# Single-modulated-pulse two-qubit gates for Rydberg atoms with noncyclic
geometric control

Zi-Yuan Chen  Key Laboratory of Atomic and Subatomic Structure and Quantum
Control (Ministry of Education), Guangdong Basic Research Center of Excellence
for Structure and Fundamental Interactions of Matter, School of Physics, South
China Normal University, Guangzhou 510006, China  Jia-Hao Liang  Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of
Education), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactions of Matter, School of Physics, South China Normal
University, Guangzhou 510006, China  Zhao-Xin Fu  Key Laboratory of Atomic and
Subatomic Structure and Quantum Control (Ministry of Education), Guangdong
Basic Research Center of Excellence for Structure and Fundamental Interactions
of Matter, School of Physics, South China Normal University, Guangzhou 510006,
China  Hong-Zhi Liu  Key Laboratory of Atomic and Subatomic Structure and
Quantum Control (Ministry of Education), Guangdong Basic Research Center of
Excellence for Structure and Fundamental Interactions of Matter, School of
Physics, South China Normal University, Guangzhou 510006, China  Ze-Rui He
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry
of Education), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactions of Matter, School of Physics, South China Normal
University, Guangzhou 510006, China  Meng Wang  Key Laboratory of Atomic and
Subatomic Structure and Quantum Control (Ministry of Education), Guangdong
Basic Research Center of Excellence for Structure and Fundamental Interactions
of Matter, School of Physics, South China Normal University, Guangzhou 510006,
China  Zhi-Wei Han  Key Laboratory of Atomic and Subatomic Structure and
Quantum Control (Ministry of Education), Guangdong Basic Research Center of
Excellence for Structure and Fundamental Interactions of Matter, School of
Physics, South China Normal University, Guangzhou 510006, China  Jia-Yi Huang
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry
of Education), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactions of Matter, School of Physics, South China Normal
University, Guangzhou 510006, China  Qing-Xian Lv<EMAIL_ADDRESS>Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of
Education), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactions of Matter, School of Physics, South China Normal
University, Guangzhou 510006, China  Yan-Xiong Du<EMAIL_ADDRESS>Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry
of Education), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactions of Matter, School of Physics, South China Normal
University, Guangzhou 510006, China  Guangdong Provincial Key Laboratory of
Quantum Engineering and Quantum Materials, Guangdong-Hong Kong Joint
Laboratory of Quantum Matter, Frontier Research Institute for Physics, South
China Normal University, Guangzhou 510006, China

###### Abstract

Arrays of neutral atoms have emerged as promising platforms for quantum
computing. Realization of high-fidelity two-qubit gates with robustness is
currently a significant important task for large-scale operations. In this
paper, we present a convenient approach for implementing a two-qubit
controlled-phase gate using Rydberg blockade. We achieve the noncyclic
geometric control with a single modulated pulse. As compared with the control
scheme by cyclic evolution that determined by dynamical parameters, the
robustness of the proposal against systematic errors will be remarkably
improved due to the geometric characteristic. Importantly, the noncyclic
geometric control reduces the gate time for small rotation angles and will be
more insensitive to the decoherence effect. We accelerate the adiabatic
control with the aid of shortcuts to adiabaticity to further shorten the
operation time. We apply our protocol to the algorithm of quantum Fourier
transformation to show the actual acceleration. Therefore, the proposed scheme
will provide an analytical waveforms for arbitrary two-qubit gates and may
have important use in the experiments of atomic arrays.

##  I Introduction

Trapped neutral atoms are attractive physical platforms for large scale
quantum information systems Saffman2010 ; Saffman2016 ; Henriet2020 ;
Browaeys2020 ; Scholl2021 ; Ebadi2021 . Large numbers of atoms can be
manipulated in such systems while maintaining excellent quantum coherence.
Furthermore, single atom initialization, quantum gates, addressing, and
readout have been demonstrated in a variety of optical trapping platforms.
Addressable single-qubit gates have been implemented with high fidelity
Xia2015 ; Wang2016 ; Wu2019 . Two-qubit gates with neutral atoms can be
implemented by driving atoms to highly excited Rydberg states which utilizes
the strong and long-range interactions Jaksch2000 . Many improvements have
been made to improve the two-qubit gate fidelity in the past decade, such as
employing new Rydberg excitation laser setups to suppress or to eliminate
intermediate state scattering, filtering laser phase noise with a high finesse
cavity, cooling an atom down to the ground vibrational state in optical
tweezers, the optimal control theory, and so on Maller2015 ; Jau2016 ;
Levine2018 ; Leseleuc2018 ; Liu2021 ; Madjarov2020 . Recently, high-fidelity
parallel entangling gates have been realized on a neutral atomic system Fu2022
; Evered2023 . The next-stage important task will be the realization of
addressable two-qubit gates, which would accommodate more robust and faster
quantum control.

