by
impelling the target atom onto an undesired state $|0\rangle$, as shown in the
lower panel of (c4).

It should be stressed that we work on the implementation of two- and three-
qubit CNOT gates by simply placing them in the vdWs-interaction regime. It is
challenging to construct a global optimization to the amplitude and phase of
the two pulses simultaneously. Especially a weak two-atom interaction will
make the condition $|\Omega_{1}(t)|>|\Omega_{2}(t)|$ unsatisfied. In that
case, the complicated atomic dynamics can not be easily understood by the map
of Fig.1(c). However, with a robust optimization of the pulse parameters, we
can always guarantee the state transformation of the system with a high
fidelity.

##  III Gate performance

###  III.1 Two-pulse scheme

To achieve desired optical pulses that work on all input states we use the
genetic algorithm to optimize an amplitude and phase profile for the driving
laser fields. The principle of optimization algorithm is given in Appendix B.
In the procedure, we treat the gate fidelity $\mathcal{F}_{2}$ (see Eq.(9)
where the subscript means the number of qubits) as a single objective
function, by which the gate error denoted by $1-\mathcal{F}_{2}$ attains its
global minimum under multi-time optimization. We begin with a guess for the
waveforms of two optical pulses, forming of an approximate Fourier-series
expansion by taking account of the symmetry of state transformation [52]:

$\displaystyle\Omega_{1}(t)=\Omega_{10}+\Omega_{11}\cos(2\pi
t/T_{g})+\Omega_{12}\sin(\pi t/T_{g})$
$\displaystyle\Omega_{2}(t)=\Omega_{20}+\Omega_{21}\cos(2\pi
t/T_{g})+\Omega_{22}\sin(\pi t/T_{g})$

where the optimized pulse has a duration of $T_{g}=1.0$ $\mu$s. In order to
meet the single constraint $|\delta\mathcal{F}_{2}|<10^{-4}$, where
$|\delta\mathcal{F}_{2}|$ is the fluctuation of fidelity, we optimize the
pulse parameters $(\Omega_{10},\Omega_{11},\Omega_{12})$ and
$(\Omega_{20},\Omega_{21},\Omega_{22})$ at the same time. Experimental setup
utilizing an AEOM to shape optical pulses, has been carried out on sub-
microsecond timescales in a solid-state quantum system [43]. An extra phase
modulation by PEOM is performed on one of them accompanied by a path locking
technique, which finally enables an arbitrary phase difference between the two
pulses.

Figure 2: From top to bottom, (a1-a2) two optimal ground-Rydberg pulses $\Omega_{1}(t)$(blue), $\Omega_{2}(t)$(red). The green-dashed curve shows the phase difference $\delta\phi=\phi_{1}-\phi_{2}$ between two pulses and a precise phase control is needed; (b1-e2) the real population dynamics under the inputs of four computational basis states $\\{|10\rangle$, $|11\rangle$, $|00\rangle$, $|01\rangle\\}$. Different central interactions $V_{0}/2\pi$=7.0 MHz($r_{0}=7.10$ $\mu$m) for (a1-e1) and $V_{0}/2\pi$=1.0 MHz($r_{0}=9.76$ $\mu$m) for (a2-e2), are used excluding the fluctuation of interactions. Table 1: Optimized pulse amplitudes and the gate fidelity $\mathcal{F}_{2}$ excluding the fluctuations to the interatomic interaction. All coefficients $(\Omega_{10},\Omega_{11},\Omega_{12})$ and $(\Omega_{20},\Omega_{21},\Omega_{22})$ are obtained under multi-optimization until the required precision $|\delta\mathcal{F}_{2}|<10^{-4}$ can be reached. The vdWs dispersion coefficient is $C_{6}/2\pi=863$ GHz$\cdot\mu m^{6}$ for state $|70S_{1/2};70S_{1/2}\rangle$ and $V_{0}=C_{6}/r_{0}^{6}$ represents the non-fluctuated vdWs interaction between two distant atoms. $r_{0}$ ($\mu m$) |  $V_{0}/2\pi$ (MHz) |  $\gamma$ (kHz) | ($\Omega_{10}$,$\Omega_{11}$,$\Omega_{12})$/2$\pi$ (MHz) | ($\Omega_{20}$,$\Omega_{21}$,$\Omega_{22})$/2$\pi$ (MHz) | $\mathcal{F}_{2}$
---|---|---|---|---|---
7.10 | 7.0 | 3.0 | (-3.9621, -0.7858, 1.5915) | (1.0942, -1.9068, -2.2182) | 0.9921
9.76 | 1.0 | 3.0 | (-0.9549, -1.9544, -0.0631) | (-2.8310, -3.6488, 0.0074) | 0.9951

To evaluate the fidelity, we first numerically solve the population dynamics
by integrating the two-atom master equation [53]

$\frac{d\hat{\rho}}{dt}=-i[\hat{\mathcal{H}},\hat{\rho}]+\hat{\mathcal{L}}[\rho]$
(7)

with $\hat{\mathcal{H}}$ the two-atom Hamiltonian. The radiative decay term of
the Lindblad operator is given by

$\hat{\mathcal{L}}[\rho]=\sum_{j=1}^{4}[\hat{L}_{j}\hat{\rho}\hat{L}_{j}^{\dagger}-\frac{1}{2}(\hat{L}_{j}^{\dagger}\hat{L}_{j}\hat{\rho}+\hat{\rho}\hat{L}_{j}^{\dagger}\hat{L}_{j})]$
(8)

where

$\displaystyle\hat{L}_{1}=\sqrt{\gamma}[|10\rangle\langle
r0|+|11\rangle\langle r1|+|1r\rangle\langle rr|]$
$\displaystyle\hat{L}_{2}=\sqrt{\gamma}[|00\rangle\langle
r0|+|01\rangle\langle r1|+|0r\rangle\langle rr|]$
$\displaystyle\hat{L}_{3}=\sqrt{\gamma}[|01\rangle\langle
0r|+|11\rangle\langle 1r|+|r1\rangle\langle rr|]$
$\displaystyle\hat{L}_{4}=\sqrt{\gamma}[|00\rangle\langle
0r|+|10\rangle\langle 1r|+|r0\rangle\langle rr|]$

separately demonstrate the spontaneous decay ($\gamma$ is the Rydberg decay
rate) of $|r\rangle\to|1\rangle$ and $|r\rangle\to|0\rangle$ of
control($\hat{L}_{1,2}$) and target($\hat{L}_{3,4}$) atoms. Relying on the
mature technology for cooling and manipulating individual 87Rb atoms, we adopt
$\gamma=3.0$ kHz in a cryogenic environment for states
$|r\rangle=|70S_{1/2}\rangle$, $|0\rangle=|5S_{1/2},F=1,m_{F}=0\rangle$,
$|1\rangle=|5S_{1/2},F=2,m_{F}=0\rangle$ in which the intermediate state
$|5P_{3/2}\rangle$ is ignored for a two-photon process (note that this is a
little different from the experimental setup in Ref.[36] where the
intermediate state is $|6P_{3/2}\rangle$). The observable function for the
two-qubit gate fidelity is defined as

$\mathcal{F}_{2}=\frac{1}{4}Tr\left[\sqrt{\sqrt{U_{\text{CNOT}}}\rho_{det}(t=T_{g})\sqrt{U_{\text{CNOT}}}}\right]$
(9)

by taking account of the average effect of four computational basis states.
$\rho_{det}$ represents the detected density matrix at $t=T_{g}$ that evolves
according to Eq.(7). We adopt numerical optimization to construct amplitude
and phase modulated pulses by which the system evolves satisfactorily,
realizing a high-fidelity CNOT gate.

