# Improving the performance of quantum approximate optimization for preparing
non-trivial quantum states without translational symmetry

Zheng-Hang Sun  Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100190, China  Yong-Yi Wang<EMAIL_ADDRESS>Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190,
China  Jian Cui<EMAIL_ADDRESS>School of Physics, Beihang University,
Beijing 100191, China  Heng Fan<EMAIL_ADDRESS>Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100190, China  Songshan
Lake Materials Laboratory, Dongguan 523808, Guangdong, China  CAS Center of
Excellence for Topological Quantum Computation, University of Chinese Academy
of Sciences, Beijing 100190, China  Beijing Academy of Quantum Information
Sciences, Beijing 100193, China

###### Abstract

The variational preparation of complex quantum states using the quantum
approximate optimization algorithm (QAOA) is of fundamental interest, and
becomes a promising application of quantum computers. Here, we systematically
study the performance of QAOA for preparing ground states of target
Hamiltonians near the critical points of their quantum phase transitions, and
generating Greenberger-Horne-Zeilinger (GHZ) states. We reveal that the
performance of QAOA is related to the translational invariance of the target
Hamiltonian: Without the translational symmetry, for instance due to the open
boundary condition (OBC) or randomness in the system, the QAOA becomes less
efficient. We then propose a generalized QAOA assisted by the parameterized
resource Hamiltonian (PRH-QAOA), to achieve a better performance. In addition,
based on the PRH-QAOA, we design a low-depth quantum circuit beyond one-
dimensional geometry, to generate GHZ states with perfect fidelity. The
experimental realization of the proposed scheme for generating GHZ states on
Rydberg-dressed atoms is discussed. Our work paves the way for performing QAOA
on programmable quantum processors without translational symmetry, especially
for recently developed two-dimensional quantum processors with OBC.

###### pacs:

Valid PACS appear here

##  I Introduction

With recent experimental developments in the controllability of noisy
intermediate scale quantum (NISQ) computers Preskill (2018); Alexeev  _et al._
(2021), more and more attention has been paid to the variational quantum
simulation (VQS) Cerezo  _et al._ (2021); Kokail  _et al._ (2019); Li and
Benjamin (2017); Carleo and Troyer (2017); Bolens and Heyl (2021); Bharti  _et
al._ (2022). One primary task of the VQS is estimating the ground-state energy
and preparing the ground state of many-body quantum systems, as target
Hamiltonians, via hybrid quantum-classical variational algorithms, which is
known as variational quantum eigensolvers (VQEs). A central ingredient of VQEs
is the design of parametrized quantum circuits. For instance, a type of VQE,
which is based on the hardware efficient ansatz comprised of layers of single-
qubit rotation gates and blocks of non-parametrized entangling gates, has been
applied to solve the ground-state energy of molecules Kandala  _et al._
(2017); Wecker  _et al._ (2015); Peruzzo  _et al._ (2014); McClean  _et al._
(2016); _et al._ (2018) and condensed-matter systems Kandala  _et al._ (2017);
Bravo-Prieto  _et al._ (2020).

An alternative approach is the VQE inspired by the quantum approximate
optimization algorithm (QAOA), which is originally employed to solve
combinatorial optimization problems, such as the MaxCut problem Farhi  _et
al._ (2014); Zhou  _et al._ (2020); Crooks (2018); Harrigan  _et al._ (2021);
Abrams  _et al._ (2020); Akshay  _et al._ (2020); Basso  _et al._ (2021);
Weidenfeller  _et al._ (2022), and the Maximum Independent Set problem Ebadi
_et al._ (2022). More recently, numerical works show that the QAOA-inspired
VQE can be used to efficiently prepare the ground states of quantum systems Ho
and Hsieh (2019); Wiersema  _et al._ (2020); Matos  _et al._ (2021); Wauters
_et al._ (2020); Vogt  _et al._ (2020); Cade  _et al._ (2020); Yao  _et al._
(2021) as well. Experimentally, the ground states of both long-range and
short-range Ising models have been generated using the QAOA on trapped-ion
quantum simulators Pagano  _et al._ (2020); Zhu  _et al._ (2020). Other
important applications of the QAOA include the demonstration of quantum
advantage Farhi and Harrow (2016); Dalzell  _et al._ (2020), and the
generation of the Greenberger-Horne-Zeilinger (GHZ) state Ho and Hsieh (2019);
Ho  _et al._ (2019).

Different from the VQE based on the hardware efficient ansatz, the quantum
circuit of the QAOA-inspired VQE belongs to a family of problem-inspired
ansatz Bharti  _et al._ (2022). More specifically, it is also called the
Hamiltonian variational ansatz Wiersema  _et al._ (2020), because the design
of the quantum circuit is closely related to the target Hamiltonian. Moreover,
it is worthwhile emphasizing that while there exist barren plateaus on the
optimization landscape of both VQEs based on the the hardware efficient ansatz
McClean  _et al._ (2018) and the QAOA, the effect of barren plateaus is weaker
for the latter Wiersema  _et al._ (2020).

The figure of merit for a VQE boils down to the distance of the VQE’s final
state from the target state. Ingredients that affect the performance of the
QAOA on the ground-state preparation have been widely explored. It has been
shown that essentially, the success of the QAOA depends on the depth of the
quantum circuit. Specifically, the ground states of one-dimensional (1D)
transverse field Ising model (TFIM) with periodic boundary condition (PBC) and
a system size $N$ can be prepared using the QAOA with a perfect fidelity
$f=1$, as long as the depth $p$ satisfies $p\geq N/2$ Ho and Hsieh (2019).
Moreover, it has been observed in reference Matos  _et al._ (2021), through
numerically simulating the ground-state preparation for the non-integrable
Ising models, that the perfect fidelity $f=1$ with $p=N/2$ is only attainable
for the scenario where the target ground state perturbs near closet free-
fermion states.