There are mainly two kinds of two-qubit gates with Rydberg interaction. The
first scheme is the Rydberg blockade with the so-called $\pi$-gap-$\pi$ pulses
sequences. It has been discussed in Jaksch2000 ; Saffman2005 ; Isenhower2011 ;
Theis2016 ; Shi2018 ; Liu2020  and realized in several experiments
Isenhower2010 ; Wilk2010 ; Zeng2017 ; Picken2018 ; Madjarov . The fidelity of
this scheme will suffer from severe decoherence between the ground state and
the Rydberg state which is indicated by the ground-to-Rydberg-state Ramsey
oscillations, although the operation period has been shorten to several
hundred nanoseconds. The second scheme is using the single-modulated pulses
off-resonant modulating driving to drive both the qubit atoms with the same
control field Sun2020 ; Sun2023 ; Levine2019 . The population of each two-
qubit basis will return to the initial one (cyclic evolution) with phases
acquired after the non-adiabatic driving, which can be used to construct the
controlled-phase gates. Usually, the dynamics of different two-qubit basis
will be asynchrony. Optimization algorithm is adopted to reshape control
waveforms to satisfy the cyclic evolution. This features that the operation
maybe sensitive to the control paramters which may influence the fidelity of
the two-qubit gates in the large-scale arrays. Therefore, realization of
robust and high-fidelity two-qubit operations in atomic arrays still need
further investigations.

In this paper, we introduce a protocol for the realization of controlled-phase
($\mathrm{C_{Z}}$) gates in atomic arrays with Rydberg interaction. As in the
Rydberg blockade region, different two-qubit basis will experience different
dynamical parameters, the geometric control with adiabatic driving would be a
natural choice which is robust against random noise and the systematic errors
Berry1984 ; wilzeck1984 ; Aharonov1987 ; Sjoqvist2008 ; Sjoqvist2016 ;
zhusl2003 ; duan2001 ; Sjoqvist2012 ; Abdumalikov2013 ; Feng2013 ; Danilin2018
. However, adiabatic evolution is inherently slow and susceptible to
dissipation. To expedite the adiabatic process, shortcut to adiabaticity (STA)
methods are considered advantageous. By applying a counter-diabatic
Hamiltonian, the errors induced by diabatic effect during fast evolution can
be mitigated Torrontegui2013 ; Liang2016 ; felix2018 ; Yu2018 ; Yan2019 ;
Vepsalainen2019 ; Vepsalainen2018 ; bhhuang ; yhchen . As a step further, the
geometric control scheme is generalized to the noncyclic case Lv2020 ; Ji2021
; Qiu2021 ; Zhang2021 . The operation time of noncyclic scheme will be much
shorter than cyclic one as long as the rotation angle of the two-qubit gates
getting smaller. We demonstrate the acceleration results of our proposal in
the algorithm of quantum Fourier transformation, a pivotal component of Shor’s
prime number factoring algorithm. Therefore, our proposal provide a fast and
robust way to realize two-qubit gate in atomic arrays with analytical control
waveforms.

The structure of this paper is as follows: In section II, The control model of
two-qubit gate with Rydberg blockade is introduced. In section III, the
proposal of controlled-phase gates with noncyclic geometric control (NCGC) and
STA is introduced. In section IV we discuss the robustness of our protocol
against random noise and systematic errors and explores its performance under
the influence of decoherence. Section V provides a comparative analysis of the
time required for the quantum Fourier transform between NCGC and the cyclic
case. We summarizes and concludes the paper in section VI.

##  II The control model of two-qubit gate with Rydberg blockade

In the following we introduce the Hamiltonian of two atoms with Rydberg
interaction, where each atom has three levels as labeled by
$\\{|0\rangle,|1\rangle,|r\rangle\\}$, $|0\rangle,|1\rangle$ are the ground
states and $|r\rangle$ is the Rydberg state. Ground state $|1\rangle$ of each
atom is coupled to Rydberg state with Rabi frequency $\Omega$, detuning
$\Delta$ and $\varphi$. The interaction between two Rydberg states of two
atoms are given by $V$. Under the single-qubit basis, the total Hamiltonian of
the two interacting atoms (as labelled by $1,2$) is described as

$\displaystyle H=(\frac{\Omega}{2}e^{i\varphi}|1\rangle_{1}\langle r|\otimes
I_{2}+I_{1}\otimes|1\rangle_{2}\langle r|+H.c.)$ (1)
$\displaystyle+\Delta(|r\rangle_{1}\langle r|\otimes
I_{2}+I_{1}\otimes|r\rangle_{2}\langle r|)+V|r\rangle_{1}\langle
r|\otimes|r\rangle_{2}\langle r|,$

where $I_{1}=|0\rangle_{1}\langle 0|+|1\rangle_{1}\langle
1|+|r\rangle_{1}\langle r|$, $I_{2}=|0\rangle_{2}\langle
0|+|1\rangle_{2}\langle 1|+|r\rangle_{2}\langle r|$, $\hbar=1$. By changing to
the two-qubit basis, the total Hamiltonian can be divided into three parts

