# Generation of complete graph states in a spin-$1/2$ Heisenberg chain with a
globally optimized magnetic field

X. X. Li  Sino-European Institute of Aviation Engineering, Civil Aviation
University of China, Tianjin 300300, China  D. X. Li  College of Physics
Science and Technology, Shenyang Normal University, Shenyang 110034, China  X.
Q. Shao<EMAIL_ADDRESS>Center for Quantum Sciences and School of
Physics, Northeast Normal University, Changchun 130024, China  Center for
Advanced Optoelectronic Functional Materials Research, and Key Laboratory for
UV Light-Emitting Materials and Technology of Ministry of Education, Northeast
Normal University, Changchun 130024, China

###### Abstract

Graph states possess significant practical value in measurement-based quantum
computation, with complete graph states that exhibit exceptional performance
in quantum metrology. In this work, we introduce a method for generating
multiparticle complete graph states using a spin-$1/2$ Heisenberg $XX$ chain
subjected to a time-varying magnetic field, which applies to a wide range of
systems. Our scheme relies exclusively on nearest-neighbor interactions
between atoms, with real-time magnetic field formation facilitated by quantum
optimal control theory. We focus specifically on neutral-atom systems, finding
that multiparticle complete graph states with $N=3\sim 6$ can be achieved in
less than $0.25~{}\mu{\rm s}$, using a hopping amplitude of
${J}/{(2\pi)}=-2.443~{}{\rm MHz}$. This assumes an initial state provided by
an equal-weight superposition of all spin states that are encoded by the
dipolar-interacting Rydberg states. Furthermore, we thoroughly address various
experimental imperfections and showcase the robustness of our approach against
atomic vibrations, fluctuations in pulse amplitude, and spontaneous emission
of Rydberg states. Taking into account the common occurrence of disturbances
in the experimental setups of neutral-atom systems, our one-step strategy to
achieve such graph states emerges as a viable alternative to techniques based
on controlled-Z gates.

##  I Introduction

Graph states, derived from mathematical graph theory [1, 2], play a central
role in quantum information theory. It describes a class of multiparticle-
entangled states including Greenberger-Horne-Zeilinger (GHZ) states and
cluster states [3]. Such states form a universal resource for measurement-
based quantum computing (MBQC) without controlled two-system quantum gates [4,
5, 6, 7, 8]. Some of these can also be widely applied in quantum teleportation
[9, 10, 11] and quantum secret sharing [12, 13]. Multiparticle complete graph
states represent that any two particles interact with each other, requiring
more projective measurements to be disentangled. When a single $\sigma_{y}$
measurement is performed at an arbitrary vertex (qubit), these states can be
converted to star graph states (GHZ type) [1, 14]. Such states exhibit
excellent performance in quantum metrology [15, 16], quantum error-correction
encoding [17], and quantum cryptography [18]. However, although complete graph
states have many advantages, less attention has been paid to preparing such
graph states experimentally, focusing mainly on linear and 2D cluster states
[19, 8, 20, 21].

The preparation of multiparticle complete graph states can be conceptually
divided into two primary categories. The first approach is applicable to
arbitrary graph states, employing the mathematically proven two-qubit
controlled-Z gate (CZ) [22, 23, 24]. However, these protocols are constrained
by the operational time and fidelity of the logic gate. On the contrary,
complete multiparticle graph states can also be generated within the framework
of cavity quantum electrodynamics [25, 26, 27]. In particular, certain
theoretical schemes [28, 28, 29] enable one-step realization, but these one-
step approaches require uniform interaction strengths between all particle
pairs, which poses challenges for experimental implementation. Recently, the
proposal of an equivalent Hamiltonian (Heisenberg $XXX$ model with a staggered
field) [30] has sparked interest in Floquet engineering for the construction
of such a Hamiltonian [31, 32]. Nevertheless, in practice, controlling the
impact of noncommutation of the Hamiltonian under periodic driving becomes
challenging as the particle number increases.

As quantum control theory progresses, optimal quantum control provides a
toolkit to craft external field shapes to perform assigned tasks, such as
state generation [33, 34, 35, 36], quantum gate implementation [37, 38], and
noisy intermediate-scale quantum technology [39]. These tools aim to achieve
the specified objectives with minimal energy and resource expenditure. The
practical realization of calculated field shapes has been facilitated by the
availability of commercially accessible arbitrary waveform generators. In this
paper, using the Gradient Ascent Pulse Engineering (GRAPE) optimization
algorithm [40, 41], we propose an alternative Hamiltonian represented as a
spin-$1/2$ Heisenberg $XX$ chain with identical nearest-neighbor interactions
under an optimized time-dependent magnetic field. Governed by this
Hamiltonian, we successfully generate complete graph states for a particle
number ranging from $N=3$ to $N=6$. Our protocol boasts two key advantages:
(i) It exclusively relies on the nearest-neighbor interactions, ensuring
robust operability and scalability in experimental setups. (ii) The
operational time is significantly reduced to less than $0.25~{}\mu{\rm s}$, a
crucial factor in mitigating the effects of decoherence.

The structure of this paper is organized as follows. In Sec. II, we initiate
our exploration with the introduction of the definition of the complete graph
state and an analysis of the corresponding scheme. Subsequently, in Sec. III,
we employ the GRAPE optimization algorithm to articulate a system-independent
realization of complete graph states in a “one-step” fashion. Transitioning to
Sec. IV, we delve into a specific physical system, the Rydberg atom array,
providing a comprehensive examination of the experiment’s feasibility. Our
results suggest that, taking into account experimental noise, the population
of the complete graph state with $N=6$ can exceed $0.91$ in a time frame of
$0.25~{}\mu{\rm s}$. Finally, Sec. V encapsulates our findings, offering a
concise summary.