In this work, we study the performance of QAOA for preparing the ground states
of several target Hamiltonians, such as the 1D and two-dimensional (2D) TFIMs
with both PBC and open boundary condition (OBC), the Ising model with random
interactions and transverse-field strengths, and the 1D Heisenberg model with
both PBC and OBC. In contrast to previous works which were mainly restricted
to target Hamiltonians with PBC, i.e., translation-invariant systems, our
results show that it is harder to perform QAOA without translation-invariant
symmetry, for instance due to the OBC or strong randomness. To more
efficiently prepare those ground states, we propose a generalized QAOA-
inspired VQE assisted by parametrized resource Hamiltonians (PRHs), and
numerically demonstrate its advantage through several concrete examples.

In addition to the ground states preparation, we also pay attention to the
generation of high-fidelity GHZ states via the QAOA. We find that the PBC is
necessary to achieve perfect fidelity of the $N$-qubit GHZ state by using the
1D quantum circuit of the conventional QAOA with a depth $N/2$, and the
fidelity significantly drops for the 1D OBC quantum circuit with the same
depth. Using the QAOA assisted by the PRH (PRH-QAOA), however, we can also
obtain the perfect fidelity employing the 1D quantum circuit with a depth of
$N/2$ even for OBC. Finally, enlightened by the PRH-QAOA, we propose a quantum
circuit with a cross-shape geometry beyond 1D, which can generate the
$N$-qubit GHZ state with perfect fidelity, requiring a shallower circuit depth
than the QAOA using the 1D quantum circuit. We also discuss its experimental
realization on Rydberg-dressed atoms.

##  II Variational quantum eigensolver based on the quantum approximate
optimization algorithm

###  II.1 Quantum circuit and algorithm

Here, we briefly introduce the quantum circuit and algorithm of the VQE
inspired by the conventional QAOA. The VQE algorithm aims to generate the
ground state of a target Hamiltonian $\hat{H}^{(T)}$, which can be often
divided into $M\geq 2$ parts, i.e.,
$\hat{H}^{(T)}=\sum_{j=1}^{M}\hat{H}^{(T)}_{j}$ with
$[\hat{H}^{(T)}_{j},\hat{H}^{(T)}_{j+1}]\neq 0$ and any two terms inside each
$\hat{H}^{(T)}_{j}$ commute with each other for all $j$. We define the unitary
matrix $U_{j}(x)=\exp(-ix\hat{H}^{(T)}_{j})$, and the quantum circuit of the
QAOA-inspired VQE can be realized by repeatedly performing cycles of
operations which consist of sequentially switching on and off each term from
$\hat{H}^{(T)}_{M}$ to $\hat{H}^{(T)}_{1}$ as

$\displaystyle
U(\textbf{x})=\prod_{n=1}^{p}[U_{1}(x_{1}^{(n)})U_{2}(x_{2}^{(n)})\ldots
U_{M}(x_{M}^{(n)})]$ (1)

with $p$ being the depth of the quantum circuit, and
$\textbf{x}=(x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{M}^{(1)},x_{1}^{(2)},\ldots,x_{M}^{(p)})$.
The parametrized final state of the quantum circuit is
$|\psi(\textbf{x})\rangle=U(\textbf{x})|\psi_{0}\rangle$ with
$|\psi_{0}\rangle$ being the initial state which can be chosen as the ground
state of $\hat{H}^{(T)}_{j}$ ($j\neq M$). Then, the energy
$E(\textbf{x})=\langle\psi(\textbf{x})|\hat{H}^{(T)}|\psi(\textbf{x})\rangle$
and the fidelity
$f(\textbf{x})=|\langle\psi_{\text{GS}}|\psi(\textbf{x})\rangle|^{2}$, where
$|\psi_{\text{GS}}\rangle$ is the ground state of $\hat{H}^{(T)}$, can be
directly obtained.

We can generate a final quantum state $|\psi(\textbf{x})\rangle$, which is
close to the ground state $|\psi_{\text{GS}}\rangle$ via a hybrid quantum-
classical procedure, where the unitary evolution $U(\textbf{x})$ is simulated
and the cost function is measured on quantum computers, and the optimization
of the parameters x is performed on classical computers. In this work, we
adopt the BFGS method, a gradient-based approach, to optimize the cost
functions.

It is worthwhile mentioning that except for preparing ground states of target
Hamiltonians, QAOA has also been employed to generate multipartite entangled
states, including the GHZ state Ho and Hsieh (2019); Ho  _et al._ (2019) and
the Dicke state Santra  _et al._ (2022). The QAOA may also have the potential
to generate other more complex state such as the quantum image Zhang  _et al._
(2013), and possibly apply to quantum image detection Chetia  _et al._ (2021).
In our work, we focus on two basic applications of the QAOA, i.e., the VQE
inspired by the QAOA and the generation of high-fidelity GHZ states.

Figure 1: (a) The infidelities $1-f$ obtained after optimization as a function
of the depth of the VQE $p$ for the 1D ferromagnetic TFIMs with both PBC and
OBC. (b) is similar to (a) but for the 2D ferromagnetic TFIMs. The inset of
(b) shows a schematic of a regular 2D lattice with $N=9$ spins (represented by
circles). For the OBC case, only the interactions represented by solid lines
are considered. For the PBC case, both the interactions represented by solid
and dashed lines are considered.

###  II.2 Performance of the QAOA-inspired VQE

We first consider the 1D ferromagnetic TFIM with $N=12$ as the target
Hamiltonian, i.e.,
$\hat{H}^{(T)}=-\sum_{i=1}^{N}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}-\lambda\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}$
for PBC ($i+N\equiv i$) and
$\hat{H}^{(T)}=-\sum_{i=1}^{N-1}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}-\lambda\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}$
for OBC. We choose $M=2$. According to equation (1), the parametrized quantum
circuit can be written as

$\displaystyle
U(\textbf{x})=\prod_{n=1}^{p}[U_{1}(x_{1}^{(n)})U_{2}(x_{2}^{(n)})]$ (2)

with

$\displaystyle
U_{2}(x_{2}^{(n)})=\exp(ix_{2}^{(n)}\sum_{i=1}^{N(N-1)}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z})$
(3)

and

$\displaystyle
U_{1}(x_{1}^{(n)})=\exp(ix_{1}^{(n)}\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}).$ (4)

The initial state $|\psi_{0}\rangle=\bigotimes_{i=1}^{N}|+\rangle_{i}$ is the
ground state of $\hat{H}^{(T)}_{1}=-\lambda\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}$
($\lambda>0$), where $|+\rangle_{i}$ is the eigenstate of
$\hat{\sigma}_{i}^{x}$ with the eigenvalue $+1$. Here, we focus on the
critical point, i.e., $\lambda=1$, separating the ferromagnetic and
paramagnetic phase for the TFIM in the thermodynamic limit Sachdev (2000).