$\displaystyle H=H_{1}+H_{2}+H_{3},$ (2) $\displaystyle
H_{1}=(\frac{\Omega}{2}e^{i\varphi}|10\rangle\langle
r0|+H.c.)+\Delta|r0\rangle\langle r0|,$ $\displaystyle
H_{2}=(\frac{\Omega}{2}e^{i\varphi}|01\rangle\langle
0r|+H.c.)+\Delta|0r\rangle\langle 0r|,$ $\displaystyle
H_{3}=[\frac{\sqrt{2}}{2}\Omega e^{i\varphi}(|11\rangle\langle
R|+|R\rangle\langle rr|)+H.c.]$ $\displaystyle+\Delta|R\rangle\langle
R|+(V+2\Delta)|rr\rangle\langle rr|,$

where $|R\rangle=(|1r\rangle+|r1\rangle)/\sqrt{2}$. As can be seen that,
$|10\rangle(|01\rangle)$ is coupled to $|r0\rangle(|0r\rangle)$ with Rabi
frequency $\Omega$ and detuning $\Delta$ while $|00\rangle$ is decoupled from
the system. Under the Blockade condition $V\gg\Omega$, $H_{3}$ can be deduced
to

$H^{\prime}_{3}=(\frac{\sqrt{2}}{2}\Omega e^{i\varphi}(|11\rangle\langle
R|+H.c.)+\Delta|R\rangle\langle R|,$ (3)

with effective Rabi frequency $\sqrt{2}\Omega$. The coupling scheme of
$|01\rangle,|10\rangle,|11\rangle$ is shown in Fig. 1(a).

An ingenious method to achieve controlled-phase gate in such system is to
drive the system cyclically with phases acquired, i.e.,
$\varphi_{11}-\varphi_{10}-\varphi_{01}=\pi$,
$\varphi_{01},\varphi_{10},\varphi_{11}$ are the phases acquired upon states
$|01\rangle,|10\rangle,|11\rangle$, respectively. Nevertheless, the dynamics
between $|01\rangle(|10\rangle)$ and $|11\rangle$ are different intrinsically
(the Rabi frequencies among them are different), and thus optimization
algorithm is adopted to design the control waveforms carefully Sun2020 ;
Sun2023 ; Levine2019 .

Figure 1:  Scheme of $C_{Z}$ gate based on noncyclic geometric quantum
control. (a) Coupling configuration of two interacting atoms under the two-
qubit basis. Under the effect of Rydberg blockade, $|10\rangle(|01\rangle)$
are coupled to $|r0\rangle(|0r\rangle)$ with Rabi frequency $\Omega$ while
$|11\rangle$ is coupled to $|R\rangle=(|1r\rangle+|r1\rangle)/\sqrt{2}$ with
$\sqrt{2}\Omega$ under the Rydberg blockade condition. All the couplings are
shifted by a detuning $\Delta$. $|00\rangle$ is decoupled from the system. (b)
Control waveforms of NCGC. Black-dashed line: $\Omega$, red-solid line:
$\varphi$. The adiabatic control can be accelerated by adding an auxiliary
detuning $\Delta_{a}$ (blue dashed-dotted line) which induces $\Omega T=4\pi$,
$T$ is the evolution period. (c), (d) Dynamics of different basis under the
driving. (c) symbols the dynamics of $|11\rangle$ while (d) symbols the ones
of $|10\rangle$ or $|01\rangle$. Blue-dashed lines: population $P^{(\eta)}$.
Red-solid lines: acquired phases $Ph^{(\eta)}$. As can be seen that all basis
return to the initial population with a $\pi$ phase acquired after the driving
and thus the $C_{Z}$ gate is achieved.  Figure 2:  Robustness of $C_{z}$ gate
realized by NCGC scheme. (a) Fidelity $F$ of $C_{z}$ gates realized by
different protocols against the variation of Rabi frequencies. (a) Fidelity
$F$ of $C_{z}$ gates realized by different protocols against the variation of
detuning. Red-solid lines: NCGC, blue-dotted lines: PM, green dashed-dotted
lines: RM, where NCGC adopts the waveforms in Eq. (10), RM adopts the ones in
Eq. (11) and PM using Eq. (12). (c) Numerical simulation of gate fidelity $F$
against random noise of Rabi frequency and detuning, in which
$\Omega_{r}(t)=\kappa^{\prime}_{1}(t)\Omega+\Omega$ and
$\Delta_{r}(t)=\kappa^{\prime}_{2}(t)\Omega+\Delta(t)$. $\kappa^{\prime}_{1}$,
$\kappa^{\prime}_{2}$ are time-varying random number and
$\kappa^{\prime}_{1}(\kappa^{\prime}_{2})\in[0,0.1]$. The fidelity of each
point was obtained by averaging 50 times.