##  II one-step Generation of complete graph states

Figure 1: (a) Multiparticle complete graph states with the number of vertices
$N=3\sim 6$. The graph connecting the two associated vertices has an edge
whenever two particles have interacted. (b) Experimental geometry. Particles
are arranged as a chain with the open boundary condition and have equivalent
interactions between the nearest particles.

Mathematically, a graph can be divided into vertices and edges, $G=\\{V,E\\}$.
Physically, such graphs can constitute the graph states, with the vertices
playing the role of physical systems, while the edges represent interactions.
As shown in Fig. 1(a), the multiparticle complete graph states represent that
there exists an interaction between two arbitrary qubits. Taking the vertices
as spin systems with $\\{|\uparrow\rangle,|\downarrow\rangle\\}$, the complete
graph states with $N$ vertices can be defined as

$|K_{N}\rangle=\frac{1}{2^{N/2}}\bigotimes_{i=1}^{N}[|\uparrow_{i}\rangle(-1)^{N-i}\prod_{j=i+1}^{N}\sigma_{z}^{j}+|\downarrow_{i}\rangle],$
(1)

with the convention $\sigma_{z}^{N+1}\equiv 1$. To generate such complete
graph states, we construct a Hamiltonian that holds a global time-dependent
magnetic field, which takes the form of ($\hbar=1$)

$H(t)=H_{\textrm{XX}}+\sum_{i=1}^{N}B(t)S^{z}_{i},$ (2)

where $S^{z}$ is the spin operator, and

$H_{\textrm{XX}}=\sum_{\langle
i,j\rangle}\frac{J}{2}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j})$
(3)

is a spin-$1/2$ Heisenberg $XX$ chain model with Pauli operators
$\sigma^{x(y,z)}$, where $\langle i,j\rangle$ indicates that the sum is taken
over all nearest neighbor pairs $(i,j)$ in the spin chain. Taking $N=3$ as an
example, the target state can be written as

$\displaystyle|K_{3}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}(|\uparrow\uparrow\uparrow\rangle+|\uparrow\uparrow\downarrow\rangle+|\uparrow\downarrow\uparrow\rangle-|\uparrow\downarrow\downarrow\rangle$
(4)
$\displaystyle+|\downarrow\uparrow\uparrow\rangle-|\downarrow\uparrow\downarrow\rangle-|\downarrow\downarrow\uparrow\rangle-|\downarrow\downarrow\downarrow\rangle),$

which presents as a superposition of eight states that can be separated into
two groups with opposite signs, i.e.
$\\{|\uparrow\uparrow\uparrow\rangle,|\uparrow\uparrow\downarrow\rangle,|\uparrow\downarrow\uparrow\rangle,|\downarrow\uparrow\uparrow\rangle\\}$
and
$\\{|\uparrow\downarrow\downarrow\rangle,|\downarrow\uparrow\downarrow\rangle,|\downarrow\downarrow\uparrow\rangle,|\downarrow\downarrow\downarrow\rangle\\}$.
$H_{\rm xx}$ described by Eq. (3) has the capability to alter the phase of the
state within the associated excitation subspace through flip-flop interactions
while ensuring the conservation of excitation numbers. Furthermore, the
introduction of a global magnetic field $\sum_{i=1}^{3}B(t)S^{z}_{i}$ allows
the accumulation of distinct phases in various excitation subspaces. The
synergistic integration of these two factors contributes to the realization of
a complete graph state of three particles.

Since the two parts ($H_{\textrm{XX}}$ and the global magnetic field term)
commute with each other, the evolution can take place under the Hamiltonian
(2) or under the $H_{\textrm{XX}}$ and the global magnetic field separately.
Theoretically, any equivalent $H_{\textrm{XX}}$ model can be applied to this
scenario. Under a global magnetic field with constant amplitude, an exact
analytical solution of $B$ can be found for $N=3$, as shown in Appendix A.
However, for $N>3$, such an exact solution is non-existent. Thus, we consider
a time-dependent magnetic field $B(t)$ and introduce the GRAPE optimization
algorithm to find the optimal solution for the system. In this way, we can
expect to obtain a multiparticle universal protocol.

Figure 2: The realization of the multiparticle complete graph states with
$N=3\sim 6$ governed by Eq. (2) initialized from state
$|\Psi_{0}\rangle=\bigotimes_{i=1}^{N}[{1}/{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)]$.
(a) and (b) respectively show evolution under the magnetic field in random and
Gaussian types for $N=3$. The dotted lines correspond to the evolution under
the guess field, while the solid lines correspond to the optimized one. Then
(c) and (d) show the evolution for $N=4$, while (e) and (f) for $N=5$, and (g)
and (h) for $N=6$. The insets show the corresponding time-dependent $B(t)$ of
the initial guess and the optimized result.

##  III The utilization of the GRAPE optimization algorithm

Given the known form of the target state $|K_{N}\rangle$, the optimization
problem, which is implemented using the gradient descent algorithm, is
translated into the search for the optimal amplitudes $B(t)$ of the magnetic
field. These amplitudes aim to guide the initial state $\rho(0)$ to $\rho(T)$
with maximum similarity to $\rho_{K}=|K_{N}\rangle\langle K_{N}|$ within a
specified time duration $T$. In our protocol, the system Hamiltonian is
segmented into two distinct parts.