To demonstrate the detailed dynamics of the system with respect to Eq.(7) we
first present the numerical results in Fig.2 without fluctuated interactions.
For completeness, we give the corresponding numerical values of relevant
parameters in Table I. Under the modulated pulses as observed in Fig.2(a1-a2),
the dynamics of $|10\rangle$($|11\rangle$) are given by the coupling of the
target atom on the
$|0\rangle(|1\rangle)\rightleftarrows|r\rangle\rightleftarrows|1\rangle(|0\rangle)$
transition, forming a two-photon Raman transfer [Fig.2(b1-c2)]. That means
except the Rydberg decay error, the major intrinsic error is related to the
transferring of the target atom, which comes from the imperfect pulse
optimization. For a chosen value of interaction $V_{0}$, the dynamics behave
quite differently. When $V_{0}$ is relatively large [$V_{0}/2\pi=7.0$ MHz,
(b1)-(c1)] it arises $|\Omega_{1}(t)|>|\Omega_{2}(t)|$ under the optimization.
For the dynamics associated with $|10\rangle$ or $|11\rangle$, the population
transfers as expected, especially the Rabi oscillation between $|10\rangle$
and $|1r\rangle$ is stronger than that between $|1r\rangle$ and $|11\rangle$.
This contrast is attributed to $|\Omega_{1}(t)|>|\Omega_{2}(t)|$. A small
$V_{0}$ [$V_{0}/2\pi=1.0$ MHz, (b2)-(c2)] leads to the comparable amplitudes
of two optimized pulses. The dynamics is a bit complicated in this case, but a
satisfactory population transfer is still achievable accompanied by irregular
oscillations.

We proceed to study a more complex case if the input is $|00\rangle$ or
$|01\rangle$ which is sensitive to the interaction. Accounting for the which-
way mechanism two atoms with different $V_{0}$ will have different excitation
channels such as Eq.(4) and Eq.(5) even though the initial state is same. For
example, see (d1), if $V_{0}/2\pi=7.0$ MHz leading to
$|\Omega_{1}|,|\Omega_{2}|<V_{0}$, due to the vdWs blockade mechanism [54,
55], the state $|00\rangle$ evolves to the middle singly-excited states
$|r0\rangle$ and $|0r\rangle$, and hence the scheme is not sensitive to the
fluctuation of interactions. In contrast, see (d2), when $V_{0}/2\pi=1.0$ MHz
and then $|\Omega_{1}|,|\Omega_{2}|\geq V_{0}$, we observe that the condition
$|\Omega_{1}|>|\Omega_{2}|$ is not satisfied. The mixture of various singly-
excited and doubly-excited Rydberg states causes unpredictable population
dynamics. The latter process involving dominant double Rydberg excitation
would be sensitive to the variation of atomic spacing. For the dynamics
associated with state $|01\rangle$[(e1) and (e2)] similar behaviors are
observed, but the only case (e1) shows an ideal dynamical evolution for the
control atom between $|01\rangle$ and $|r1\rangle$ [see Eq.(6)]. As a
consequence, the optimal pulse amplitudes are mainly determined by the
interaction energy $V_{0}$ which arises a changeable dynamics, even if the
input is same.

In general, a large interaction energy with $V_{0}>|\Omega_{1}|>|\Omega_{2}|$
can trigger a finite excitation blockade in the vdWs regime, allowing one of
the atoms to be excited at any time (Fig.2(e) of Ref.[38] showed a collective
single-atom excitation with an enhanced Rabi frequency for $|62D_{3/2}\rangle$
of rubidium atoms when $r_{0}$ is about 4.0 $\mu$m). While a weaker
interaction makes the atoms redistributed in various energy levels. The
simulated gate fidelity including intrinsic decay and optimization error, is
$\mathcal{F}_{2}=0.9921$ for $V_{0}/2\pi=7.0$ MHz and $\mathcal{F}_{2}=0.9951$
for $V_{0}/2\pi=1.0$ MHz.

###  III.2 Fluctuation of interactions

Figure 3: Average gate fidelity $\bar{\mathcal{F}}_{2}$ as a function of the
standard deviation $\sigma_{x}$ for different central interactions
$V_{0}/2\pi=1.0$ MHz(stars), 7.0 MHz(crosses). During each measurement the
real interatomic distance $r$ is random satisfying a three-dimensional
Gaussian distribution with mean values $x_{0}=(C_{6}/V_{0})^{1/6}$,
$y_{0}=z_{0}=0$ and standard derivations $\sigma_{x,y,z}$. Here
$\sigma_{y}=\sigma_{z}=0.27$ $\mu$m and $\sigma_{x}$ can be widely adjusted.
Each point denotes an average of 500 measurements. Relevant optimal parameters
basing on no fluctuated interactions are given in Table I.

In fact, the real interatomic distance $r$ cannot be strictly fixed during the
gate operation, leading to the fluctuation of interactions. In the numerical
simulation, we model the distance $r$ by a three-dimensional Gaussian
distribution with mean values $x_{0}=(C_{6}/V_{0})^{1/6}$, $y_{0}=z_{0}=0$.
The distance errors(standard deviation) are estimated to be
$\sigma_{y}=\sigma_{z}=0.27$ $\mu$m and $\sigma_{x}\in[0,1.5]$ $\mu$m is a
tunable value, which agree with the experimental results reported in Ref.
[11]. During each measurement we extract a random position $r_{i}$ according
to the Gaussian distribution, giving rise to a fluctuated interaction $V_{i}$.
By applying $V_{i}$ instead of $V_{0}$, we re-calculate the gate fidelity
$\mathcal{F}_{2,i}$ for the $i$th measurement. The final average fidelity is
provided by

$\bar{\mathcal{F}}_{2}=\sum_{i=1}^{\mathcal{N}}{\mathcal{F}_{2,i}}/\mathcal{N}.$
(10)

Figure 3 presents the $\bar{\mathcal{F}}_{2}$ as a function of $\sigma_{x}$
under a sufficient number of measurements, $\mathcal{N}=500$. Since all pulse
parameters have been optimally obtained under a non-fluctuated interaction
$V_{0}$, they are unable to be exactly appropriate for the real population
evolution with a fluctuated interaction $V_{i}$. Hence the reduction of
$\bar{\mathcal{F}}_{2}$ with respect to $\sigma_{x}$ is clearly observed no
matter what the $V_{0}$ is. Besides owing to the suppression of double Rydberg
excitation by a finite vdWs energy shift, in the case of a larger $V_{0}$, the
gate fidelity $\bar{\mathcal{F}}_{2}$ shows a good robustness to the increase
of $\sigma_{x}$. It is apparent that the average fidelity can maintain a
higher value, $\bar{\mathcal{F}}_{2}\approx 0.9327$, even at
$\sigma_{x}=1.5\mu$m for $V_{0}/2\pi=7.0$ MHz. Yet this value will reduce to
0.8857 for $V_{0}/2\pi=1.0$ MHz.

###  III.3 Improved one-pulse scheme

Figure 4: Realization of an one-pulse two-qubit CNOT gate. From top to bottom,
(a1) the optimized pulse shape(blue-solid) and phase variation(red-dashed)
under $V_{0}/2\pi=7.0$ MHz. Coefficients $(\Omega_{0},\Omega_{1},\Omega_{2})$
are explicitly shown in the picture. (b1-e1) Numerically-simulated population
dynamics of four computational basis states
$\\{|10\rangle,|11\rangle,|00\rangle,|01\rangle\\}$ excluding the effect of
fluctuated interactions. Similarly, (a2-d2) show the results for
$V_{0}/2\pi=1.0$ MHz. (f) Average fidelity $\bar{\mathcal{F}}_{2}$ with
respect to the variation of the standard deviation $\sigma_{x}$ for
$V_{0}/2\pi=$1.0 MHz(stars), 7.0 MHz(crosses). Each point denotes an average
of 500 measurements.

To obtain a minimal model for the targeted gate the two optical pulses can
even have same amplitudes generated by one AEOM (see Fig.1(a) and Sec.VI for
more details). This so-called one-pulse scheme only requires one optical pulse
taking a similar form of

$\Omega(t)=\Omega_{0}+\Omega_{1}\cos(2\pi t/T_{g})+\Omega_{2}\sin(\pi
t/T_{g}).$ (11)

We seek appropriate values $(\Omega_{0},\Omega_{1},\Omega_{2})$ for achieving
a satisfactory gate performance. During the gate implementation, the global
coupling over both qubits between $|0\rangle$ and $|r\rangle$, as well as the
local transition from $|1\rangle$ to $|r\rangle$ for the target qubit, are
played by a same-shaped pulse with the Rabi frequency $\Omega(t)$. The laser
is frequency modulated by an AOM to produce the zeroth- and first-order
diffractions which achieves a spatial separation for driving two separated
atomic qubits. The one-pulse scheme can strongly prohibit the unwanted phase
control between the two pulses, promising for the true realization of a
minimal two-qubit CNOT gate.