##  III $\mathrm{C_{Z}}$ gate by noncyclic geometric control with shortcut to
adiabaticity

In the following we introduce how to realize the controlled-phase gate with
noncyclic geometric control. Here we introduce the labelling of
$|\mu^{(1)}_{+}\rangle=|r0\rangle$, $|\mu^{(2)}_{+}\rangle=|0r\rangle$,
$|\mu^{(3)}_{+}\rangle=|R\rangle$, and $|\mu^{(1)}_{-}\rangle=|10\rangle$,
$|\mu^{(2)}_{-}\rangle=|01\rangle$, $|\mu^{(3)}_{-}\rangle=|11\rangle$. Then
the eigenstates of Hamiltonian $H_{1},H_{2},H^{\prime}_{3}$ can be written as

$\displaystyle|\lambda^{(\eta)}_{+}\rangle=\sin(\frac{\theta^{(\eta)}}{2})e^{i\varphi}|\mu^{(\eta)}_{+}\rangle+\cos(\frac{\theta^{(\eta)}}{2})|\mu^{(\eta)}_{-}\rangle,$
(4)
$\displaystyle|\lambda^{(\eta)}_{-}\rangle=\cos(\frac{\theta^{(\eta)}}{2})|\mu^{(\eta)}_{+}\rangle-\sin(\frac{\theta^{(\eta)}}{2})e^{-i\varphi}|\mu^{(\eta)}_{-}\rangle,$

$\eta=1,2,3$, $\theta^{(1)}=\theta^{(2)}=\arctan(\Omega/\Delta)$,
$\theta^{(3)}=\arctan(\sqrt{2}\Omega/\Delta)$. The corresponding eigenvalues
are derived as $E^{(\eta)}_{\pm}$=$\pm\hbar\Omega^{(\eta)}$, among which
$\Omega^{(1)}=\Omega^{(2)}=\sqrt{\Omega^{2}+\Delta^{2}},\Omega^{(3)}=\sqrt{2\Omega^{2}+\Delta^{2}}$.

The dynamics of $H_{\eta}(H^{\prime}_{\eta})$ can be treated as the two-level
systems $\\{|01\rangle,|r0\rangle\\}$, $\\{|10\rangle,|0r\rangle\\}$,
$\\{|11\rangle,|R\rangle\\}$ governed by control fields with the
parameterizations as
$\textbf{B}^{\eta}=({\sin\theta^{(\eta)}\sin\varphi,\sin\theta^{(\eta)}\sin\varphi,\cos\theta^{(\eta)}})$.
To realize the controlled-phase gate which is independent of $\Omega$, we may
adopt $\theta^{(1)}=\theta^{(2)}=\theta^{(3)}=\pi/2$ by setting $\Delta=0$.
Namely, we only modulate the phase of the control field. To drive the system
evolve adiabatically, we need to satisfy
$\dot{\varphi}\ll\mathrm{Min}[(E^{(\eta)}_{+}-E^{(\eta)}_{-})]/\hbar=2\Omega^{(1)}$,
$\mathrm{Min}$ symbols the function that returns the minimal values. Now we
assume that the initial state upon each subsystem is given by
$|\Psi^{(\eta)}_{0}\rangle=b_{1}|\lambda^{(\eta)}_{+}\rangle+b_{2}|\lambda^{(\eta)}_{-}\rangle$.
The construction of geometric gates can be separated into two stages: firstly,
modulating the phase adiabatically from $\varphi^{(0)}$ to
$\varphi^{(0)}+\Delta\varphi$, $\varphi^{(0)}$ is the initial phase of the
control field. The components of $|\Psi^{(\eta)}_{0}\rangle$ that evolve along
$|\lambda_{\pm}^{(\eta)}\rangle$ will pick up a total phase as
$\alpha^{(\eta)}_{\pm}=\delta^{(\eta)}_{\pm}+\beta^{(\eta)}_{\pm}$, where the
geometric phases are given by
$\delta^{(\eta)}_{\pm}=i\int_{0}^{T}\langle\lambda^{(\eta)}_{\pm}|\frac{d}{dt}|\lambda^{(\eta)}_{\pm}\rangle
dt=\pm\Delta\varphi/2$ and the dynamical phases are given by
$\beta^{(\eta)}_{\pm}=-\int_{0}^{T}\langle\lambda^{(\eta)}_{\pm}|H_{\eta}|\lambda^{(\eta)}_{\pm}\rangle
dt=\mp\int_{0}^{T}\Omega^{(\eta)}dt$. The evolution operator
$U^{(\eta)}_{1}=U(\theta:\pi/2,\varphi:\varphi^{(0)}\rightarrow\varphi^{(0)}+\Delta\varphi)$
under the basis
$\\{|\lambda^{(\eta)}_{+}\rangle,|\lambda^{(\eta)}_{-}\rangle\\}$ is given by
Lv2020

$U^{(\eta)}_{1}=\left(\begin{array}[]{cc}e^{i(\delta_{+}+\beta_{+})}&0\\\
0&e^{i(\delta_{-}+\beta_{-})}\end{array}\right).$ (5)

Then we flip the control field by a $\pi$ phase and driving
$\textbf{B}^{\eta}$ adiabatically from $\varphi^{(0)}+\pi+\Delta\varphi$ to
$\varphi^{(0)}+\pi+2\Delta\varphi$. The evolution operator
$U^{(\eta)}_{2}=U(\theta:\pi/2,\varphi:\varphi^{(0)}+\pi+\Delta\varphi\rightarrow\varphi^{(0)}+\pi+2\Delta\varphi)$
is given by