$H(t)=H_{\textrm{XX}}+B(t)H_{z},$ (5)

where $H_{z}=\sum_{i=1}^{N}S^{z}_{i}$ is referred to as the control component.
The quantum control landscape, depicting the expectation value of a Hermitian
observable operator $\rho_{K}$ at a given time $T$, is represented as [42]

$\Phi[B(\cdot)]=\textrm{Tr}[\rho(0)\rho_{K}(T)],$ (6)

where $\rho_{K}(T)=U^{{\dagger}}(T,0)\rho_{K}U(T,0)$. The variation of the
observable operator $\rho_{K}$ is expressed as

$\delta\rho_{K}(t)=\int_{0}^{t}[\rho_{K}(t^{\prime}),H_{z}(t^{\prime})]\delta
B(t^{\prime})dt^{\prime},$ (7)

where $H_{z}(t^{\prime})=-iU^{{\dagger}}(t^{\prime},0)H_{z}U(t^{\prime},0)$.
This results in the fundamental equation

$\frac{\delta\rho_{K}(T)}{\delta
B(t^{\prime})}=[\rho_{K}(T),H_{z}(t^{\prime})],~{}~{}\forall t^{\prime}<T.$
(8)

Consequently, the corresponding gradient is given by

$g[B(t)]=\frac{\partial\Phi[B(\cdot)]}{\partial
B(t^{\prime})}=\textrm{Tr}[[\rho(0),\rho_{K}(T)]H_{z}(t^{\prime})].$ (9)

Therefore, the performance function $\Phi[B(\cdot)]$ can be enhanced by
selecting

$B(k)\rightarrow B(k)+\alpha g[B(t)],$ (10)

where $\alpha$ denotes the learning rate, subject to variation based on the
number of iterations $k$.

Guided by Eq. (2), the determination of $B(t)$ involves considering two forms
of guess fields. One takes on a Gaussian profile, defined as

$B_{\textrm{G}}=\frac{B_{0}}{\sqrt{2\pi}\sigma}e^{-\frac{t_{g}^{2}}{2\sigma^{2}}},$
(11)

with $B_{0}=J$, $\sigma=0.1$, and $t_{g}=[-0.5,0.5]$, while the other
manifests as a random sequence comprising random numbers $\xi$ within the
range $[0,1]$ noted as $B_{\rm G}=B_{0}\xi$. The guess fields in the GRAPE
optimal algorithm consist of discrete points with a time interval of $\delta
t=T/n$, where $n$ takes values of $\\{100,10\\}$ for Gaussian and random
types, respectively. Fig. 2 illustrates the optimized results for $N={3\sim
6}$. Within the GRAPE optimal algorithm, the maximum population of the target
state is achievable after optimization, irrespective of the shape of the guess
field, once the evolution time is specified. The corresponding evolution time
$T$ and the optimized population $P(N)_{O}$ for varying particle numbers are
presented in Table 1.

Table 1: The evolution time and the populations of the multiparticle complete graph states governed by Eq. (2). Target state |  Evolution time (units of $J$) |  Population
---|---|---
$|K_{3}\rangle$ |  2.3 |  1
$|K_{4}\rangle$ |  2.808 |  0.9931
$|K_{5}\rangle$ |  3.386 |  0.9710
$|K_{6}\rangle$ |  3.952 |  0.9346

The results demonstrate the successful generation of multiparticle complete
graph states under the influence of a spin-$1/2$ Heisenberg $XX$ chain in a
time-varying magnetic field optimized by the GRAPE algorithm. Consequently,
the preparation of multiparticle complete graph states can be achieved
mathematically in a “one-step” manner through Eq. (2). This protocol is
universally applicable in systems capable of nearest-neighbor interactions
characteristic of the spin-$1/2$ Heisenberg $XX$ chain, including
superconducting quantum circuits [43, 44, 45, 46] and ion traps [47, 48, 49,
50, 51]. In the following part, we will delve into the detailed methodology
for preparing the aforementioned multiparticle complete graph states within a
neutral-atom system, closely considering experimental parameters.

##  IV A specific implementation based on Rydberg atom array

Figure 3: The realization of the multiparticle complete graph states with
$N=3\sim 6$ governed by the Hamiltonian $H_{\textrm{sys}}$ initialized from
state
$|\Psi_{0}\rangle=\bigotimes_{i=1}^{N}[{1}/{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)]$.
(a) and (b), respectively, show evolution under the magnetic field in Gaussian
and random types for $N=3$. The dotted lines correspond to the evolution under
the guess field, whereas the solid lines correspond to the optimized one. Then
(c) and (d) show the evolution for $N=4$, while (e) and (f) for $N=5$, (g) and
(h) for $N=6$. The insets show the corresponding time-dependent $B(t)$ of the
initial guess and the optimized result, where $B_{0}=2\pi\times 1~{}{\rm
MHz}$.

The Rydberg atom array, characterized by robust programmability, has recently
emerged as a powerful candidate in quantum simulations, spurred by
breakthroughs in experimental technology [52, 53]. Individual Rydberg atoms,
for example, can be placed in an optical tweezers array [54, 55]. Leveraging
the long-range interactions inherent in highly excited Rydberg states, the
Rydberg atom array finds widespread applications in quantum state transmission
[56], quantum topology research [57, 58, 59], and artificial gauge field [60,
61]. Notably, recent works, such as those in Refs. [62, 63], have realized a
primary quantum computer employing 256 Rydberg atoms, demonstrating
proficiency in various quantum tasks.