As shown in figure. 1(a), for the 1D TFIM with PBC, one can achieve a perfect
fidelity $f$ ($1-f\simeq 10^{-15}$ to machine precision) with a depth $p\geq
N/2$ Ho and Hsieh (2019). However, for the TFIM with OBC, the perfect fidelity
is not achievable with a depth up to $12$, indicating that the performance of
the QAOA-inspired VQE is relevant to the boundary condition of the target
Hamiltonian. Next, we study the TFIM on a 2D square lattice with $N=3\times 3$
sites as the target Hamiltonian, i.e., $\hat{H}^{(T)}=-\sum_{\langle
i,j\rangle}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}-\lambda\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}$
with $\langle i,j\rangle$ referring to the nearest-neighbor sites $i$ and $j$
on the square lattice [see the inset of figure. 1(b)]. Similar to the 1D case,
we also consider $\lambda=3.05$, as a critical case near the phase transition
point Jordan  _et al._ (2008); Blöte and Deng (2002). The results for both OBC
and PBC are displayed in figure. 1(b). It is shown that for both the 1D and 2D
models, the infidelities for PBC are larger than those for OBC when the
circuit depth is shallow, whereas it is the opposite for deep circuits.

Moreover, for the 2D TFIM with both PBC and OBC, a perfect fidelity is not
achievable. This can be interpreted by the fact that the 2D TFIM can not be
mapped to free fermions via the Jordan-Wigner and Bogoloibov transformations
Pfeuty (1970), and this result generalizes the conjecture of reference Matos
_et al._ (2021) to 2D systems, suggesting that the perfect fidelity obtained
from the QAOA-inspired VQE relies on the target Hamiltonian with a free-
fermion-like ground state.

##  III Improvement of the variational quantum eigensolver

###  III.1 Basic idea

For the QAOA-inspired VQE, the design of the quantum circuit is closely
related to the target Hamiltonian $\hat{H}^{(T)}$. Here, we explore whether
the performance of the VQE can be improved if we adopt a resource Hamiltonian
$\hat{H}^{(R)}$, which is not identically to the target Hamiltonian. The
resource Hamiltonian $\hat{H}^{(R)}$ is then parametrized as
$\hat{H}^{(R)}=\hat{H}^{(R)}(\textbf{y})$, where
$\textbf{y}=(y_{1},y_{2},\ldots,y_{L})$. The quantum circuit based on the
parametrized resource Hamiltonian (PRH) $\hat{H}^{(R)}$ reads

$\displaystyle
U(\textbf{x},\textbf{y})=\prod_{n=1}^{p}[U_{1}^{(R)}(x_{1}^{(n)},\textbf{y})\ldots
U_{M}^{(R)}(x_{M}^{(n)},\textbf{y})],$ (5)

where $U_{j}^{(R)}(x,\textbf{y})=\exp[-ix\hat{H}^{(R)}_{j}(\textbf{y})]$, and
$\hat{H}^{(R)}(\textbf{y})=\sum_{j=1}^{M}\hat{H}^{(R)}_{j}(\textbf{y})$ with
$[\hat{H}^{(R)}_{j}(\textbf{y}),\hat{H}^{(R)}_{j+1}(\textbf{y})]\neq 0$. We
note that the parameters y are fixed all the way through the evolution, i.e.,
$n=1,2,\ldots,p$ in equation (5), and so that cannot be absorbed to the
parameters x. Next, by optimizing the cost functions, for example the energy
and the fidelity, in the parameter space $\textbf{x}\bigcup\textbf{y}$, it is
possible to obtain a lower energy or higher fidelity than the conventional
QAOA-inspired VQE with $\hat{H}^{(R)}(\textbf{y})\equiv\hat{H}^{(T)}$. In this
paper, the generalized QAOA assisted by the PRH is labeled as PRH-QAOA for
short.

Actually, it has been numerically shown that employing the 1D Heisenberg model
as the resource Hamiltonian, one can efficiently prepare the ground state of a
modified Haldane-Shastry Hamiltonian, providing a revelation as to making the
resource Hamiltonian different from the target one in designing QAOA-inspired
VQEs Wiersema  _et al._ (2020). We emphasize, however, that in reference
Wiersema  _et al._ (2020), the Heisenberg model as the resource Hamiltonian
$\hat{H}^{(R)}$ is non-parametrized. Different from reference Wiersema  _et
al._ (2020), here we consider a PRH $\hat{H}^{(R)}=\hat{H}^{(R)}(\textbf{y})$,
and will later find out the most appropriate $\hat{H}^{(R)}$ for preparing
ground states of other $\hat{H}^{(T)}$. More technical details of the QAOA-
inspired VQE with 1D Heisenberg models being the resource Hamiltonian can be
found in Appendix A.

Figure 2: (a) The optimized parameter $J^{\text{opt}}_{i,i+1}$ as a function
of the spatial index $i$ obtained from the quantum circuit equation (5) with a
depth $p=6$. (b) The optimized parameter $h^{\text{opt}}_{i}$ as a function of
the spatial index $i$ obtained from the quantum circuit equation (5) with a
depth $p=6$. (c) is similar to (a) but for the quantum circuit with a depth
$p=12$. (d) is similar to (b) but for the quantum circuit with a depth $p=12$.
(e) The infidelities $1-f$ obtained after optimization as a function of the
depth $p$. The inset of (e) shows the ratio $R$ defined by equation (8) as a
function of the depth $p$.