$U^{(\eta)}_{2}=\left(\begin{array}[]{cc}e^{i(\delta_{+}+\beta_{-})}&0\\\
0&e^{i(\delta_{-}+\beta_{+})}\end{array}\right).$ (6)

Thus, the evolution operator of the union operations is derived as

$U^{(\eta)}=U^{(\eta)}_{2}U^{(\eta)}_{1}=\left(\begin{array}[]{cc}e^{i\Delta\varphi}&0\\\
0&e^{-i\Delta\varphi}\end{array}\right).$ (7)

Here we have used the relationship of
$\delta^{(\eta)}_{+}=-\delta^{(\eta)}_{-}$ and
$\beta^{(\eta)}_{+}=-\beta^{(\eta)}_{-}$. It can be seen that $U^{(\eta)}=U$
is only depends on the value of $\Delta\varphi$ (has no dependence with
$\Omega$) and is thus purely geometric. The proposed scheme is also noncyclic
as $\Delta\varphi$ can be set from $0$ to $\pi$. A smaller $\Delta\varphi$
refers to a shorter evolution time $T$ against the same adiabatic condition,
and thus the control can be sped up when the rotating angle $\Delta\varphi$
become smaller.

Figure 3:  Fidelity $F$ of $C_{z}$ gate against Rydberg interacting strength
$V$. Red lines: NCGC, blue lines: PM. Solid lines: $T=500\mu$s, dashed lines:
$T=250\mu$s. The robustness of NCGC scheme against $V$ will be enhanced when
the operation time is shorten while the robustness of PM scheme will weaken.

Now we turn to the case of two-qubit gate upon the two-qubit basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$. According to Eq.(7),
states $|01\rangle,|10\rangle,|11\rangle$ will pick up the same phases
$\Delta\varphi$ after the noncyclic geometric evolution and the evolution
operator will be given by

$U_{p}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&e^{-i\varphi_{01}}&0&0\\\
0&0&e^{-i\varphi_{10}}&0\\\ 0&0&0&e^{-i\varphi_{11}}\end{array}\right)\\\ ,$
(8)

where $\varphi_{11}=\varphi_{10}=\varphi_{01}=\Delta\varphi$. By setting
$\Delta\varphi=\pi$, we arrive at the $\mathrm{C_{Z}}$ gate that satisfying
the condition $\varphi_{11}-\varphi_{10}-\varphi_{01}=\pm\pi$.

To further accelerate the protocol, we adopt the STA scheme through adding
auxiliary Hamiltonian
$H_{1}^{(\eta)}(t)=i\underset{k=\pm}{\sum}(|\partial_{t}\lambda^{(\eta)}_{k}\rangle\langle\lambda^{(\eta)}_{k}|-\langle\lambda^{(\eta)}_{k}|\partial_{t}\lambda^{(\eta)}_{k}\rangle|\lambda^{(\eta)}_{k}\rangle\langle\lambda^{(\eta)}_{k}|)$.
According to Eq.(4), the expression of $H_{1}^{(\eta)}$ upon the subspace
$\\{|\lambda^{(\eta)}_{+}\rangle,|\lambda^{(\eta)}_{-}\rangle\\}$ will be
given by

$H_{1}^{(\eta)}(t)=\frac{1}{2}\left(\begin{array}[]{cc}\Delta_{a}&0\\\
0&-\Delta_{a}\end{array}\right),$ (9)

$\Delta_{a}=\dot{\varphi}$. As can be seen that all the adiabatic controls and
their acceleration in subspace
$\\{|\lambda^{(\eta)}_{+}\rangle,|\lambda^{(\eta)}_{-}\rangle\\}$ can be
chosen the same set of control parameters $(\Omega,\Delta,\varphi)$, and thus
each two-level system can be manipulated simultaneously.

In Fig. 1(b) we plot the control waveform of controlled-phase gate, where the
Rabi frequency is set to be $\Omega_{0}=2\pi\times 4$MHz and detuning
$\Delta_{0}=0$. $V=2\pi\times 500$MHz is adopted to confirm the blockade
condition. The formulation of phases are given by

$\varphi_{0}=\left\\{\begin{array}[]{lr}0.5a\pi[1-\cos(\frac{\pi
t}{\tau})],(0<t\leq\tau)&\\\
\pi+0.5a\pi\\{3-\cos[\frac{\pi(t-\tau)}{\tau}]\\},(\tau<t<T)\end{array},\right.$
(10)