Within the Rydberg atom array, the Rydberg dipole-dipole interaction serves as
a direct means to achieve $H_{\textrm{XX}}$ [64, 65]. Additionally, the
magnetic field $B(t)$ can be equivalently obtained through the Stark effect
induced by a large detuned time-dependent laser field coupled transition
between the intermediate state $|p\rangle$ and one of the Rydberg states [66,
67]. It is crucial to note that the inevitable long-range interactions among
Rydberg atoms, such as van der Waals (vdW) forces and the long-range dipolar
interactions, should also be taken into consideration, with respect to the
realistic physical system.

###  IV.1 The realization of multiparticle complete graph states

We consider a one-dimensional chain geometry with open periodic boundary
conditions, as shown in Fig. 1(b), in which $N$ atoms are contained in
tweezers that are equidistantly spaced $R$ along the quantized $z$ axis. For
the purpose of illustration, states with opposite parity are transformed into
pseudo-spin states, using the ${}^{87}\textrm{Rb}$ atom as an example,
$|\uparrow\rangle=|80S_{1/2},m_{j}=1/2\rangle$ and
$|\downarrow\rangle=|79P_{3/2},m_{j}=1/2\rangle$ [68]. The resonant dipole-
dipole interaction gives rise to the Heisenberg $XX$ Hamiltonian

$H_{\textrm{XX}}=\sum_{\langle
i,j\rangle}\frac{V_{\textrm{dip}}(\theta)}{2}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j}),$
(12)

where $V_{\textrm{dip}}(\theta)=C_{3}(1-3\cos^{2}\theta)/R^{3}$ with $\theta$
the polar angle between the quantization $z$ axis and the vector direction of
atomic connection, and the constant $C_{3}$ is defined as $2\pi\times
8.780~{}{\rm GHz}\cdot\mu{\rm m}^{3}$ [69]. For $\theta=0$ and
$R=19.3~{}\mu{\rm m}$, we have $V_{\textrm{dip}}\simeq-2\pi\times 2.443~{}{\rm
MHz}$. Additionally, we account for inevitable interactions, including long-
range vdW interactions and dipolar interactions, which are characterized as
error terms.

$\displaystyle H_{\textrm{err}}$ $\displaystyle=$
$\displaystyle\sum_{j>i}(U^{\uparrow}_{ij}|\uparrow_{i}\uparrow_{j}\rangle\langle\uparrow_{i}\uparrow_{j}|+U^{\downarrow}_{ij}|\downarrow_{i}\downarrow_{j}\rangle\langle\downarrow_{i}\downarrow_{j}|)$
(13)
$\displaystyle+\sum_{j>i+1}\frac{V_{\textrm{dip}}^{ij}}{2}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j}),$

where $U^{\uparrow(\downarrow)}_{ij}=-C^{\uparrow(\downarrow)}_{6}/R_{ij}^{6}$
with $C^{\uparrow}_{6}=-2\pi\times 4161.55~{}{\rm GHz}\cdot\mu{\rm m}^{6}$ and
$C^{\downarrow}_{6}=2\pi\times 3452.60~{}{\rm GHz}\cdot\mu{\rm m}^{6}$, and
$V_{\rm dip}^{ij}=C_{3}(1-3\cos^{2}\theta)/R_{ij}^{3}$. $R_{ij}$ denotes the
distance between the $i$th and $j$th atom.

$H_{\rm sys}=H_{\rm XX}+H_{\rm err}+\sum_{i=1}^{N}B(t)S_{i}^{z}.$ (14)

Using $N=3\sim 6$ as illustrative examples, we depict the evolutions governed
by Eq. (14). In Fig. 3, we specifically show the evolutions under the initial
guess and optimized fields in Gaussian and random types. Within this
framework, the populations in the target state, denoted as $P(N)_{O}$, exhibit
multiple peak points corresponding to different values of $T$ (as shown in the
picture in Appendix B). By scanning the time within $0.25~{}\mu{\rm s}$, we
successfully obtain the optimized fields under $H_{\textrm{sys}}$ for various
numbers of particles. The inset shows the time-dependent $B(t)$ of the
application, where solid lines represent the optimized pulse, and dotted lines
represent the guess pulse. The evolution time $T$ and the populations of the
target states are summarized in Table 2. Although the population of the target
state decreases as $N$ increases, we can still prepare the six-particle
complete state with a population greater than $0.92$.

Table 2: The evolution time and the populations of the multiparticle complete graph states corresponding to the system constructed by ${}^{87}{\rm Rb}$ atom array arranged in a line with polar angle $\theta=0$. Target state |  Evolution time ($\mu{\rm s}$) |  Population
---|---|---
$|K_{3}\rangle$ |  0.141 |  0.9989
$|K_{4}\rangle$ |  0.172 |  0.9920
$|K_{5}\rangle$ |  0.203 |  0.9728
$|K_{6}\rangle$ |  0.233 |  0.9294

Recently, a quantum circuit utilizing the Rydberg atom array has been
demonstrated to generate a complete graph state of five particles [70]. When
coupled with quantum annealing algorithms, the fidelity of the target state
can be increased from approximately $0.85$ to $0.999$ by increasing
computational depth. However, this increased computational depth is
accompanied by prolonged execution times, leading to an increased
susceptibility to decoherence effects. On the contrary, our protocol enables
the generation of the target state in a single step, regardless of the number
of atoms involved. This approach significantly reduces operational time while
preserving high levels of fidelity, which presents notable advantages in
mitigating decoherence effects.