Simulation in Figure 4 shows the population dynamics of four states
$\\{|10\rangle,|11\rangle,|00\rangle,|01\rangle\\}$ under an optimized pulse.
After an optimization with $T_{g}=1.0$ $\mu$s we reach an amplitude and phase
modulated pulse as displayed in (a1-a2). We note that a peak Rabi frequency of
similar magnitude was used for implementing the Rydberg adiabatic gates [18,
22]. The dynamics of $|10\rangle$ (and $|11\rangle$) is affected by the
$|0\rangle\to|r\rangle\to|1\rangle$(and $|1\rangle\to|r\rangle\to|0\rangle$)
transition of the target atom, forming a three-level coupled system. The
resulting coherent population conversion between two ground states
$|10\rangle$ and $|11\rangle$ shows strong oscillations through a two-photon
process mediated by $|1r\rangle$. Also, due to the symmetric waveform of
$\Omega(t)$ the dynamics of $|10\rangle$ and $|11\rangle$ share a same
behavior as time going [see (b1-c1) and (b2-c2)]. The population evolution of
state $|00\rangle$ ($|01\rangle$) strongly depends on the strength of the
interaction because of the which-way mechanism, which leads to a wide
distribution of populations on various middle Rydberg states including
$|0r\rangle$, $|r0\rangle$, $|rr\rangle$, and $|r1\rangle$. Finally, while
excluding the fluctuation of interactions, we show that the one-pulse two-
qubit CNOT gate realized has a fidelity of $\mathcal{F}_{2}\sim 0.9923$ for
$V_{0}/2\pi=7.0$ MHz and $\mathcal{F}_{2}\sim 0.9935$ for $V_{0}/2\pi=1.0$ MHz
in a cryogenic environment.

To obtain more practical numbers we also add extra fluctuations to the
interaction $V_{0}$ characterized by position error $\sigma_{x,y,z}$ in the
three spatial dimensions. As shown in Fig.4(f), $\sigma_{x}$ is changed while
$\sigma_{y}=\sigma_{z}=0.27$ $\mu$m. The simulation of the average fidelity
$\bar{\mathcal{F}}_{2}$ shows a similar tendency as in the two-pulse case
where $\bar{\mathcal{F}}_{2}$ decreases as $\sigma_{x}$ grows. Here the scheme
with $V_{0}/2\pi=7.0$ MHz is more sensitive to the fluctuations of
interaction, because the fast-oscillating excitation of $|rr\rangle$
substantially reduces the gate fidelity. So taking into account the position
error with $\sigma_{x}=1.5$ $\mu$m, $\sigma_{y(z)}=0.27$ $\mu$m, the resulting
average fidelity $\bar{\mathcal{F}}_{2}$ is decreased to $\sim 0.8043$ for
$V_{0}=7.0$ MHz and $\sim 0.8322$ for $V_{0}=1.0$ MHz after 500 measurements.

We demonstrate an unprecedented realization of a two-qubit CNOT gate with only
one optical pulse, which is enabled by a robustly-optimized pulse waveform
that works for all input states. Different from in a two-pulse scheme, it is
not required here to face the technical difficulties coming from a careful
alignment of the two pulses on the time axis and a precise control of their
phase difference. Therefore, our scheme can offer a promising route to
implement a fast and convenient two-qubit CNOT gate. On the other hand, a pure
vdWs interaction is assumed between two well-separated qubits, instead of a
strong dipole-dipole interaction, which immunizes the gate to any leakage
error from nearby Rydberg states. A detailed proof for a pure vdWs interaction
when $r_{0}=7.10$ $\mu$m or $9.76$ $\mu$m is given in Appendix A.

##  IV Error estimates

###  IV.1 Intrinsic errors

Rydberg decay error. Except for the position error characterized by standard
deviations $\sigma_{x,y,z}$, the simulated gate fidelity is mostly limited by
the Rydberg decay error which increases with the rate of decays. Yet in our
scheme for any state dynamics, due to the smooth waveform of the pulse
amplitude modulation, the total duration for the Rydberg population of four
input states can be reduced. That fact differs from a piecewise Cz gate
sequence adopted in a Rydberg-blockade-based CNOT gate, where the control atom
persists on the Rydberg state during the pulse duration applied to the target
atom, arising a large decay loss. e.g. in Ref. [56], an estimation of decay
error shows $\mathcal{E}_{sp}=2.6\times 10^{-3}$ for trapped
$97d_{5/2}$($\tau\sim 300$ $\mu$s) rubidium atoms at a temperature of 175
$\mu$K. And this error increases to $7.5\times 10^{-3}$ for $66s$($\tau\sim
130$ $\mu$s) cesium atoms in a room temperature [11].

Figure 5: Estimated decay error $\mathcal{E}_{sp}$ vs the Rydberg decay rate
$\gamma$ under (a1) $V_{0}/2\pi=7.0$ MHz and (a2) $V_{0}/2\pi=1.0$ MHz. (b)
Calculated optimization imperfection $\mathcal{E}_{opt}$ as a function of the
vdWs interaction strength $V_{0}$ where $\gamma=0$. For each $V_{0}$ all pulse
coefficients have to be re-optimized, see more details given in Table III. (c)
Average optimization error $\bar{\mathcal{E}}_{opt}$ under 500 measurements
taking account of the fluctuated interactions. Here $\sigma_{x}=1.5$ $\mu$m,
$\sigma_{y}=\sigma_{z}=0.27$ $\mu$m. The results from the one(two)-pulse
scheme are marked by red stars(green dots), respectively.

In our protocol, to estimate the influence of the Rydberg-state decay we
numerically compute the gate error with respect to the variation of the decay
rate ($\gamma\sim 1/\tau$) where $\tau$ is the Rydberg lifetime. In
Fig.5(a1-a2), we show the results by shifting the curve starting from zero in
order to exclude the influences from other imperfections. For each value
$\gamma$ we use same pulse parameters as described in Table I and Fig.
4(a1-a2). The fluctuation of the qubit positions is ignored for showing a pure
relationship. Compared to the Rydberg CNOT gates that work in a strong
blockade regime the decay error of our gate has a similar relationship which
shows $\mathcal{E}_{sp}\propto\gamma$ (Eq.1 of Ref.[15] explicitly showed
$\mathcal{E}_{sp}\propto 1/\tau$ in a controlled-phase gate where $\tau$ is
the radiative lifetime of the Rydberg level), yet benefiting from a lower
value. The reason is that the smooth modulation of the pulse amplitudes keeps
small Rydberg population in $|rr\rangle$, $|\beta r\rangle$, $|r\beta\rangle$
with $\beta\in\\{0,1\\}$. Based on relevant parameters in [57], a numerical
estimation arises $\mathcal{E}_{sp}\approx 3.5\times 10^{-3}$
($\mathcal{E}_{sp}\approx 6.0\times 10^{-3}$) above the room temperature for
$V_{0}/2\pi=7.0$ MHz ($V_{0}/2\pi=1.0$ MHz) where $\tau\sim 100$ $\mu$s. When
the atom temperature is set to $50$ $\mu$K with $\tau\sim 400$ $\mu$s this
error has a clear decrease leading to $\mathcal{E}_{sp}\approx 8\times
10^{-4}$ ($\mathcal{E}_{sp}\approx 1.6\times 10^{-3}$), accordingly. By
comparing these values we conclude that, the spontaneous emission from the
Rydberg states is one of the major limitations on the gate fidelity in the
weak interaction regime, however it has been well controlled to a compatible
level with previous gate protocols.

Resilience to the change of vdWs interaction. Furthermore, we need to explore
the influence of different interatomic interactions on the gate performance,
which mainly depends on the optimization. That differs from a Rydberg blockade
gate where a residual blockade error can be deeply reduced by a huge dipole-
dipole interaction between high-lying Rydberg states (The second term in Eq.2
of Ref. [58] showed the intrinsic error of a Rydberg blockade gate is
proportional to $1/V_{0}^{2}$). In other words the gate should work in the
strong dipole-dipole interaction regime with a small qubit spacing where its
interaction strength is large enough as compared with the peak Rabi
frequencies of all driving pulses [17]. However, our gate works in the weak
vdWs-interaction regime with a limited blockade strength. Owing to the
exclusion of nearby two-atom Rydberg states we could study the influences of
different interaction strengths based on a pure vdWs environment.