###  III.2 Application 1: one-dimensional transverse-field Ising model with
open boundary condition as the target Hamiltonian

In figure 1(a), we have seen the performance of conventional QAOA on 1D TFIM
with OBC is less efficient. Here, we apply the PRH-QAOA to improve the
performance of the VQE for the 1D TFIM with OBC. The target Hamiltonian is
still
$\hat{H}^{(T)}=-\sum_{i=1}^{N-1}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}-\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}$
with $N=12$, and we adopt a more generic Ising model with OBC as the resource
Hamiltonian, which is divided into
$\hat{H}^{(R)}(\textbf{y})=\hat{H}_{1}^{(R)}(\textbf{h})+\hat{H}_{2}^{(R)}(\textbf{J})$
with

$\displaystyle\hat{H}_{1}^{(R)}(\textbf{h})=-\sum_{i=1}^{N}h_{i}\hat{\sigma}_{i}^{x},$
(6)

and

$\displaystyle\hat{H}_{2}^{(R)}(\textbf{J})=-\sum_{i=1}^{N-1}J_{i,i+1}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}.$
(7)

That is we generalize the $\hat{H}^{(R)}$ to allow for parametrized site-
dependent interactions and fields therein. It is obvious that the PRH covers
the target Hamiltonian, being the $\hat{H}^{(R)}$ in the conventional QAOA, as
a subset with $J_{i,i+1}=J$ and $h_{i}=h,~{}\forall j$. With the above
resource Hamiltonian, the parameter space of the quantum circuit (5) is
$\textbf{x}\bigcup\textbf{y}$ with x being the same as the parameters in
equations (2)-(4), and
$\textbf{y}=\textbf{h}\bigcup\textbf{J}=(h_{1},h_{2},\ldots,h_{N})\bigcup(J_{1,2},J_{2,3},\ldots,J_{N-1,N})$.
With $h_{i}>0$ for all $i=1,2,\ldots,N$, the initial state is chosen as
$|\psi_{0}\rangle=\bigotimes_{i=1}^{N}|+\rangle_{i}$. Then, we can numerically
obtain the final state
$|\psi(\textbf{x},\textbf{y})\rangle=U(\textbf{x},\textbf{y})|\psi_{0}\rangle$,
and calculate the corresponding cost functions.

After optimizing the fidelity as the cost function, we demonstrate that for
the 1D TFIM with OBC, the system with uniform interaction and magnetic field
is not the optimal choice. We present two examples of the optimized h and J in
figures 2(a)-(d). It is noted that there exists an inversion symmetry for the
optimized parameters, i.e., $J^{\text{opt}}_{i,i+1}\equiv
J^{\text{opt}}_{N-i,N-i+1}$ and $h^{\text{opt}}_{i}\equiv
h^{\text{opt}}_{N-i+1}$.

In figure 2(e), we plot the infidelities $1-f$ after optimization for the VQE
based on the quantum circuit equation (1) and equation (5), known as the
conventional QAOA and the PRH-QAOA, respectively. Moreover, to quantify the
improvement of the PRH-QAOA-inspired VQE, we define a ratio

$\displaystyle
R=\frac{(1-f^{C})-(1-f^{PRH})}{1-f^{C}}=\frac{f^{PRH}-f^{C}}{1-f^{C}}$ (8)

with $f^{PRH}$ and $f^{C}$ being the fidelity after optimization of the VQE
based on the PRH-QAOA and conventional QAOA, respectively, and plot the value
of $R$ as a function of the depth of quantum circuit $p$ in the inset of
figure 2(e). In comparison with the VQE based on the conventional QAOA, one
can see the PRH-QAOA-inspired VQE is more efficient for preparing the ground
state of the 1D TFIM with OBC.

###  III.3 Application 2: two-dimensional transverse-field Ising model as the
target Hamiltonian

Figure 3: (a) The infidelities $1-f$ obtained after optimization as a function
of the depth $p$ for the 2D ferromagnetic TFIM. (b) The values of optimized
parameters $J^{\text{opt}}_{i,j}$ (near the solid and dashed lines) and
$h^{\text{opt}}_{i}$ (inside the circles) with the depth $p=5$. (c) The ratio
$R$ defined by equation (8) as a function of the depth $p$.

We now employ the PRH-QAOA-inspired VQE to generate the ground states of 2D
ferromagnetic TFIM with $N=9$ and $\lambda=3.05$. Here, we focus on the model
with OBC since the open-boundary system is experimentally more relevant for 2D
superconducting circuits Wu  _et al._ (2021); Arute  _et al._ (2019);
Braumüller  _et al._ (2022) and Rydberg quantum simulators Labuhn  _et al._
(2016); Lienhard  _et al._ (2018); Guardado-Sanchez  _et al._ (2018); Ebadi
_et al._ (2021). We consider the resource Hamiltonian
$\hat{H}^{(R)}(\textbf{y})=\hat{H}_{1}^{(R)}(\textbf{h})+\hat{H}_{2}^{(R)}(\textbf{J})$
with $\hat{H}_{1}^{(R)}(\textbf{h})$ being the same as equation (6) and

$\displaystyle\hat{H}_{2}^{(R)}(\textbf{J})=-\sum_{\langle
i,j\rangle}J_{i,j}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z},$ (9)

which describes ferromagnetic Ising interactions on a 2D square lattice.

As shown in figures 3(a) and (c), a lower infidelity $1-f$ can be obtained
using the VQE based on the PRH-QAOA than the $1-f$ obtained from the
conventional QAOA. We present an example of the optimized y for the resource
Hamiltonian in figure 3(b). It is revealed that the optimized y preserves both
the inversion symmetry and rotation symmetry.

###  III.4 Remarks

We apply the PRH-QAOA-inspired VQE to the 2D TFIM with PBC and find out that
$\hat{H}^{(R)}(\textbf{y})\equiv\hat{H}^{(T)}$ is the optimal choice, namely
generalizing the QAOA to admit site-dependent parameter in the $\hat{H}^{(R)}$
does not help for the target Hamiltonians with PBC.