###  IV.2 Experimental feasibility analysis

This part focuses on experimental feasibility by examining the experimental
imperfections and providing a concise design for an experiment procedure using
the Rydberg atom array. This includes the production of the initial state and
the final decoupling process.

####  IV.2.1 The influence of experimental errors

Here, we rigorously assess the scheme by taking into account experimental
imperfections, such as spontaneous radiation of Rydberg levels, as well as the
discrepancy between the dipole-dipole interaction $V_{\textrm{dip}}$ and the
magnetic field $B(t)$. The associated numerical simulations employ Gaussian-
type optimal pulses.

(i) Examining the impact of spontaneous emission of Rydberg states. To analyze
the influence of spontaneous radiation, we introduce an empty ground state
$|g\rangle$, under the assumption that all Rydberg states will spontaneously
radiate to this unoccupied state. At a temperature of $0.1~{}\rm K$, the
lifetimes of the Rydberg states $|80S_{1/2}\rangle$ and $|79P_{3/2}\rangle$
are $\tau_{\uparrow,\downarrow}=0.569,1.1~{}{\rm ms}$, respectively. Thus, the
corresponding master equation can be described as

$\displaystyle\mathcal{L}[\rho]$ $\displaystyle=$ $\displaystyle-i[H_{\rm
sys},\rho]+\sum_{i=1,j=\uparrow,\downarrow}^{N}\frac{\gamma_{j}}{2}(2s_{i}\rho
s_{i}^{+}-s_{i}^{+}s_{i}\rho$ (15) $\displaystyle-\rho s_{i}^{+}s_{i}),$

where $\gamma_{\uparrow(\downarrow)}$ is the decay rate of the Rydberg state
$|\uparrow(\downarrow)\rangle$ equals to $1/\tau_{\uparrow(\downarrow)}$, and
$s_{i}=|g\rangle_{i}\langle j|$ denotes the decay channel of the $i$th atom.
Through numerical simulation, dissipation is observed to lead to a reduction
in the population of the target state by values of
$\\{0.0006,0.0009,0.0010,0.0017\\}$, corresponding to $N$ ranging from three
to six, respectively.

(ii) Addressing the impact of the mismatch between the dipole-dipole
interaction $V_{\textrm{dip}}$ and the magnetic field $B(t)$. This discrepancy
may arise from sources: the instability in the distance between atoms and the
fluctuation of the magnetic field.

Figure 4: The influence of the mismatch of the dipole interaction
$V_{\textrm{dip}}$ and the magnetic field $B(t)$ caused by atomic vibration.
(a) Populations of the complete multiparticle graph states with range error
$\delta r$ under the Hamiltonian $H_{\textrm{sys}}(\delta r)$. (b) and (c)
Average evolution results for $N=4,6$ considering a true fluctuation where the
position distribution is Gaussian with variance
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}\simeq\\{193.5,193.5,1242.9\\}~{}{\rm
nm}$. Note that the light-red region shows the results of 50 stochastic
simulations, and the dashed lines correspond to the average results.

Figure 5: The influence of the mismatch of $V_{\textrm{dip}}$ and $B(t)$
caused by the fluctuation of the magnetic field. (a)-(d), respectively, show
the average evolution results (dashed line in blue) governed by Eq. (18). Note
that the light-red region shows the results of 50 stochastic simulations,
while the inner pictures present the magnetic field of one of the experiments.

To investigate the first, we introduce $V_{\textrm{dip}}(\delta
r)=-2C_{3}/(R+\delta r)^{3}$ with a range error $\delta r\in[-300,300]~{}{\rm
nm}$ for a preliminary estimate. The Hamiltonian $H(t)$ can be rewritten as

$H_{\delta V}(t)=\sum_{\langle i,j\rangle}\frac{V_{\textrm{dip}}(\delta
r)}{2}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j})+\sum_{i=1}^{N}B(t)S^{z}_{i}.$
(16)

It should be noted that as the vdW interaction also depends on the distance
between the atoms, the Hamiltonian $H_{\textrm{err}}$ should likewise be
formulated as

$\displaystyle H^{\textrm{err}}_{\delta V}$ $\displaystyle=$
$\displaystyle\sum_{j>i}[U^{\uparrow}_{ij}(\delta
r)|\uparrow_{i}\uparrow_{j}\rangle\langle\uparrow_{i}\uparrow_{j}|+U^{\downarrow}_{ij}(\delta
r)|\downarrow_{i}\downarrow_{j}\rangle\langle\downarrow_{i}\downarrow_{j}|]$
(17) $\displaystyle+\sum_{j>i+1}\frac{V_{\textrm{dip}}^{ij}(\delta
r)}{2}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j}),$