In our scheme, for the dynamics associated with $|10\rangle$ and $|11\rangle$
the double excited Rydberg state $|rr\rangle$ does not receive any population
because the control atom in $|1\rangle$ is not coupled by the laser pulses.
Yet, the population dynamics with states $|00\rangle$ and $|01\rangle$ are
indeed impacted by using different interaction strengths. Luckily, thanks to
the which-way mechanism there always exists a possible route for the evolution
of population dynamics no matter what the interaction is. Especially in the
case of a weaker interaction all populations are distributed in a number of
Rydberg levels [e.g. see Fig.2(d2),(e2)], mixing with multi-transferred
routes. That is the reason why our protocol can show a resilient insensitivity
to the change of interaction strengths. The smooth modulation of the pulse
amplitudes via optimization helps to suppress the rotation error and keep a
minimal population in $|rr\rangle$. Except for the decay error, another
intrinsic limitation on our gate fidelity comes from the optimization
imperfection due to the limited precision of algorithm, as estimated in
Fig.5(b). This optimization error $\mathcal{E}_{opt}$ has shown its
insensitivity to the change of $V_{0}$ for both one- and two-pulse schemes.

During a single measurement without interaction fluctuations, the optimization
error $\mathcal{E}_{opt}$ can sustain around $\sim 0.01$ for any $V_{0}$
values. We note that the optimization is more efficient at $V_{0}/2\pi=1.0$
MHz(smaller) and $7.0$ MHz(larger) because of the which-way mechanism.
Furthermore, by carrying out sufficient measurements with position
fluctuations: $\sigma_{x}=1.5\mu$m, $\sigma_{y(z)}=0.27\mu$m, we can reach an
average optimization error $\bar{\mathcal{E}}_{opt}$ as displayed in Fig.5(c).
We observe that, in the one-pulse case(red stars), $\bar{\mathcal{E}}_{opt}$
remains to be around 0.15 for any $V_{0}$ because the population on state
$|rr\rangle$ shows little change [see Fig.4(d1-e2)]. While turning to the case
of two pulses(green dots), the excitation process in a larger-interaction
regime with $V_{0}/2\pi>5.0$ MHz acquires a big suppression to the population
of $|rr\rangle$, so as to be insensitive to the fluctuation of qubit
separation. Thus the $\bar{\mathcal{E}}_{opt}$ can reach as low as $\sim
0.058$(an average value when $V_{0}/2\pi=(6.0,7.0)$ MHz), which is much
smaller than the average value $\sim 0.179$ given by the one-pulse scheme in
the same regime. These results conclude that, provided the spontaneous
emission from the Rydberg level has been suppressed via a higher-lying Rydberg
level or a cryogenic environment, the intrinsic obstacle for a high gate
fidelity is only constrained by the precision of optimization.

###  IV.2 Technical errors

Figure 6: Imperfections of the gate fidelity caused by (a1-a2) the motional
dephasing effect under different temperatures $T_{a}\in[0,50]$ $\mu$K and
(b1-b2) the fluctuation of laser amplitudes where
$\delta_{\Omega}=\delta\Omega/0.5\Omega_{1(2)}^{pp}$
($\delta_{\Omega}=\delta\Omega/0.5\Omega^{pp}$ for the one-pulse scheme)
stands for the ratio between the maximal deviation $\delta\Omega$ and half of
the peak-peak value of optimized pulse amplitudes. Each point denotes an
average of 500 samplings. Results from the two-pulse and one-pulse schemes are
marked by green dots and red stars respectively without fluctuated
interactions. Insets of (b1-b2) amplify the data solved from the two-pulse
case.

In what follows, we analyze the gate performance due to some technical
imperfections, such as the thermal motion of atoms and the fluctuation from
laser amplitudes. During the gate operation, the atomic thermal motion at a
finite temperature will induce an inevitable Doppler-dephasing to the
excitation of the Rydberg state, which can be estimated as a phase change of
the two-photon Rabi frequencies in the two-pulse case, obeying

$\Omega_{1}\rightarrow\Omega_{1}e^{i\Delta_{1}t},\Omega_{2}\rightarrow\Omega_{2}e^{i\Delta_{2}t}$
(12)

Note that this replacement turns to be $\Omega\rightarrow\Omega e^{i\Delta t}$
for the one-pulse scheme. The detunings $\Delta_{1(2)}$ and $\Delta$ in the
phase factor of the Rabi frequencies can be approximately characterized by a
Gaussian distribution around its desired value $\bar{\Delta}=0$ with the
standard deviation $\sigma_{\Delta}$. Typically,
$\sigma_{\Delta}=\bm{k}\bm{v}$ where $\bm{k}=\sum_{j}\bm{k}_{j}$ is the
overall wave vector and $\bm{v}$ is the root-mean-square velocity. Although a
Doppler-free three-photon transition was proposed by satisfying $\bm{k}=0$
with a specific starlike planar geometry of three optical fields [59], there
is still no way to entirely eliminate the Doppler effect in a two-photon
transition which is mostly used. Ref. [60] presents a detailed discussion for
how to suppress the Doppler-dephasing induced by atomic thermal motion via two
counterpropagating sets of fields. Presently, to minimize this effect, we also
use counterpropagating beams at 780nm and 480nm for the transition
$|5S_{1/2}\rangle\to|5P_{3/2}\rangle\to|70S_{1/2}\rangle$, leading to
$\sigma_{\Delta}=(\bm{k}_{480}-\bm{k}_{780})|\bm{v}|=kv_{rms}$. Here
$k=5\times 10^{6}$ $m^{-1}$ and $v_{rms}=\sqrt{k_{B}T_{a}/M}$ where $k_{B}$ is
the Boltzmann constant and $M$ is the atomic mass. The atomic temperature
$T_{a}$ is assumed to change within the range of $T_{a}\in[0,50]$ $\mu$K for a
cryogenic environment. We note that similar magnitudes of an atomic
temperature have been reached experimentally [61, 62, 63].

The imperfection of the overall gate fidelity represented by
$\mathcal{F}_{2}(T_{a}=0)-\mathcal{F}_{2}(T_{a})$ as a function of $T_{a}$ is
shown in Fig. 6(a1-a2). For a given temperature $T_{a}$ we randomly adopt a
detuning from the $T_{a}$-dependent Gaussian distribution to simulate the
influence of phase error on the population evolution. Each point denotes the
result obtained by averaging over 500 samplings. The numerical results reveal
that the two-pulse scheme has stronger robustness to the thermal-motion
induced dephasing. Even at $T_{a}=50$ $\mu$K corresponding to
$\sigma_{\Delta}=0.3498$ MHz, the gate imperfection can be suppressed to $\sim
0.0111$ no matter what the interatomic interaction is. On the contrary, the
population dynamics of the one-pulse scheme are more sensitive to the extra
phase fluctuations of the Rabi frequency, resulting in a larger motion-induced
dephasing error. For example the imperfection of gate fidelity increases to be
$\sim 0.0318(0.0813)$ for $V_{0}/2\pi=7.0(1.0)$ MHz at $T_{a}=50$ $\mu$K. In
experiment, to suppress the motional dephasing between the ground state and
the Rydberg state, one efficient way is to cool atoms via Raman sideband
cooling which leads to a temperature about 10 $\mu$K or below [64, 65, 66]. In
that case the standard deviation $\sigma_{\Delta}$ can be decreased to 0.155
MHz (for $T_{a}=10$ $\mu$K) which satisfies the condition of
$\sigma_{\Delta}/\Omega\ll 1$ for suppressing the motion-induced dephasing as
proposed in [60]. At $T_{a}=10$ $\mu$K our numerical simulation shows the
Doppler dephasing error can be dramatically lowered to a level of $\sim
10^{-5}$ and safely ignored for all cases. Additionally, advanced laser
technology that depends on low-noise laser sources can further eliminate phase
errors [39].