Additionally, we also consider the Heisenberg model as the target Hamiltonian,
and study the performance of both VQE inspired by the conventional QAOA and
PRH-QAOA. The results and corresponding discussions are presented in Appendix
A. The Heisenberg model is an integrable model exactly solved using the Bethe
ansatz Franchini (2017). However, the Heisenberg model cannot be mapped to
free fermions, and thus the perfect fidelity cannot be obtained via the VQE
with a $N/2$ depth of quantum circuit. The dependence of the performance of
the conventional QAOA-inspired VQE on the boundary condition is the same as
that we have revealed in the TFIMs. For the Heisenberg model with OBC being
the target Hamiltonian, the PRH-QAOA-inspired VQE can also obtain a higher
fidelity of the ground state than the conventional QAOA.

Moreover, in Appendix B, we present the results of the energy as the cost
function, the 2D TFIM with other values of $\lambda$, as well as the 2D TFIM
with triangular geometry, as the target Hamiltonians. The additional data
displayed in Appendix A and B demonstrate that the improvement of the VQE
based on the PRH-QAOA is potential to be generalized to other cases.

Figure 4: (a) The infidelities $1-f$ obtained after optimization as a function
of the depth of the VQE $p$ for the randomized Ising models (10). (b) The
ratio $R$ defined in equation (8) as a function of the depth $p$. (c) An
example of the optimized parameter $J_{i,i+1}^{\text{opt}}$ and the
corresponding parameter $J_{i,i+1}^{(T)}$ in the target Hamiltonian (10),
obtained from the PRH-QAOA-inspired VQE with a depth $p=6$, as a function of
the spatial index $i$. (d) is similar to (c) but for the the optimized
parameter $h_{i}^{\text{opt}}$ and the corresponding parameter $h_{i}^{(T)}$
in equation (10).

##  IV Generating ground states of Ising models with random interactions and
transverse-field strengthes

After revealing the influence of boundary condition on the performance of
QAOA, both for the conventional and the PRH-assisted scenarios, we now
consider another source that can break the translational invariance, i.e., the
randomness. We focus on the TFIM with PBC as the target Hamiltonian

$\displaystyle\hat{H}^{(T)}=-\sum_{i=1}^{N}J_{i,i+1}^{(T)}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}-\sum_{i=1}^{N}h_{i}^{(T)}\hat{\sigma}_{i}^{x},$
(10)

where the Ising interactions $J_{i,i+1}^{(T)}$ and the transverse-field
strengthes $h_{i}^{(T)}$ are drawn from a uniform distribution
$[(2-D)/2,(2+D)/2]$ with $D$ being the disorder strength. If $D\rightarrow 0$,
the Hamiltonian (10) becomes translation-invariant. With the increase of $D$,
one can explore how the breakdown of the translational invariance influences
the performance of the VQE.

Here, for the conventional QAOA-inspired VQE, the quantum circuit can be
written as equation (2) with
$U_{2}(x_{2}^{(n)})=\exp(ix_{2}^{(n)}\sum_{i=1}^{N}J_{i,i+1}^{(T)}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z})$
and
$U_{1}(x_{1}^{(n)})=\exp(ix_{1}^{(n)}\sum_{i=1}^{N}h_{i}^{(T)}\hat{\sigma}_{i}^{x})$.
For the PRH-QAOA-inspired VQE, the quantum circuit can be described by
equation (5) with the resource Hamiltonian
$\hat{H}^{(R)}=\hat{H}_{1}^{(R)}(\textbf{h})+\hat{H}_{2}^{(R)}(\textbf{J})$,
where $\hat{H}_{1}^{(R)}(\textbf{h})$ is the same as equation (6), and
$\hat{H}_{2}^{(R)}(\textbf{J})=-\sum_{i=1}^{N}J_{i,i+1}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}$
with $\textbf{J}=(J_{1,2},J_{2,3},\ldots,J_{12,1})$ for $N=12$. We consider
three disorder strengthes $D=1$, $0.1$, and $0.01$.

The results obtained from the VQE based on the conventional QAOA and PRH-QAOA
are plotted in figure 4(a), where the infidelities are obtained by averaging
$10$ samples of $J_{i,i+1}^{(T)}$ and $h_{i}^{(T)}$ for each $D$. It is shown
that the performance of the VQEs is always better for smaller $D$ regardless
of the conventional QAOA or PRH-QAOA, and for each given $D$, the VQE based on
PRH-QAOA always outperforms the conventional one. Moreover, as shown in figure
4(b), when the depth $p\geq 6$, the improvement becomes significant. In
figures 4(c) and (d), we also present an example of optimized parameters in
the resource Hamiltonian $\hat{H}^{(R)}$.

##  V Generating the Greenberg-Horne-Zeilinger state via the improved quantum
approximate optimization algorithm

The GHZ states, as a type of multipartite entangled states, are central for
numerous quantum-based technologies, ranging from quantum teleportation Zhao
_et al._ (2004), quantum metrology Giovannetti  _et al._ (2011) to quantum
error-correcting codes Knill (2005). Experimental generation of high-fidelity
GHZ states has been achieved on various quantum platforms, including
superconducting circuits Song  _et al._ (2019, 2017); Barends  _et al._
(2016); Wei  _et al._ (2020), optomechanical-like quantum systems Wu  _et al._
(2022), quantum optical platforms Sahu (2018); Pan  _et al._ (2012), trapped
ions Monz  _et al._ (2011); Pogorelov  _et al._ (2021) and Rydberg atoms Omran
_et al._ (2019); Graham  _et al._ (2022).

In neutral-atom quantum processors, GHZ states with fidelity $f>0.5$ have been
prepared via the optimal control of the adiabatic dynamics Omran  _et al._
(2019); Cui  _et al._ (2017). In quantum optical platforms, using the Hong-Ou-
Mandel interference, multi-photon GHZ states can be generated Yao  _et al._
(2012). Digital-gate-model quantum circuits with CNOT gates designed for GHZ
states can also be implemented on Rydberg atoms Graham  _et al._ (2022) and
superconducting circuits Neeley  _et al._ (2010). Nowadays, how to generate
high-fidelity and large-scale GHZ states remains a central topic in quantum
physics, and new schemes of preparing multi-qubit GHZ states are timely in
need.