where $U^{\uparrow(\downarrow)}(\delta
r)=-C^{\uparrow(\downarrow)}_{6}/(R_{ij}+\delta r)^{6}$, and $V_{\rm
dip}^{ij}=C_{3}(1-3\cos^{2}\theta)/(R_{ij}+\delta r)^{3}$. In Fig. 4(a), we
present the evolution results governed by $H_{\textrm{sys}}(\delta
r)=H_{\delta V}(t)+H^{\textrm{err}}_{\delta V}$, with $N$ varying from three
to six, respectively. The population of target states $P(N)_{O}$ consistently
exceeds 0.9 for $\delta r\leq 300~{}{\rm nm}$. In experimental settings, the
vibration of atoms near the ideal position, induced by nonzero temperature,
follows the Maxwell-Boltzmann distribution and is associated with the atomic
temperature as well as the parameters of the tweezer beam and trap. According
to Ref. [71], the atomic temperature can be cooled to $5.2~{}\mu{\rm K}$ in a
$50~{}\mu{\rm K}$ trap. Setting the corresponding parameters of the laser
beams as wavelength $\lambda_{f}=830~{}{\rm nm}$, typical beam power
$P_{f}=174~{}\mu{\rm W}$, and waist ($1/e^{2}$ intensity radius)
$\omega_{f}=1.2~{}\mu{\rm m}$, we estimate a position distribution in Gaussian
with variance
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}\simeq\\{193.5,193.5,1242.9\\}~{}{\rm
nm}$. Thus, in Fig. 4(b) and Fig. 4(c), we further assess the robustness of
the protocol with $N=4$ and $N=6$ as realistically as possible, considering a
three-dimensional random vibration under these estimated parameters. The
distance between the atoms $R$ is redefined as
$D=|\textbf{R}_{i}-\textbf{R}_{i+1}|$, where
$\textbf{R}_{i}=(x_{i},y_{i},z_{i})$. Since the evolution time $T$ is less
than $0.25~{}\mu{\rm s}$, we reasonably consider only one group of random
fluctuations ($\sigma_{x,y,z}$) error throughout the evolution process here.
Fig. 4(b) and 4(c) respectively illustrate the population of the target states
$|K_{4}\rangle$ and $|K_{6}\rangle$, where the light red portions represent
the results of 50 stochastic simulations, and the dashed lines in dark blue
correspond to the average outcomes. The average populations of $|K_{4}\rangle$
and $|K_{6}\rangle$ in this test are $0.9728$ and $0.9187$, respectively. With
advancements in experimental methods, the actual vibration of the atoms in the
trap is expected to be smaller than our estimated parameters [72, 73], which
is more conducive to the realization of our scheme.

For the latter reason, the fluctuation of the magnetic field intensity is
introduced by a time-dependent fluctuation $\delta B$. It is assumed to follow
normal distribution functions with standard deviations $\sigma(t)\simeq
2\pi\times 0.5~{}{\rm MHz}$. Thus, we have the system Hamiltonian

$H_{\delta B}(t)=H_{\rm XX}+H_{\rm err}+\sum_{i=1}^{N}[B(t)+\delta
B]S^{z}_{i}.$ (18)

As shown in Fig. 5, we present the evolution results governed by the
Hamiltonian $H_{\delta B}(t)$ averaged over 50 realizations. It is evident
that the fluctuation in the global magnetic field has a negligible impact on
our scheme.

####  IV.2.2 Mapping between Rydberg states and ground states

Figure 6: The decoupling process (dashed lines) and the relevant levels of
${}^{87}{\rm Rb}$ atom. The ground states can be chosen as $|5S_{1/2}\rangle$
hyperfine clock states, such as $|0\rangle\equiv|F=1,m_{F}=0\rangle$,
$|1\rangle\equiv|F=2,m_{F}=0\rangle$, while the auxiliary Rydberg states can
be chosen as $|r\rangle=|78S_{1/2},m_{j}=1/2\rangle$.

For the neutral-atom system, ensuring the stability of the target state
typically involves encoding it in the ground states $|0\rangle$ and
$|1\rangle$, which are the hyperfine clock states of $|5S_{1/2}\rangle$.
Moreover, given the natural existence of dipole-dipole interactions between
Rydberg states, the scheme should be implemented in three steps, that is, the
initial state preparation, the core evolution governed by $H_{\textrm{sys}}$,
and the decoupling process. In this section, we present separate feasibility
proposals for the initial state preparation and decoupling process.

(i) Preparation of the initial state. In this protocol, the initial state is
selected as

$|\Psi_{N}(0)\rangle=\bigotimes_{i=1}^{3}[\frac{1}{\sqrt{2}}(|\uparrow_{i}\rangle+|\downarrow_{i}\rangle)].$
(19)

To excite multiple atoms to Rydberg states without being constrained by the
Rydberg blockade, the process can be initiated by driving the ground state
$|0\rangle\rightarrow|\uparrow\rangle$ through a two-photon resonance with an
effective Rabi frequency $\Omega$ and an evolution time $t=\pi/\Omega$.
Subsequently, the transition from $|\uparrow\rangle$ to $|\downarrow\rangle$
is achieved sequentially using a microwave pulse with Rabi frequency
$\Omega_{\textrm{mw}}^{a}$ and duration
$t_{\textrm{mw}}=\pi/(2\Omega_{\textrm{mw}}^{a})$. This technique to prepare
the initial state is well established in quantum simulations [74, 32, 57].

(ii) Dynamical decoupling process: As dipole-dipole interactions naturally
persist, the system evolution under the Hamiltonian $H_{\textrm{XX}}$ cannot
be voluntarily stopped by controlling external electric or magnetic fields. To
address this challenge, we select an auxiliary Rydberg level $|r\rangle$
characterized by non-dipole-dipole interactions, such as
$|78S_{1/2},m_{j}=1/2\rangle$. Illustrated by the dashed line in Fig. 6, the
dynamics can be decoupled by driving atoms in states $|\downarrow\rangle$ to
this auxiliary Rydberg level via a microwave pulse $\Omega_{\textrm{mw}}^{b}$.
Given that the Rabi frequency of the microwave pulse can reach $200~{}{\rm
MHz}$ [31], this process is rapid. By combining this step with the two-photon
resonance process with an effective Rabi frequency $\Omega$, we can further
drive the Rydberg levels to hyperfine clock states with an evolution time
$t=\pi/\Omega$, thus obtaining complete graph states encoded by ground states
of the ${}^{87}{\rm Rb}$ atom.