In addition to the motion-induced dephasing error, we also investigate the
influence of the fluctuation of the laser amplitude on the gate fidelity.
Generally speaking, such gate protocols based on optimization require an exact
knowledge of the modified pulse shapes so as to be sensitive to the
fluctuations of laser amplitudes. This feature shares similarities with
adiabatic pulses [51]. Especially, the modified pulse shapes are crucial for
the success of the gate procedure. To study this effect, we assume a constant
fluctuation $\delta\Omega$ on the Rabi frequency and keep its waveform
unchanged, resulting in $\Omega_{1(2)}(t)\to\Omega_{1(2)}(t)+\delta\Omega$ for
the two-pulse scheme and $\Omega(t)\to\Omega(t)+\delta\Omega$ for the one-
pulse scheme. More precisely the deviation $\delta\Omega$ is defined by
$\delta\Omega=0.5\Omega_{1(2)}^{pp}\delta_{\Omega}$, where
$\Omega_{1(2)}^{pp}$ is the peak-peak value of the modified(optimized) pulse
amplitudes and $\delta_{\Omega}\in[0,0.05]$ is a coefficient for relative
deviation. A similar denotation with
$\delta_{\Omega}=\delta\Omega/0.5\Omega^{pp}$ is used for the one-pulse
scheme. For each $\delta_{\Omega}$ the numerical result is shown in
Fig.6(b1-b2) where we impose a random deviation within the range of
$[-\delta\Omega,+\delta\Omega]$ on the optimized pulse amplitudes, and carry
out the gate procedure. Every point in the figure is an average result over
sufficient gate implementations. From the results in Fig.6(b1) and (b2) we can
find the imperfection of the gate fidelity is only $\sim 10^{-4}$(see the
insets of (b1-b2)) which certifies the stronger robustness of our two-pulse
protocol to the fluctuation of laser amplitudes. A dramatic increase of the
gate imperfection with $\delta_{\Omega}$ is observed in the one-pulse case as
a consequence of a larger peak-peak amplitude $\Omega^{pp}$. With the relative
deviation $\delta_{\Omega}$ up to $5\%$ we show that the imperfection becomes
$\sim 0.05$($0.005$) for $V_{0}/2\pi=7.0(1.0)$ MHz. We expect a long-term
stable and precisely-adjustable Rydberg laser system in the experiment, which
can deeply reduce this technical error [67].

##  V Three-qubit Toffoli gate

So far we focus on a straightforward implementation of two-qubit CNOT gates.
Next, we proceed to describe how to realize a three-qubit Toffoli gate with
the same optimization algorithm. Consider three atoms arranged in a line where
two outer control atoms constrain the behavior of the central target atom, we
assume the nearest neighboring interaction is $V_{0}/2\pi=7.0$ MHz, leading to
a weak next-nearest neighboring interaction $V_{0}/2^{6}=2\pi\times 0.11$ MHz
between two control atoms. Its influence is avoidable when one adopts a two-
dimension triangular arrangement [55], but for generality, we describe the
gate in the line arrangement. Apparently, if any control atom is prepared in
the idle state $|1\rangle$, the three-qubit inputs can be directly reduced to
the state same as in the two-qubit gate, which are

$|10\beta\rangle\Rightarrow|0\beta\rangle,|01\beta\rangle\Rightarrow|0\beta\rangle,|11\beta\rangle\Rightarrow|1\beta\rangle.$
(13)

with $\beta\in\\{0,1\\}$. Discussions for implementing a two-qubit CNOT gate
have been given in Sec. III. So here we only pay attention to the most
different case if the input states are $|000\rangle$ and $|001\rangle$, which
means two outer control atoms acquire a same ground-Rydberg coupling by the
optical pulses. Intuitively the weak next-nearest neighboring interaction
between them would facilitate a simultaneous excitation onto state
$|rr\rangle$ by following the ideal transformation of

$|00\beta\rangle\to|rr\beta\rangle\to|00\beta\rangle$ (14)

prohibiting the excitation of the target atom. Nevertheless, the realistic
three-atom dynamics is very complicated due to the leakage of the target atom
onto other unwanted states under the globally-optimized laser pulses.
Therefore we use numerical optimization based on same genetic algorithm to
construct amplitude and phase-modulated pulses for the three-qubit Toffoli
gate.

Utilizing optimization algorithm we numerically explore the realization of a
three-qubit Toffoli protocol, in which the gate fidelity is assumed as

$\mathcal{F}_{3}=\frac{1}{8}\text{Tr}\left[\sqrt{\sqrt{U_{\text{Toffoli}}}\rho_{det}(t=T_{g})\sqrt{U_{\text{Toffoli}}}}\right]$
(15)

with the transformation matrix given by

$U_{\text{Toffoli}}=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1&0\end{array}\right)\\\ $
(16)

and the eight computational basis states $\\{|000\rangle$, $|001\rangle$,
$|010\rangle$, $|011\rangle$, $|100\rangle$, $|101\rangle$, $|110\rangle$,
$|111\rangle\\}$. The density matrix $\rho_{det}$ in Eq.(15) is solved from
the master equation with two-body interactions. The optimal pulses have a
duration of $T_{g}=1.2$ $\mu$s. Notice that this three-qubit Toffoli gate is
achieved straightforwardly with a pure vdWs energy shift $V_{0}/2\pi=7.0$ MHz
between two nearest atoms. No extra local single-qubit operations are required
that leads to the total gate time being as short as $1.2$ $\mu$s. Note that in
Ref. [68] a fast three-qubit Toffoli gate was proposed via three-body Förster
resonance with a multi-pulse sequence of 2.46 $\mu$s-duration.

Figure 7: Realization of a three-qubit Toffoli gate. (a1) The optimal pulse
amplitudes and the relative phase $\delta\phi$ in the two-pulse scheme.
(b1-c1) The corresponding time-dependent population dynamics of the input
states $|000\rangle$ and $|001\rangle$. The nearest-neighbor interaction is
$V_{0}/2\pi=7.0$ MHz and the decay rate from Rydberg states is $\gamma=3.0$
kHz. Similarly, (a2-c2) show the results of the one-pulse scheme. State
evolutions of $|r00\rangle$ and $|0r0\rangle$ are same and marked by same
linetype. Without considering the fluctuation of interactions, the overall
gate fidelity is $\mathcal{F}_{3}=0.9738$(two-pulse) and
$\mathcal{F}_{3}=0.9039$(one-pulse).

For the sake of clarity in Fig. 7 we only show the population dynamics when
the input states are $|000\rangle$ or $|001\rangle$. First, we observe that
when the target atom is initialized in state $|0\rangle$, i.e. $|000\rangle$,
the simultaneous excitation of three atoms caused by $\Omega_{1}(t)$ will
arise a competitive effect. Because two control atoms experience a dynamical
swapping of

$|rr0\rangle\rightleftarrows\frac{1}{\sqrt{2}}(|0r\rangle+|r0\rangle)\otimes|0\rangle$
(17)

before returning back to $|000\rangle$ [lower panels, Fig. 7(b1-b2)]. During
this period the target atom is also possibly excited to $|r\rangle$ due to the
use of a global pulse [upper panels, Fig. 7(b1-b2)]. If the input state is
$|001\rangle$, the competitive effect between the doubly- and singly-excited
Rydberg states for two control atoms persists in the dynamics [lower panels,
Fig. 7(c1-c2)]. In addition, we find that state $|001\rangle$ also couples to
$|000\rangle$ via a two-photon process, accompanied by a swapping of the
target atom between $|001\rangle$ and $|000\rangle$ mediated by $|00r\rangle$
[upper panels, Fig.7(c1-c2)]. The central physics here is that the two control
qubits, undergoing a simultaneous excitation, can effectively manipulate the
behavior of the target atom. The major limitations on the attainable fidelity
of a three-qubit Toffoli gate still come from the decay error from the
transient Rydberg population, as well as the imperfection in optimization.
With intrinsic errors included, the gate fidelity would be
$\mathcal{F}_{3}=0.9738$(or 0.9039) for the two-pulse(one-pulse) scheme.

It is apparent that the optimization algorithm becomes inefficient, especially
for the one-pulse case, leading to a relatively poor gate fidelity. That is
caused by the presence of fast and irregular population oscillations among
various middle states. To our knowledge, most of the evolutionary methods
including the genetic algorithm we used, adopt random policies to search for
the global maximum. And they perform insufficiently when too many random
processes are involved at the same time, e.g. the one-pulse scheme. While
there is no silver bullet, we think the way of reinforcement learning may be
an exceptive candidate for solving this problem in which the final fidelity of
quantum gate can be highly determined by the choice of a suitable value
function [69, 70, 71, 72]. More details for this issue can be found in
Appendix.B.