###  V.1 One-dimensional quantum circuit

Figure 5: (a) The infidelities $1-f$ with $f$ being the GHZ state fidelity,
obtained after optimization, as a function of the depth $p$. The resource
Hamiltonian is chosen as the 1D ferromagnetic TFIM with OBC or PBC. The inset
of (a) plots the $1-f$ as a function of the depth $p$ in a linear y-axis. (b)
The optimized parameter $J^{\text{opt}}_{i,i+1}$ as a function of the spatial
index $i$ obtained from the PRH-QAOA with a depth $p=6$. (c) is similar to (b)
but for the optimized parameter $h^{\text{opt}}_{i}$. (d) The infidelities
$1-f$ with $f$ being the GHZ state fidelity, obtained after optimization, as a
function of the depth $p$. (e) The values of optimized parameters
$J^{\text{opt}}_{i,j}$ (near the solid and dashed lines) and
$h^{\text{opt}}_{i}$ (inside the circles) with the depth $p=2$. (f) The
geometry of the Ising interactions (presented as the solid lines) of the
resource Hamiltonian with $N=9$. The circles in (f) denote the location of
qubits. The hollow circles in (c) denote the central qubit.

The GHZ state defined as

$\displaystyle|\text{GHZ}\rangle\equiv\frac{\bigotimes_{i=1}^{N}|\uparrow\rangle_{i}+\bigotimes_{i=1}^{N}|\downarrow\rangle_{i}}{\sqrt{2}},$
(11)

with $|\uparrow(\downarrow)\rangle_{i}$ being the eigenstate of
$\hat{\sigma}_{i}^{z}$ with the eigenvalue $+1(-1)$, is a ground state of the
ferromagnetic Ising Hamiltonian $\hat{H}^{(T)}=-\sum_{\langle
i,j\rangle}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}$. We have demonstrated
that the QAOA circuit equations (2-4) can generate the ground state of
transverse field Ising models. Here, we study the performance of same QAOA
circuit $U(\textbf{x})=\prod_{n=1}^{p}[U_{1}(x_{1}^{(n)})U_{2}(x_{2}^{(n)})]$
with $U_{1}(x_{1}^{(n)})=\exp(ix_{1}^{(n)}\sum_{i=1}^{N}\hat{\sigma}_{i}^{x})$
and
$U_{2}(x_{2}^{(n)})=\exp(ix_{2}^{(n)}\sum_{i=1}^{N(N-1)}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z})$
for preparing the GHZ state (11), with the cost function

$\displaystyle f\equiv|\langle\text{GHZ}|\psi(\textbf{x})\rangle|^{2}$ (12)

being the fidelity of the GHZ state.

It has been numerically shown that a $N$-qubit GHZ state with a perfect
fidelity $f\simeq 1$ can be generated via the QAOA with the PBC and a depth
$p=N/2$ Ho and Hsieh (2019) [also see the results in figure 5(a)]. Moreover,
we show that a perfect fidelity cannot be achieved when employing the OBC in
equation (2). As displayed in figure 5(a), with the OBC, a 12-qubit GHZ state
with a perfect fidelity can be generated if the depth of the QAOA is $p=12$,
much deeper than the required depth $p=N/2=6$ when the PBC is adopted.

Next, in order to generate a GHZ state with a perfect fidelity using a
shallower depth quantum circuit with OBC, we consider the PRH-QAOA with the
circuit
$U_{2}(x_{2}^{(n)},\mathbf{J})=\exp(ix_{2}^{(n)}\sum_{i=1}^{N}J_{i,i+1}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z})$,
$U_{1}(x_{1}^{(n)},\mathbf{h})=\exp(ix_{1}^{(n)}\sum_{i=1}^{N}h_{i}\hat{\sigma}_{i}^{x})$,
and
$U(\textbf{x},\mathbf{J},\mathbf{h})=\prod_{n=1}^{p}[U_{1}(x_{1}^{(n)},\mathbf{h})U_{2}(x_{2}^{(n)},\mathbf{J})]$,
which is the same as the equation (5) and the resource Hamiltonian equations
(6) and (7). In Appendix C, we display a quantum circuit representation of the
aforementioned PRH-QAOA protocol for generating GHZ states. The results of
infidelity $1-f$ is plotted in figure 5(a), and the optimized parameters in
the resource Hamiltonian are displayed in figures 5(b) and (c). It is seen
that a 12-qubit GHZ state with $f\simeq 1$ can be generated via the PRH-QAOA
with minimal depth $p=6$.

###  V.2 Quantum circuit beyond one-dimensional geometry

Figure 6: (a) The infidelities $1-f$ with $f$ being the GHZ state fidelity,
obtained after optimization, as a function of the depth $p$, with the system
size $N=13$ and $N=17$. (b) The geometry of the Ising interactions (presented
as the solid lines) of the resource Hamiltonian with $N=13$. (c) is similar to
(b) but for the system size $N=17$.

We also study the performance of QAOA for generating GHZ states using a 2D
quantum circuit with a size $N=3\times 3$, i.e., the TFIM on 2D square lattice
as the resource Hamiltonian [see the inset of figure 1(b)]. We first focus on
the conventional QAOA for 2D quantum circuits with both OBC and PBC, in
comparison with the results of 1D quantum circuit with PBC. The results of the
GHZ state fidelity are shown in figure 5(d). For a 1D PBC quantum circuit with
$N=9$, the fidelity reaches a value $1-f\sim 10^{-11}$ when $p=4$, and a
perfect fidelity $1-f\sim 10^{-15}$ can be achieved when $p\geq 5$. For a 2D
PBC quantum circuit, a perfect fidelity is attainable when $p\geq 4$. However,
with a depth $p\geq 5$, for the 2D OBC quantum circuit, the obtained GHZ state
fidelities are much smaller than those for the PBC quantum circuit.

Here, we consider the PRH-QAOA with a 2D OBC quantum circuit. The resource
Hamiltonian is the summation of equation (6) and equation (9). As shown in
figure 5(d), a 9-qubit GHZ state with perfect fidelity can be generated by the
algorithm with a quite shallow depth $p=2$ of the quantum circuit. The
optimized parameters are presented in figure 5(e). It is revealed that there
are four ferromagnetic Ising interactions whose strength is 0, marked by the
dashed lines in figure 5(e), in the resource Hamiltonian equation (9).
Moreover, for the resource Hamiltonian equation (6), the strength of
transverse field of the central qubit is 0.