Figure 7: A complete evolution via the resonant dipole-dipole interaction for
$N=3$ with initial state $|000\rangle$. The dashed and dotted lines
respectively correspond to the initial state preparation and the decoupling
process, and the solid line presents the core evolution with the optimized
$B(t)$ in Gaussian type. The parameters are set as $\Omega=2\pi\times 4~{}{\rm
MHz}$, $\Omega_{\rm mw}^{a}=2\pi\times 70~{}{\rm MHz}$, and
$\Omega_{\textrm{mw}}^{b}=2\pi\times 200~{}{\rm MHz}$.

Fig. 7 illustrates a complete evolution with $N=3$. To enhance observability,
we depict the evolution of the target states for each process, where
$|\psi_{1}\rangle=\bigotimes_{i=1}^{3}[{1}/{\sqrt{2}}(|\uparrow_{i}\rangle-i|\downarrow_{i}\rangle)]$,
$|\psi_{2}\rangle={1}/{(2\sqrt{2})}\bigotimes_{i=1}^{3}[|\uparrow_{i}\rangle(-1)^{3-i}\prod_{j=i+1}^{3}\sigma_{z}^{j}-i|\downarrow_{i}\rangle]$,
$|\psi_{3}\rangle={1}/{(2\sqrt{2})}\bigotimes_{i=1}^{3}[|\uparrow_{i}\rangle(-1)^{3-i}\prod_{j=i+1}\sigma_{z}^{j}-|r_{i}\rangle]$,
and
$|\psi_{4}\rangle={1}/{(2\sqrt{2})}\bigotimes_{i=1}^{3}[-|0_{i}\rangle(-1)^{3-i}\prod_{j=i+1}\sigma_{z}^{j}+|1_{i}\rangle]$.
The dashed and dotted lines correspond to the initial state preparation and
the decoupling process, respectively, while the solid line represents the core
evolution with the optimized $B(t)$ in Gaussian type. Under the parameters
$\Omega=2\pi\times 4~{}{\rm MHz}$, $\Omega_{\rm mw}^{a}=2\pi\times 70~{}{\rm
MHz}$ and $\Omega_{\textrm{mw}}^{b}=2\pi\times 200~{}{\rm MHz}$, the
population of the target state can reach about $0.9916$ with a total evolution
time $t_{\textrm{tot}}=0.3971~{}\mu{\rm s}$. Given that the independent
evolution process of the core operation can reach perfection with $N=3$, the
error caused by the preparation of the initial state and the return driving is
approximately $0.0074$.

It should be noted that this three-step scheme is specifically designed for
the Rydberg atom array system. In other systems, such as superconducting
systems, the mapping process mentioned here may no longer be necessary.

Table 3: Error estimation of population for the multiparticle complete graph states with $N=3\sim 6$ corresponding to the system constructed by Rydberg atom array. Error budget | $|K_{3}\rangle$ | $|K_{4}\rangle$ | $|K_{5}\rangle$ | $|K_{6}\rangle$
---|---|---|---|---
Initial preparing and decoupling | 0.0074
Spontaneous radiation | 0.0006 | 0.0009 | 0.0010 | 0.0017
Atomic vibration ($\pm 100~{}{\rm nm}$) | 0.0009 | 0.0016 | 0.0020 | 0.0038
$P(N)_{O}$ | 0.99 | 0.9821 | 0.9624 | 0.9165

As shown in Table. 3, we make an estimation on the population of complete
states with $N=3\sim 6$ considering the experimental errors to make a
comparison with the other protocol. In recent experiments, the fidelity of the
two-qubit entanglement gate has been improved to $99.5\%$ [75, 76]. Based on
this analysis, we can estimate the time and fidelity necessary to achieve
complete graph states using CZ gates for comparison. For $N=3$, a minimum of
three CZ gates are required, with an estimated fidelity of approximately
$0.985$. For $N=6$, 15 CZ gates are needed, with an estimated fidelity of
$0.928$, demonstrating performance comparable to our approach. However, for
the gate-based approach, although a single gate operation time can be less
than $300~{}{\rm ns}$, such CZ gate operations cannot be performed
simultaneously. The evolution time will increase rapidly with increasing
number of particles $N$. On the contrary, the evolution time required by our
protocol is shorter, which can avoid some unknown errors. This reduction is
crucial from the perspective of decoherence.

##  V summary

In summary, we have introduced an efficient single-step technique for
generating multiparticle complete graph states. Our approach relies solely on
a constant nearest-neighbor interaction strength and a time-varying magnetic
field which is optimized using the GRAPE algorithm. This methodology is
applicable to various physical systems capable of implementing a spin-$1/2$
Heisenberg $XX$ chain, such as superconducting systems, trapped ion systems,
and neutral atom systems. We have thoroughly investigated the protocol’s
execution in the Rydberg atom array system, evaluating its resilience in the
face of experimental errors. Our numerical findings underscore the robustness
of our approach against challenges such as atomic position inaccuracies and
pulse oscillations. Our one-step solution significantly reduces operating time
while maintaining a relatively high estimated fidelity. We anticipate that
this discovery will quickly contribute to the practical implementation of
quantum computing and quantum error correction in neutral-atom systems.

Figure 8: Evolution governed by the Hamiltonian (20) with $N=3$. (a) The
intensity of the constant amplitude pulse $B$ satisfies the relationship shown
in Eq. (31). (b) Scanning results for the population of the target state with
different parameters.