##  VI Discussion and Conclusion

To implement a real one-pulse CNOT gate, one may follow the sketch as
illustrated in Fig.1(a). By using an AEOM together with an arbitrary waveform
generator, the amplitude of the single laser beam can be modulated as
requested according to the optimization [43]. A subsequent AOM can spatially
separate the single pulse into the zeroth-order and the first-order
diffraction pulses with a hyperfine frequency difference $\omega_{01}\sim$
GHz. Ideally, the zeroth-order field serves as pulse 1 that globally excites
both atoms, realizing the off-resonant transition from $|0\rangle$ to the
intermediate state e.g. $|5P_{3/2}\rangle$. While the first-order field
deviates from the zeroth-order one in space which couples states $|1\rangle$
and $|5P_{3/2}\rangle$ of the target atom, serving as the local pulse 2. Here
we assume the coupling strengths between $|0\rangle$ and $|1\rangle$ with the
intermediate state are same so as to share a same pulse waveform. The
additional 480 nm laser is left on making transitions of
$|5P_{3/2}\rangle\to|70S_{1/2}\rangle$ for both atoms. In fact even if the
dipole matrix elements between $|0\rangle$ and $|1\rangle$ with the
intermediate state are slightly different, a same effective two-photon Rabi
frequency may still be attainable by using an adjustable detuning to the
intermediate state [73]. In experiment it remains possible to generate same-
shaped pulses based on two independent lasers with a fixed frequency
difference which requires extra frequency locking technique [74]. A more
detailed study of the implementation of one-pulse CNOT gates, taking into
account the practical light-atom coupling mechanism with real energy levels,
will be the subject of a future work.

In conclusion, we have presented an experimentally-accessible proposal of the
two-qubit CNOT gate and the three-qubit Toffoli gate with weakly-interacting
Rydberg atoms using a minimal number of optical pulses. The key lies in a
careful optimization of the pulse amplitudes and phases in advance, which
leads to the scheme being straightforward for implementation. We adopt a
genetic algorithm to perform global optimization which ensures not only a fast
gate duration, $\sim 1.0$ $\mu$s for the two-qubit gate($\sim 1.2$ $\mu$s for
the three-qubit gate), but also a high gate fidelity. Under practical
parameters excluding the fluctuations of interaction, we show that it is
possible to create a two-qubit CNOT gate by two(one) optical pulses with a
fidelity of 0.9951(0.9935) when two atoms are separated by 9.76 $\mu$m, and a
three-qubit Toffoli gate with a fidelity about 0.9738(0.9039) if the nearest
spacing is 7.10 $\mu$m. The more compact scheme with only one pulse achieves a
slightly smaller fidelity, because of the imperfection in the optimization
algorithm (see Appendix A).

Besides, we systematically investigate various intrinsic and technical errors
of the scheme. Compared to the earlier multi-pulse proposals in the strong
blockade regime it is apparent that our entangled gate using simultaneous
excitation with fewer pulses works well in the vdWs-interaction regime, which
waives the requirement of fast switching of multiple lasers at different
locations and greatly reduces the complexity of gate implementation.
Nevertheless, we also find that our optimization method becomes insufficient
while dealing with the most exotic case where the three-qubit Toffoli gate is
achieved by one pulse (the fidelity is only 0.9039). Because various ground-
Rydberg transitions and interactions make the three-atom dynamics very
complicated. In our future work, we will seek more powerful optimization
algorithms for the one-pulse Toffoli gate, hopefully achieving a fidelity
greater than 0.99.

###### Acknowledgements.

We are grateful to Shilei Su for help with the numerical simulation, to Jiefei
Chen and Jinxian Guo for useful discussions to the experimental design. This
work is supported by the National Key Research and Development Program of
China under Grant No. 2016YFA0302001; by the NSFC under Grants No. 12174106,
No.11474094, No.11104076 and No.11654005, by the Science and Technology
Commission of Shanghai Municipality under Grant No.18ZR1412800, by the
Shanghai Municipal Science and Technology Major Project under Grant No.
2019SHZDZX01, the Shanghai talent program.

##  Appendix A Proof of the vdWs interactions

Table 2: Non-resonant dipole-dipole processes between $|rr\rangle=|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$ and other nearby Rydberg pairs $|r_{c_{j}}r_{t_{j}}\rangle$ with $j\in(1,2,3,4)$, which are $|r_{c_{1}}r_{t_{1}}\rangle=|70P_{3/2},m_{J}=3/2;69P_{3/2},m_{J}=3/2\rangle$, $|r_{c_{2}}r_{t_{2}}\rangle=|70P_{3/2},m_{J}=3/2;69P_{1/2},m_{J}=-1/2\rangle$, $|r_{c_{3}}r_{t_{3}}\rangle=|69P_{3/2},m_{J}=3/2;70P_{1/2},m_{J}=-1/2\rangle$, $|r_{c_{4}}r_{t_{4}}\rangle=|70P_{1/2},m_{J}=-1/2;69P_{1/2},m_{J}=-1/2\rangle$ obtained from [48]. The Förster energy defects $\delta_{j}$ with respect to state $|r_{c_{j}}r_{t_{j}}\rangle$ are $(\delta_{1},\delta_{2},\delta_{3},\delta_{4})/2\pi=(0.71,1.01,0.99,1.29)$ GHz. $B_{j}=C_{3}^{(j)}/r_{0}^{3}$ represents the coupling strength with the dispersion coefficients given by $C_{3}^{(j)}/2\pi=(7.94,6.37,6.59,5.28)$ GHz$\cdot\mu$m3. $\mathcal{E}_{j}$ stands for the population leakage due to single coupled pair between $|rr\rangle\rightleftarrows|r_{c_{j}}r_{t_{j}}\rangle$, and $\bar{\mathcal{E}}$ gives the average population leakage from state $|rr\rangle$ when all channels $|r_{c_{j}}r_{t_{j}}\rangle$ are considered.  $r_{0}$($\mu m$) |  $B_{1}/2\pi$(MHz) | $\mathcal{E}_{1}$ |  $B_{2}/2\pi$(MHz) | $\mathcal{E}_{2}$ |  $B_{3}/2\pi$(MHz) | $\mathcal{E}_{3}$ |  $B_{4}/2\pi$(MHz) | $\mathcal{E}_{4}$ | $\bar{\mathcal{E}}$
---|---|---|---|---|---|---|---|---|---
4.89 | 67.90 | $1.7\times 10^{-2}$ | 54.45 | $5.7\times 10^{-3}$ | 56.33 | $6.4\times 10^{-3}$ | 45.17 | $2.5\times 10^{-3}$ | $2.5\times 10^{-2}$
7.10 | 22.18 | $1.9\times 10^{-3}$ | 17.79 | $6.1\times 10^{-4}$ | 18.40 | $6.9\times 10^{-4}$ | 14.76 | $2.6\times 10^{-4}$ | $2.9\times 10^{-3}$
9.76 | 8.54 | $2.8\times 10^{-4}$ | 6.85 | $9.1\times 10^{-5}$ | 7.08 | $1.0\times 10^{-4}$ | 5.68 | $3.9\times 10^{-5}$ | $4.3\times 10^{-4}$

To search for an appropriate interatomic distance $r_{0}$ that can be used for
a weak vdWs interaction between two distant atoms, we numerically simulate the
population dynamics of state $|rr\rangle$ due to the non-resonant interaction
with pairs of Rydberg states $|r_{c_{j}}r_{t_{j}}\rangle$. Here we apply four
dominant states, yet in principle we have to sum up sufficient adjacent states
at the same time which leads to a second-order level shift of $|rr\rangle$
represented by the coefficient
$C_{6}\approx\sum_{j}(C_{3}^{(j)})^{2}/\delta_{j}$ [20]. Because this non-
resonant interaction is only relevant to the population of state $|rr\rangle$
we choose it as the initial state in the calculation.