The resource Hamiltonian with the parameters obtained after optimization leads
to a Ising model with a cross-shape geometry of the interactions, as plotted
in figure 5(f). In Appendix C, we also display a quantum circuit
representation of the PRH-QAOA protocol with cross-shape geometry for
generating GHZ states. The size of the quantum circuit with the cross-shape
geometry is scalable, i.e., $N=5+4m$ $(m=0,1,\ldots)$ [see figure 6(b) and
(c)]. Furthermore, the transverse-field strength of the central spin of the
resource Hamiltonian is fixed as 0. As shown in figure 6(a), 13-qubit and
17-qubit GHZ state with perfect fidelity can be generated by the algorithm
with the 3-depth and 4-depth quantum circuit, respectively, indicating that
employing the Ising model with the special geometry and parameters as the
resource Hamiltonian, the QAOA quantum circuit with a depth $m$ can generate
$(5+4m)$-qubit GHZ state with perfect fidelity.

We note that the shallow depth of the quantum circuit with the ability of
generating perfect-fidelity GHZ states benefits from the cross-shape geometry
of the resource Ising Hamiltonian plotted in figures 5(f), 6(b), and 6(c). In
Appendix D, it is seen that using the square-lattice Ising model with $N=16$,
both the conventional QAOA and PRH-QAOA can not generate the perfect-fidelity
GHZ state with a 4-depth or shallower depth quantum circuit.

Moreover, we emphasize that all the aforementioned QAOA circuits preparing GHZ
states have the global $\mathbb{Z}_{2}$ symmetry given by the parity operator
$\hat{P}=\prod_{i=1}^{N}\hat{\sigma}_{i}^{x}$. With the initial state
$|\psi_{0}\rangle=\otimes_{i=1}^{N}|+\rangle_{i}$ satisfying
$\langle\psi_{0}|\hat{P}|\psi_{0}\rangle=+1$, the symmetry ensures that the
final states of the QAOA circuits are in the same symmetry sector
$\langle\hat{P}\rangle=+1$. Consequently, the generated ground state of
ferromagnetic Ising Hamiltonian $\hat{H}^{(T)}=-\sum_{\langle
i,j\rangle}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}$ is the GHZ state (11)
belonging to the sector $\langle\hat{P}\rangle=+1$, instead of those ground
states out of the sector, such as $\otimes_{i=1}^{N}|\uparrow\rangle_{i}$.

###  V.3 Rydberg dressing scheme of realizing the improved QAOA for generating
the Greenberg-Horne-Zeilinger state

Figure 7: (a) Schematic representation of the relevant energy level for a
single Rydberg-dressed atom. (b) Schematic representation of Ising dynamics.
(c) Schematic representation of the spin echo for realizing Ising dynamics
using the unitary dynamics of $\hat{H}_{D}$, i.e., $U_{NN}$, and the single-
qubit operation $X_{\pi/2}=\exp[-i(\pi/2)\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}]$.
(d) An array of mixed-species atoms beyond 1D geometry enlightened by figures
5(e) and (f) with $N=9$. Three typical interactions $J_{0}$, $J_{1}$, and
$J_{2}$ are relevant. (e) An array of mixed-species atoms with $N=17$.

Here, we discuss the experimental realization of generating the GHZ state via
the PRH-QAOA with the cross-shape geometry in figures 5(f), 6(b) and (c), in
Rydberg-dressed neutral atoms Borish  _et al._ (2020); Zeiher  _et al._
(2017); Jau  _et al._ (2016), accompanied with recent developed mixed-species
atom arrays Sheng  _et al._ (2022); Singh  _et al._ (2022).

We first consider the generation of the 9-qubit GHZ state. The realization of
this protocol requires an Ising interaction $-\sum_{\langle
i,j\rangle}J_{i,j}\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{j}$ with $\langle
i,j\rangle$ meaning the nearest-neighbor sites $i$ and $j$ on the lattice in
figure 5(f). For Rydberg-dressed quantum processors, qubits are encoded in two
hyperfine ground states $|\downarrow\rangle$ and $|\uparrow\rangle$, with one
of the ground state $|\uparrow\rangle$ coupled to a Rydberg state $|R\rangle$
via a laser field of detuning $\delta$ and Rabi frequency $\Omega$ Borish  _et
al._ (2020); Zeiher  _et al._ (2017); Jau  _et al._ (2016) [also see figure
7(a)]. When $\Omega/\delta\ll 1$, the Rydberg-dressed state can induce a
pairwise energy shift dependent on the distance between two atoms, which can
be described by the Hamiltonian

$\displaystyle\hat{H}_{D}=\sum_{\langle
i,j\rangle}V_{i,j}\hat{n}_{i}\hat{n}_{j},$ (13)

where $\hat{n}_{i}=|\uparrow\rangle_{i}\langle\uparrow|_{i}$,
$\hat{n}=(\hat{\sigma}^{z}+\textbf{1})/2$ with 1 being the two-dimensional
identity matrix, and the strength of Rydberg-dressed interaction reads

$\displaystyle
V_{i,j}=V_{0}\frac{R_{C}^{6}}{|\textbf{r}_{i}-\textbf{r}_{j}|^{6}+R_{C}^{6}}$
(14)

with the positions of atoms $\textbf{r}_{i,j}$,
$V_{0}=\hbar\Omega^{4}/8\delta^{3}$, $R_{C}=|C_{6}/2\hbar\delta|^{1/6}$, and
$C_{6}$ being the van der Waals coefficient of the Rydberg state. We define
the unitary dynamics of $\hat{H}_{D}$ as $U_{NN}(t)=\exp(-it\hat{H}_{D})$. By
using a spin echo Borish  _et al._ (2020); Zeiher  _et al._ (2017) shown in
figure 7(c), one can realize the Ising dynamics [see figure 7(b)]

$\displaystyle U_{ZZ}(\gamma)=\exp[-i\gamma\sum_{\langle
i,j\rangle}(V_{i,j}/4)\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{j}].$ (15)