Figure 9: The multiple peaks corresponding to different $T$. (a) The optimized
magnetic fields $B(t)$ under different evolution time $T$. (b) The
corresponding evolution governed by $H_{\rm sys}$ with $N=3$.

## acknowledgment

The authors thank Professor Re-Bing Wu of Tsinghua University for his guidance
on the GRAPE optimal algorithm. The anonymous reviewers are also thanked for
constructive comments that helped improve the quality of this article. This
work is supported by the NFSC through Grant No. 12174048, the Fundamental
Research Funds for the Central Universities (Grant No. 3122023QD26) and the
Research Start-up Funds of the Shenyang Normal University (Grants No.
BS202301, No. BS202303, No. BS202314).

##  Appendix A A protocol for $N=3$ under constant amplitude magnetic field

In the three-particle case, the preparation of the complete graph state is
particularly unique. Governed by the Hamiltonian with a time-independent
global magnetic field, represented as

$H_{\textrm{con}}=\sum_{\langle
i,j\rangle}J_{ij}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j})+\sum_{i=1}^{3}BS^{z}_{i},$
(20)

the system has definite solutions to realize the corresponding complete state.
According to Schrödinger equation,

$i\frac{d|\Psi(t)\rangle}{dt}=H|\Psi(t)\rangle,$ (21)

we have

$\displaystyle|\Psi(t)\rangle$ $\displaystyle=$
$\displaystyle\psi_{1}(t)|\downarrow\downarrow\downarrow\rangle+\psi_{2}(t)|\downarrow\downarrow\uparrow\rangle+\psi_{3}(t)|\downarrow\uparrow\downarrow\rangle$
(22)
$\displaystyle+\psi_{4}(t)|\downarrow\uparrow\uparrow\rangle+\psi_{5}(t)|\uparrow\downarrow\downarrow\rangle+\psi_{6}(t)|\uparrow\downarrow\uparrow\rangle$
$\displaystyle+\psi_{7}(t)|\uparrow\uparrow\downarrow\rangle+\psi_{8}(t)|\uparrow\uparrow\uparrow\rangle,$

where

$\psi_{1}(t)=\frac{e^{-\frac{3itB}{2}}}{2\sqrt{2}},$ (23)
$\psi_{2}(t)=\frac{e^{-i\frac{1}{2}t(4\sqrt{2}J+B)}[2+\sqrt{2}-(\sqrt{2}-2)e^{4i\sqrt{2}Jt}]}{8\sqrt{2}},$
(24)
$\psi_{3}(t)=\frac{e^{-i\frac{1}{2}t(4\sqrt{2}J+B)}[1+\sqrt{2}-(\sqrt{2}-1)e^{4i\sqrt{2}Jt}]}{4\sqrt{2}},$
(25)
$\psi_{4}(t)=\frac{1}{4}e^{\frac{itB}{2}}(\sqrt{2}\cos{\frac{4Jt}{\sqrt{2}}}-i\sin{\frac{4Jt}{\sqrt{2}}}),$
(26)
$\psi_{5}(t)=\frac{e^{-i\frac{1}{2}t(4\sqrt{2}J+B)}[2+\sqrt{2}-(\sqrt{2}-2)e^{4i\sqrt{2}Jt}]}{8\sqrt{2}},$
(27)
$\psi_{6}(t)=\frac{1}{4}e^{\frac{itB}{2}}(\sqrt{2}\cos{\frac{4Jt}{\sqrt{2}}}-2i\sin{\frac{4Jt}{\sqrt{2}}}),$
(28)
$\psi_{7}(t)=\frac{1}{4}e^{\frac{itB}{2}}(\sqrt{2}\cos{\frac{4Jt}{\sqrt{2}}}-i\sin{\frac{4Jt}{\sqrt{2}}}),$
(29) $\psi_{8}(t)=\frac{e^{\frac{3itB}{2}}}{2\sqrt{2}},$ (30)

with assumption of $J_{ij}=J$ is a positive real number. By solving the
equations, we find that when

$B=\frac{4J(1-4C_{1})}{\sqrt{2}(2C_{1}-2C_{2}-1)},~{}~{}C_{2}\geq C_{1}\in Z,$
(31)

the triatomic complete graph state with a global phase $-i$ can be realized at
$t=\sqrt{2}\pi[1+2(C_{2}-C_{1})]/(4J)$, as shown Fig. 8(a). However, this is
just a particular set of solutions. When considering other global phases,
there may exist more solutions. As shown in Fig. 8(b), it is obtained by
scanning the population of the target state under different parameters. We can
find that three of the peaks correspond exactly to our analytical results,
while more peaks also exist. However, in the case of a system with more than
three qubits, such a solution is non-existent, and the application of the
time-dependent magnetic field with quantum optimal control is necessary.

##  Appendix B The multi-selectivity of evolution time

Within this protocol, populations in the target state exhibit multiple peaks
corresponding to different values of $T$. Taking $N=3$ as an example, Fig. 9
presents the evolution results with $T=\\{0.141,0.42,0.696\\}~{}\mu{\rm s}$,
respectively. We can find that under these three evolution times, the
population of the target state will reach the peak which corresponds to three
good optimization results. As we know, a shorter evolution time is more
beneficial for the suppression of atomic decoherence. Thus, after
comprehensively considering the evolution time, the population of the target
state, and the strength of the magnetic field, we choose the parameters of the
first peak for simulation in the main text.

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