We consider four dominant leakage channels as shown in Fig. 8(a) where the
coupling strengths $B_{j}$ and the corresponding Förster defects $\delta_{j}$
are given in Table II using the ARC open source library [75]. In Fig. 8(b) we
show the time-dependent population dynamics of state $|rr\rangle$ with four
leakage channels under different two-atom distances $r_{0}=(4.89,7.10,9.76)$
$\mu$m. It is clearly noted that the average leakage of population is around
$\bar{\mathcal{E}}\sim 0.025$ for $r_{0}=4.89$ $\mu$m which will cause a big
effect on the optimal gate fidelity which is typically larger than $>99\%$ if
the fluctuation of interactions is excluded. In this case it is inappropriate
to assume a pure vdWs interaction for state $|rr\rangle$. However, if the two-
atom distance is increased to be $r_{0}=7.10$ $\mu$m or $9.76$ $\mu$m which
are larger than Rydberg blockade distance, the population missing from state
$|rr\rangle$ acquires a dramatic reduction. Depending on a full calculation
including all(four) leakage channels we show the imperfection due to non-
resonant dipole-dipole interactions can be at the level of
$\bar{\mathcal{E}}\sim 2.9\times 10^{-3}$ for $r_{0}=7.10$ $\mu$m and
$\bar{\mathcal{E}}\sim 4.3\times 10^{-4}$ for $r_{0}=9.76$ $\mu$m, which are
smaller than the intrinsic error during the gate execution. In the article we
would assume two well-separated atoms with a distance of $7.10$ $\mu$m and
9.76 $\mu$m, arising a vdWs interaction of $V_{0}=C_{6}/r_{0}^{6}=7.0$ MHz and
$1.0$ MHz as used in our calculation.

Figure 8: (a) Several dominant leakage channels of the two-atom Rydberg states
for the detuned dipole-dipole interactions where
$|rr\rangle$=$|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$ is initially
occupied. $B_{j}$ and $\delta_{j}$($j=1,2,3,4$) represent the coupling
strength and the Förster energy defect between $|rr\rangle$ and
$|r_{c_{j}r_{t_{j}}}\rangle$. (b) Population dynamics of state $|rr\rangle$
taking account into four nearby two-atom Rydberg states. Different central
distances $r_{0}=4.89$ $\mu m$(green dash-dotted), $7.10$ $\mu m$(red dashed)
and $9.76$ $\mu m$(black-solid) are used. Detailed parameters can be found in
Table II.

##  Appendix B Numerical optimization algorithm

Figure 9: Graphic diagram for the genetic optimal algorithm where the dashed
box shows the core of algorithm that connects with the physical model.

Genetic algorithm description. Utilizing numerical optimization procedures we
can obtain a set of optimal pulse parameters which approximately implements
the gate with a high performance. We adopt genetic algorithm(GA) in this
article. GA is a kind of random search algorithm inspired by Darwin’s theory
of natural selection and evolution. It can realize heuristic search for a
complex search space via simplified genetic process, which usually consists of
the population initialization, fitness assessment, selection, crossover and
mutation. The aim of GA’s usage in our study is to find out a set of
parameters in the given population that can make one of the individuals in the
population has the best fitness. The fitness is determined according to the
objective function. GA is only determined by the fitness function that has no
limit on the domain of definition, which robustly extends its generality. GA
starts optimization from a series of temporary results and iterates them
simultaneously which allows it to avoid local maximum and realize parallel
computation easily. Thus, GA offers a global optimization within the given
search region for all parameters, without the need for parameter
initialization. However, Rudolph has shown that the traditional canonical GA,
which only contains crossover, mutation and selection, can hardly realize a
global optimization because the crossover and mutation operators may destroy
the high-fitness scheme of an individual [76]. Thus, Ref. [77] proposed an
elitism scheme to overcome this which is verified to get a global convergence.
In our work, we adopt the method of Ref.[77] which can always preserve the
best individual without crossover or mutation to the next generation.

According to the procedure diagram as shown in Fig.9 we start by assuming an
objective function i.e. the targeted gate fidelity $\mathcal{F}_{n}$ with $n$
denoting the number of optimizations. We ignore the subscripts 2 or 3 as
defined in Eq.(9) and (15) for the qubit number. Then we give a reasonable
initial search region for all pulse parameters $\Omega_{\alpha\beta}s$. For
example, in the two-pulse scheme the set of parameters is
$(\Omega_{10},\Omega_{11},\Omega_{12})$ and
$(\Omega_{20},\Omega_{21},\Omega_{22})$. The number of loop iterations in the
algorithm is set to be 20$\times$50\. Here 20 is for the population size and
50 means the evolutionary generation number. The maximum of objective function
is $\mathcal{F}_{n,max}=1$. By solving the master equation (7) with random
pulse parameters $\Omega_{\alpha\beta}$s which are obtained from the initial
search region, we start to perform 20$\times$50 iterations by GA and get a
gate fidelity of $\mathcal{F}_{1}$ after the first optimization. To improve
the precision, we continuously perform the $(n)$th optimization where the new
search region is given depending on the optimized pulse parameters obtained
after the $(n-1)$th optimization. $n$ is the number of optimizations. Finally,
if the criterion
$\delta\mathcal{F}=|\mathcal{F}_{n}-\mathcal{F}_{n-1}|<10^{-4}$ is met the
iteration is terminated; otherwise, we restart the $(n+1)$th optimization
until this criterion is reached. All numerical results presented in the paper
are obtained by multi-optimization with same precision.

Table 3: For constructing a two-qubit CNOT gate with one pulse or two pulses, the optimized pulse parameters and the corresponding gate fidelities are presented with different vdWs interactions. Here $C_{6}/2\pi=863$ GHz$\cdot\mu m^{6}$, $\gamma=3.0$ kHz, $\mathcal{F}_{2,one(two)}$ represents the two-qubit gate fidelity in the one(two)-pulse scheme, and all frequency parameters are in unit of MHz. Numerical plot of these results is shown in Fig.5(b).  $V_{0}/2\pi$ |  $r_{0}$($\mu m$) | ($\Omega_{0}$,$\Omega_{1}$,$\Omega_{2})/2\pi$ | $\mathcal{F}_{2,one}$ | ($\Omega_{10}$,$\Omega_{11}$,$\Omega_{12})/2\pi$ |  $(\Omega_{20}$,$\Omega_{21}$,$\Omega_{22})/2\pi$ | $\mathcal{F}_{2,two}$
---|---|---|---|---|---|---
1.0 | 9.76 | (-5.8283,-5.5942,1.3558) | 0.9935 | (-0.9549,-1.9544,-0.0631) | (-2.8310, -3.6488,0.0074) | 0.9951
2.0 | 8.69 | (-8.0524, -2.3823, -6.2005) | 0.9938 | (-3.2197, -1.6527, -4.3272) | (-5.5704, 0.5088, -5.5704) | 0.9897
3.0 | 8.13 | (-0.8549,-15.0568,4.5848) | 0.9783 | (-6.4985,7.2516,0.2753) | (-0.0947, -3.0809, 0.5679) | 0.9752
4.0 | 7.74 | (-2.6977,-15.8330,7.5137) | 0.9810 | (1.6474,-2.2875,1.2089) | (0.5896, 1.7279,0.0084) | 0.9723
5.0 | 7.46 | (6.1341,-11.1408,9.1814) | 0.9840 | (-1.4798,-1.0744, -4.0572) | (-1.2613,4.4488,5.7295) | 0.9804
6.0 | 7.24 | (-0.5748, -15.1009, 8.7235) | 0.9899 | (-1.4115, -2.2769, -2.3879) | (0.5608, -1.6526, -1.5915) | 0.9834
7.0 | 7.10 | (-0.7619, -15.7833, 8.9923) | 0.9923 | (-3.9621, -0.7858, 1.5915) | (1.0942, -1.9068, -2.2182) | 0.9921

Failure of the genetic algorithm. Furthermore, we show some details about why
GA performed badly when implementing the three-qubit Toffoli gate with one
pulse, and why we expect reinforcement learning(RL) as a candidate for this
problem. To our knowledge, evolutionary methods including GA, apply some
static policies which interact over a period of time or one agent’s life with
a separate instance of the environment [69]. In other words it means that the
son generation could only perceive and interact with what its parents left.
Thus, those evolutionary methods are effective only if the space of policy in
optimization is small enough or could be cut apart into a set of small spaces.
Unfortunately, the search space is hard to be separated in our system. On the
other hand, those methods with more randomness are always seeking for various
policies or “paths”, in order to refine the results. In general, a traditional
evolutionary method is difficult to obtain sufficient information if the
process of evolution is highly random in some specific tasks. For example, in
the case of three-qubit Toffoli gate with one pulse as described in
Fig.7(a2-c2), the poor performance of states $|000\rangle$ and $|001\rangle$
arises from the fact that GA can not acquire sufficient evolutionary
information from the fast- and irregular-oscillating dynamics.