Here we set a typical evolution period $T=500$ns and $\tau=T/2$ Evered2023 ,
$a$ is a constant that determined the rotation angle. To fulfill the condition
of $\mathrm{C_{Z}}$ gate, we set $a=1$ ($\Delta\varphi=\pi$). As shown in Fig.
1(c) and 1(d), when the system is prepared upon the state
$|\Psi_{0}^{(\eta)}\rangle=|11\rangle$ ($|01\rangle,|10\rangle$), the
population $P^{(\eta)}$ (blue-dashed lines) will return to the initial value
after the evolution. The relative phases between the initial states
$|\Psi_{0}^{(\eta)}\rangle$ and the final states $|\Psi_{f}^{(\eta)}\rangle$
are calculated by
$Ph^{(\eta)}=\mathrm{Arg}\langle\Psi_{f}^{(\eta)}|\Psi_{0}^{(\eta)}\rangle$,
$\mathrm{Arg}$ returns the phase angles of complex numbers. As shown by the
red-solid lines in Fig. 1(c) and 1(d), all of the above states will pick up a
$\pi$ phase and thus the controlled-phase gate can be achieved. It should be
noted that in our calculations $\Omega T=4\pi$ that is not fulfill the
adiabatic condition, nevertheless, the system will evolve along the adiabatic
pathes after adding the auxiliary Hamiltonian $H_{1}^{(\eta)}$. Therefore,
fast and geometric control can be realized in the proposed scheme.

Figure 4:  Numerical simulation of gate fidelity $F$ against the evolution
time $T$ under the effect of decay and dephasing. The simulation include the
decay from upper energy level $|\lambda^{(\eta)}_{+}\rangle$ to lower energy
level $|\lambda^{(\eta)}_{-}\rangle$ and upper energy level
$|\lambda^{(\eta)}_{+}\rangle$ to the energy level outside the system
$|W\rangle$, as well as a dephasing between the upper energy level
$|\lambda^{(\eta)}_{+}\rangle$ and lower energy level
$|\lambda^{(\eta)}_{-}\rangle$.

##  IV Performance of $\mathrm{C_{Z}}$ gate of noncyclic geometric scheme

In the actual experiments, quantum control of the system will be influenced by
the offset of the control parameters ($\Omega,\Delta,\varphi$) and the
decoherence effect. Here we test the performance of NCGC protocol. In Fig.2 we
can study the influence of offset of Rabi frequency and detuning and make a
comparison with the other two protocols: single-modulated pulses off-resonant
modulating driving Sun2020  and time optimal gate Evered2023 . The off-
resonant modulating driving scheme can be realized by modulating the Rabi
frequency (RM) with a detuning as given by:

$\displaystyle\Omega_{1}=\sum\limits_{\nu=1}^{4}\beta_{\nu}[b_{\nu,n}(t/T)+b_{n-\nu,n}(t/T)],$
(11) $\displaystyle\Delta_{1}=2\pi\times 3.512\text{MHz},$
$\displaystyle\varphi_{1}=0,$

$\beta_{1}$=1.419 MHz, $\beta_{2}$=0 MHz, $\beta_{3}$=5.076 MHz,
$\beta_{4}$=13.425 MHz. $b_{v,n}$ is the $v$th Bernstein basis polynomial of
degree $n$ and we set $n=8$. The time optimal scheme is realized by modulating
the control phases (PM) as given by:

$\displaystyle\Omega_{2}=2\pi\times 4.6\,\text{MHz},$ (12)
$\displaystyle\Delta_{2}=0,$ $\displaystyle\varphi_{2}=A\cos(\omega
t-\varphi_{2}^{(0)}),$

$A=2\pi\times 0.1122$ rad, $\omega=0.1431\Omega_{2}$,
$\varphi^{(0)}_{2}=-0.7318$ rad. All the coefficients in Eq. (10) and (11) are
determined by the optimized algorithm. Due to the finite coherence time of the
Rydberg states, we set the evolution time of the three cases (NCGC, PM and RM)
to be $T=500$ ns. The Rydberg interacting strength is given by $V=2\pi\times
500$ MHz. It can be seen that the control waveforms of NCGC is quite similar
to the ones of PM. However, by dividing the operations into two parts, the
dynamical phase can be cancelled and the waveforms of arbitrary controlled-
phase gates can be obtained analytically. Furthermore, the adiabatic control
of NCGC is accelerated by shortcut to adiabaticity which make it less
sensitive to the decoherence effect.

In Fig. 2, we compare the robustness of the three schemes (NCGC, PM and RM)
against the systematic errors. The fidelity is defined as
$F=|tr({U_{t}}{U_{z}}^{\dagger})|/16$, $U_{z}$ is the ideal matrix of
$\mathrm{C_{Z}}$ gate (Eq. (8) with
$\varphi_{11}-\varphi_{10}-\varphi_{01}=\pm\pi$) while ${U_{t}}$ is the
evolution operator of the protocols which calculated by
$U_{t}=\mathcal{T}e^{-iH_{p}t}$, $H_{p}=H_{1}+H_{2}+H^{\prime}_{3}$,
$\mathcal{T}$ symbols time-ordering operator. $tr$ symbols the trace of the
matrix. The deviation of the parameters are introduced by
$\Omega^{\prime}_{k}=(1+\kappa_{1})\Omega_{k}$,
$\Delta^{\prime}_{k}=\Delta_{k}+\kappa_{2}\Omega_{k},k=0,1,2$,
$\kappa_{1}(\kappa_{2})\in[-0.1,0.1]$. The numerical results are shown in Fig.
2(a) and 2(b) where the control parameters are listed in Eq. (10), (11) and
(12), red-solid lines: NCGC, blue-dotted lines: PM, green dashed-dotted lines:
RM. As can be seen that the noncyclic geometric control does not affected by
the deviation of the Rabi frequency since the variation of Rabi frequency only
affect the accumulated dynamic phases and the dynamic phase is finally
eliminated. The NCGC scheme is also robust against the offset of detuning as
shown in Fig. 2(b). Therefore, the intrinsic geometric control is robust
against systematic errors as compared with the optimal controls that are only
optimized at some special settings.