In the future we expect the RL method can become a candidate for optimizing
parameters in constructing a high-fidelity Toffoli gate with one pulse.
Compared to evolutionary methods, RL includes dynamical policies with the help
of value functions. As a consequence, RL can learn the feature of the
environment by interacting with it, and can use more information to find out a
better outcome than evolutionary methods. The targeted fidelity of quantum
gates is determined by the selection of an appropriate value function.
Therefore in RL by using a proper reward policy it is desirable to improve the
performance of a one-pulse Toffoli gate protocol [78]. Recently, we propose a
new deep learning method with a hybrid network for solving the high-frequency
oscillating population dynamics in a double-well potential, paving one-step
closer to this target [79].

##  Appendix C Optimized pulse parameters under different central interactions

As a supplement to Fig.5(b), Table 3 presents the optimized pulse parameters
as well as the simulated gate fidelity within an adjustable vdWs interaction
regime $V_{0}/2\pi\in[1.0,7.0]$ MHz. No fluctuated interaction is considered.

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# Expression Classification using Concatenation of Deep Neural Network for the
3rd ABAW3 Competition

Kim Ngan Phan
Dept. of Artificial Intellegence Convergence
Chonnam National University
Gwangju, South Korea
<EMAIL_ADDRESS>Hong-Hai Nguyen
Dept. of Artificial Intellegence Convergence
Chonnam National University
Gwangju, South Korea
<EMAIL_ADDRESS>Van-Thong Huynh
Dept. of Artificial Intellegence Convergence
Chonnam National University
Gwangju, South Korea
<EMAIL_ADDRESS>Soo-Hyung Kim
Dept. of Artificial Intellegence Convergence
Chonnam National University
Gwangju, South Korea
<EMAIL_ADDRESS>Corresponding author

###### Abstract

For computers to recognize human emotions, expression classification is an
equally important problem in the human-computer interaction area. In the 3rd
Affective Behavior Analysis In-The-Wild competition, the task of expression
classification includes 8 classes including 6 basic expressions of human faces
from videos. In this paper, we perform combination representation from RegNet,
Attention module, and Transformer Encoder for the expression classification
task. We achieve 35.87 % for F1-score on the validation set of Aff-Wild2
dataset. This result shows the effectiveness of the proposed architecture.

##  1 INTRODUCTION

Understanding Affective Behavior is playing an essential role in the
interaction between computers and humans [12]. This interaction makes it
possible for computers to understand human behaviors and emotions and
feelings. For many years, scientists have been working to build an intelligent
and automated machine that can understand and serve humans in many fields of
health, education, and services. Emotion recognition system allows receiving
many different data sources such as biological signals, visual or documents.
Visual data directly depicts interpretations of emotions through facial
expressions, thus playing an important role in emotion classification. In
1969, Ekman proposed six basic emotions in [2] including anger, disgust, fear,
happiness, sadness, and surprise. They are used popular but are not sufficient
to express complex human emotional states. In 2022, the 3rd Affective Behavior
Analysis In-The-Wild (ABAW) Competition is organized with the major target of
building and improving the emotional recognition ability of machines [5, 12,
6, 8, 11, 10, 7, 9, 17]. The competition consists of 4 challenges: valence-
arousal estimation, expression classification, action unit (AU) detection,
multi-task-learning. Each task corresponds to huge datasets with different
sizes from Aff-Wild2 Database [5, 12, 6, 8, 11, 10, 7, 9, 17]. They include
videos and cropped and aligned frames that contain annotations in terms. We
perform the expression classification task in this paper. In [3], the author
proposes the pre-trained ResNet model that combines spatio-channel attention
and efficient channel attention to get robust features. In [15], the authors
propose the multi-task learning technique for the pre-trained Resnet model,
and they use the Focal Loss to counter the imbalanced dataset. In this study,
we propose the combination of representative features from deep learning
networks for the expression classification task. We opt RegNet as the backbone
of our network. The Attention Mechanism and the Transformer Encoder play a
role as the embedding layer to extract good representations from the backbone.
Our model archives better performance than the baseline on validation set.
Section 2 describes the proposed method. The training details and results are
reported at the last of the paper.

##  2 PROPOSED METHOD

In [16], the authors introduce Multi-Head Attention has several attention
layers in parallel. The output of attention layers is a weighted sum of the
value, which the weight is computed by softmax function of Scaled Dot-Product
Attention [16]. The attention outputs are independent that are concatenated
and linearly transformed with the expected dimension. The multi-head allows
for attending to parts of the sequence differently. Fig 1 is overview of
Multi-Head Attention. The authors recommend the Transformer architecture that
has the encoder-decoder structure to build global dependencies between input
and output. The Transformer Encoder including a stack of multi-head self-
attention mechanisms and feed-forward networks can map an input sequence to
representation features. In this paper, we use Multi-Head Attention and
Transformer Encoder as the embedded layer to generate sequence
representations. We also employ the pre-trained model RegNet [14] as the
backbone for the proposed network. We use the weight of RegNet with 1.6GF
architecture on ImageNet dataset [1] to extract feature of images. The pre-
trained model takes the input size of the image as 112x112x3. The backbone is
extracted with 888 features by the flatten layer of the pre-trained model. We
reshape the backbone to (batch size, sequence, feature) and fed it into the
Transformer Encoder. In this work, we opt for the length of the sequence to be
64. In addition, the backbones are connected to the Multi-Head Attention
mechanism of the Transformer Encoder. We concatenate all backbones, output of
attention mechanism, and Transformer Encoder module to get final
representation features for the expression classification task. Dropout layer
with 50% information and dense layers of 8 neurons are used for final output
corresponding to 8 expressions of human. Fig 2 describe overview of our
architecture.

Figure 1: Overview of Multi-Head Attention Module. Figure 2: Overview of our
architecture.

##  3 EXPERIMENTS

###  3.1 Dataset

The large-scale in-the-wild Aff-Wild2 database contains 548 videos with
approximately 2.7 million frames. The training process is conducted on the
Aff-Wild2 database. Data is used containing annotations on 6 basic expressions
including Anger, Disgust, Fear, Happiness, Sadness, Surprise, plus Neutral
state, and plus Other which denotes emotional expressions other than the 6
basic states. The participants are provided with frames in RGB color space
from the database. The images are cropped and aligned with an input size of
112x112 from the video.

###  3.2 Training Details

The network is trained on the Pytorch framework. We use Adam optimization [4]
to update the weights. We opt Focal Loss [13] for the classification task of
eight emotions. The training process automatically finds the best learning
rate. The batch size of 16 on the train set and validation set and the epoch
of 30 is used for the training process. In this work, the final result is
evaluated across the average F1 Score of 8 emotion categories:

$F_{1}^{final}=\frac{\sum{F_{1}^{expr}}}{8}$ where $F_{1}^{expr}$ is $F_{1}$
score of each expression.

Parameters | Attention | Transformer Encoder
---|---|---
Depth | 1 | 1
Head Number | 2 | 2
Head Dimension | 64 | 64
Feedforward Dimension | - | 512
Dropout Value | 0. | 0.
Table 1: Hyperparameter of our architecture.

Model | Backbone | Module | Classifier |  $F_{1}$ Score (%)
---|---|---|---|---
Baseline [5] | VGG16 | - | FC | 23
Our Model | RegNet | Attention | FC | 35.87
|  | Transformer |  |

Table 2: Result of our model on validation set.

###  3.3 Results

In the baseline [5], the authors perform the pre-trained VGG16 network on the
VGGFACE dataset and get softmax probabilities for the 8 expression
predictions. They archive the F1 score of 0.23 on validation set of Aff-Wild2.
In the proposed model, we not only use the backbone but also combine
additional outputs after passing backbones through the attention module and
transformer encoder. As a result, this combination has a lot of information so
that the dropout layer is applied immediately after. Our model was 35.87% F1
score on validation set which is better than the baseline result. This shows
the effectiveness of the attention module as an embedding layer to extract the
salient information of the backbone.

##  4 CONCLUSION

In the 3rd Affective Behavior Analysis in-the-wild (ABAW3) Competition, we
have the opportunity to contribute research results in the field of human-
computer interaction. Our proposed model is performed for the expression
classification task based on the Aff-Wild2 database. We recommend combining
representative features from layer branches including RegNet, Attention
mechanism, and Transformer Encoder are used as embedding layers. We get a
better 35.87 % F1 Score than the baseline result for this task.

## acknowledgments

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (NRF-2020R1A4A1019191) and Basic
Science Research Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education (NRF-2021R1I1A3A04036408).

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