Here we investigate the robustness NCGC scheme against the random noise. The
numerical results are shown in Fig. 2(c) where the random terms are introduced
by $\Omega_{r}(t)=\kappa^{\prime}_{1}(t)\Omega+\Omega$ and
$\Delta_{r}(t)=\kappa^{\prime}_{2}(t)\Omega+\Delta(t)$. $\kappa^{\prime}_{1}$,
$\kappa^{\prime}_{2}$ are time-varying random number and
$\kappa^{\prime}_{1}(\kappa^{\prime}_{2})\in[0,0.1]$. The parameters settings
are the same with Fig. 1. Since the proposed geometric gate is achieved by the
accumulation of the noncyclic geometric phases with vanishing dynamical
phases, the fidelity of the protocol keep a fidelity of over 0.9999 even if
the amplitude of noise reach 10$\%$ of the Rabi frequency, which shows strong
noise robustness.

Figure 5:  Numerical simulation of total evolution time of quantum Fourier
transformation versus the number of qubits $N$. Blue-dotted line: cyclic
scheme, red-solid line: noncyclic scheme. The NCGC scheme rise slower than
cyclic scheme as the amount of two-qubit gates with small rotating angles
increase polynomially.

In Fig.3 we study the influence of Rydberg interacting strength $V$ upon the
manipulation. To include the effect of finite interaction strength, we
calculate the evolution operator using the fully Hamiltonian, that is,
$U_{H}=\mathcal{T}e^{-iHt}$, $H=H_{1}+H_{2}+H_{3}$. We make a comparison
between NCGC scheme and the PM scheme which adopt the parameters in Eq. (10)
and Eq. (12), respectively. When the evolution time $T=500\mu$s, the fidelity
of NCGC (red-solid line) and PM (blue-solid line) will exceed 0.995 when
$V>2\pi\times 400$ MHz, nevertheless, the NCGC scheme would be more sensitive
to the decreasing of strength $V$ as compared with PM. The situation will be
different when the evolution time is shorten to $T=250\mu$s. As can se seen
that the robustness of NCGC (red-dashed line) against $V$ will be enhanced
while the one of PM (blue-dashed line) drops. One can check that the Rabi
frequency of PM scheme should be increase to $\Omega_{2}=2\pi\times 9.2$ MHz
in this case to keep $\Omega_{2}T$ unchange to certificate the cyclic
evolution. A larger Rabi frequencies makes the PM scheme more sensitive to
$V$. On the other hand, the Rabi frequency of NCGC scheme will be hold as the
acceleration is achieved by tilting the detuning $\Delta_{a}$ in Eq.(9). The
decrease of time $T$ would reduce the influence of the imperfect blockade.
Therefore, a low ratio of $\Omega_{k}/V$ at a given evolution time is an
essential condition to realize high fidelity two-qubit gate.

In the actual experiment, all the gate protocol will suffer from decoherence.
In Fig.4 we simulate the relationship between fidelity of $\mathrm{C_{Z}}$
gate and the operation time. The simulation include the decay from upper
energy level $|\lambda^{(\eta)}_{+}\rangle$ to lower energy level
$|\lambda^{(\eta)}_{-}\rangle$ and upper energy level
$|\lambda^{(\eta)}_{+}\rangle$ to the energy level outside the system
$|W\rangle$, as well as a dephasing between the upper energy level
$|\lambda^{(\eta)}_{+}\rangle$ and lower energy level
$|\lambda^{(\eta)}_{-}\rangle$. We consider a fully Hamiltonian
$H=H_{1}+H_{2}+H_{3}$ that include finite Rydberg blockade. We introduce
decoherence through master equation which can be written as Lindblad1976 ;
Johansson2012

$\dot{\rho}=-i[H,\rho]+\underset{n}{\sum}2L_{n}\rho
L_{n}^{\dagger}-L_{n}^{\dagger}L_{n}\rho-\rho L_{n}^{\dagger}L_{n},$ (13)

in which $\rho$ is the density matrix of current state,
$L_{1}=\sqrt{\gamma_{1}}(|11\rangle\langle R|+|10\rangle\langle
r0|+|01\rangle\langle 0r|)$ is decay from upper energy level to lower energy
level, $L_{2}=\sqrt{\gamma_{2}}(|W\rangle\langle R|+|W\rangle\langle
r0|+|W\rangle\langle 0r|)$ is decay from upper energy level to the energy
level outside the system, $L_{3}=\sqrt{\gamma_{3}}(|R\rangle\langle
R|-|11\rangle\langle 11|+|r0\rangle\langle r0|-|10\rangle\langle
10|+|0r\rangle\langle 0r|-|01\rangle\langle 01|)$ is the dephase between upper
energy level and lower energy level. Here we adopt a typical dephasing rates
as $\Gamma_{1}=1$kHz, $\Gamma_{2}=4$kHz, $\Gamma_{3}=30$kHz and
$\gamma_{n}=\Gamma_{n}^{2}/\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}+\Gamma_{3}^{2}}$
(i.e., Rydberg states with $n=79$) Liu12021 . The parameters are the same with
the ones in Fig. 1. The fidelity is defined as
$F=\sqrt{tr(\rho_{ideal}\rho)}$, $\rho_{ideal}$ is the ideal density matrix.
As shown in Fig.4, the fidelity of NCGC will be decrease as gate time
increase. To attain a fidelity of over 0.99, the operation time of the
protocol should be less than 560ns while a fidelity of 0.999 require a gate
time less than 80ns. Given a Rabi frequency $\Omega$ and Rydberg interacting
strength $V$, gate time of NCGC can be shortened by tilting detuning
$\Delta_{a}$ in Eq.(9), with the weaken of robustness against the systematic
errors.

##  V Acceleration of noncyclic evolution

In the following we discuss the advantages of using noncyclic evolution.
According to the NCGC scheme, the adiabatic condition requires
$\dot{\varphi}\ll 2\Omega$, or equivalently, $\Delta\varphi\ll 2\Omega T$.
Given that $\Omega T=K\Delta\varphi/2$ (Usually, $K\gg 10$ to fulfill the
adiabatic condition, however, $K\sim 1$ after speeding up by the shortcut to
adiabaticity), $T=K\Delta\varphi/2/\Omega$. This results that the evolution
time $T$ will be shorten by $n$ times when the rotating angle $\Delta\varphi$
is reduced by $n$ times. Therefore, the NCGC scheme offer an analytical method
to speed up the control under small angles rotation, without the need of
increasing the Rabi frequencies (that has no conflict with the blockade
condition $V\gg\Omega$).

As a step further, the NCGC may have potential use in the algorithm of quantum
Fourier transformation (the key constitution in Shor’s prime number factoring
algorithm) of $N$ qubits, where $N$ single-qubit gate (Hadamard gate) and
$(L-1)L/2$ two-qubit controlled-phase gate $U_{p}$ with phase variation of
$\Delta\varphi_{L}=2\pi/2^{L}$, $L=1,2,...,N-1$, are needed. In Fig.5 we
calculate the total time that the quantum Fourier transformation cost and make
a comparison between the cyclic scheme and the noncyclic one. The controlled-
phase gate of noncyclic scheme is realized by the NCGC as described in Fig. 1.
Considering the $C_{z}$ gate and the single-qubit rotation gate
$R_{\Delta\varphi}$ form a universal set of quantum computation, the
controlled-phase gate $U_{p}$ of cyclic scheme is accomplished by three gates
as note

$U_{p}=R_{\Delta\varphi}U_{z}R_{\Delta\varphi},R_{\Delta\varphi}=\left(\begin{array}[]{cc}\cos\Delta\varphi&\sin\Delta\varphi\\\
\sin\Delta\varphi&-\cos\Delta\varphi\end{array}\right).$ (14)

Here we assume that the operation time of single-qubit gates (Hadamard gates
and the rotation gates) and the controlled-phase gate $U_{z}$ are all set to
be $T_{N}=250$ ns while the operation time of $U_{p}$ realized by NCGC is
decrease by $T_{N}/(2^{L})$ for $\Delta\varphi_{L}$. As shown in Fig.5, when
the number of qubit $N$ increases, the NCGC scheme (red-solid line) shows a
great advantage in comparison to cyclic scheme (blue-dashed line) as the
amount of two-qubit gates with small rotating angles increase polynomially.
Therefore, NCGC provides a much faster way to realize controlled phase gate
with small rotation angle, which is of great significance great significance
to realize quantum Fourier transformation in large scale of qubit.

##  VI Conclusion

In summary, we have proposed a scheme to realize two-qubit controlled-phase
gate with geometric phases in noncyclic evolution. The adiabatic control can
be achieved by modulating the phases with constant Rabi frequencies while the
acceleration can be tilting the detuning according to shortcut to
adiabaticity. The protocol shows a great robustness against the systematic
errors, random noise and decoherence. Furthermore, the protocol shows great
potentials in the algorithm (i.e., the quantum Fourier transformation) with
small rotating angles due to the noncyclic evolution. Therefore, our proposal
provide an experimentally feasible and robust way to realize quantum
computation in atomic arrays with Rydberg interactions.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grants No. 12074132, No. 12225405, No. 12247123, and No. U20A2074), NSF of
Guangdong province (Grant No. GDZX2303006), the China Postdoctoral Science
Foundation(Grant No. 2022M721222, No. 2023T160233). Z. Y. Chen and J. H. Liang
contribute equally to this work.

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