According to equation (14), non-local interactions should be considered. Here,
we take into account the additional interactions presented as $J_{0}$ in
figure 7(d). The nearest-neighbor interactions includes $J_{1}$ and $J_{2}$ in
figure 7(d). Both the interactions $J_{1}$ and $J_{2}$ can be adjusted by
changing the corresponding intra-atom distance. In addition to the distance,
the interaction $J_{1}$ can also be adjusted by choosing different species
type-2 [see figure 7(d)] and different Rydberg states, which essentially
results in different coefficient $C_{6}$. In short, the interactions $J_{0}$,
$J_{1}$, and $J_{2}$ can be separately adjusted. Moreover, the Rydberg
dressing quantum processor with mixed-species atoms Sheng  _et al._ (2022);
Singh  _et al._ (2022) allows the single-qubit gate performed on the central
qubit, i.e., the type-2 atom in figure 7(d), and a different single-qubit
operation on other qubits [the type-1 atom in figure 7(d)] except for the
central one.

Enlightened by the results shown in figure 5(e), for the QAOA, the resource
Hamiltonian is chosen as

$\displaystyle\hat{H}^{(R)}_{1}=-\sum_{i\in\\{\text{type-1}\\}}\hat{\sigma}_{i}^{x},$
(16)

and

$\displaystyle\hat{H}^{(R)}_{2}(J_{0},J_{1},J_{2})=-\sum_{\langle
i,j\rangle}J_{i,j}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z},$ (17)

with $J_{i,j}\in\\{J_{0},J_{1},J_{2}\\}$. Based on equation (5), the quantum
circuit with a depth $p=2$ can be written as

$\displaystyle
U(\textbf{x},\textbf{y})=U_{1}^{(R)}(x_{1}^{1})U_{2}^{(R)}(x_{2}^{1},\textbf{y})U_{1}^{(R)}(x_{1}^{2})U_{2}^{(R)}(x_{2}^{2},\textbf{y})$
(18)

with $\textbf{x}=(x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2})$ and
$\textbf{y}=(J_{0},J_{1},J_{2})$. The initial state is
$|\psi_{0}\rangle=\otimes_{i=1}^{N}|+\rangle_{i}$. The preparation of the
initial state $|\psi_{0}\rangle$ and the unitary evolution $U_{1}^{(R)}(x)$
can be experimentally realized by imposing the microwave pulse on the qubits
encoded by the hyperfine ground state $|\uparrow\rangle$ and
$|\downarrow\rangle$ of neutral atoms Borish  _et al._ (2020); Zeiher  _et
al._ (2017) [also see figure 7(a)]. The unitary evolution
$U_{2}^{(R)}(x,\textbf{y})$ is the same as equation (15), which can be
realized by the dressing pulse and the spin echo [see figures 7(a)-(c)]. After
optimization, we obtain the parameters for generating the GHZ state with
perfect fidelity, i.e., $\textbf{y}^{\text{opt}}=(4/3,1,1/3)$ and
$\textbf{x}^{\text{opt}}=(\pi/4,3\pi/4,3\pi/4,3\pi/4)$.

Finally, we generalize the results of $N=9$ to larger size $N=13$ and $17$
[see figure. 7(e) for the geometry of the lattice with $N=17$]. Generating the
perfect-fidelity GHZ with $N=13$ and $17$ requires the depth of the quantum
circuit $p=3$, and $p=4$, respectively. For $N=13$ and $p=3$, the optimized
parameters are $\textbf{y}^{\text{opt}}=(4/3,1,1/3)$ and
$\textbf{x}^{\text{opt}}=(x_{1}^{1},x_{2}^{1},x_{3}^{1},x_{1}^{2},x_{2}^{2},x_{3}^{2})=(\pi/4,\pi/4,3\pi/4,3\pi/4,3\pi/4,3\pi/4)$.
Moreover, for $N=17$ and $p=4$, $\textbf{y}^{\text{opt}}=(4/3,1,1/3)$ and
$\textbf{x}^{\text{opt}}=(x_{1}^{1},x_{2}^{1},x_{3}^{1},x_{4}^{1},x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{4}^{2})=(\pi/4,\pi/4,\pi/4,3\pi/4,3\pi/4,3\pi/4,3\pi/4,3\pi/4)$.

##  VI Discussion

We have studied the performance of the QAOA-inspired VQE for target
Hamiltonians with and without translation-invariant symmetry. For systems
without the symmetry, including the models with OBC or strong randomness, the
performance of the VQE is less efficient, in comparison with the case of
translation-invariant target Hamiltonians. In general, QAOA steers a
controllable quantum system, which we call the resource Hamiltonian in this
paper. Previous studies in the literature mainly focused on the QAOA-inspired
VQE, restricted to the scenario that the resource Hamiltonian is exactly
identical to the target Hamiltonian. We generalize the QAOA to allow the PRH
to be possibly different from the target Hamiltonian. If the translation-
invariant symmetry in the target Hamiltonian is broken, one can use the PRH-
QAOA-inspired VQE to achieve higher fidelity of the target ground state or
lower energy corresponding to the target Hamiltonian. Our work provides an
essential step toward an application of the PRH-QAOA to preparing ground
states of more generic models. An example is the quantum Sherrington-
Kirkpatrick (QSK) model with the all-to-all random Ising interactions
Sherrington and Kirkpatrick (1975); Schindler  _et al._ (2022); Das and
Chakrabarti (2008); Mukherjee  _et al._ (2015). Obviously, the translation-
invariant symmetry is absent for the QSK model, and the performance of PRH-
QAOA for generating the ground states of the QSK model is worthy of future
considerations.

Except for the QAOA, shortcuts to adiabaticity, being the fast routes to the
final results of quantum annealing, is another typical way of obtaining ground
states of target Hamiltonians Wu  _et al._ (2021); Guéry-Odelin  _et al._
(2019). However, near the quantum critical point, the adiabatic state
preparation is less efficient Campbell  _et al._ (2015). An interesting future
task is to study whether the PRH-QAOA outperforms the shortcuts to
adiabaticity for preparing quantum critical states.

In the NISQ era, quantum computers are not perfectly translation-invariant,
due to their OBC Harrigan  _et al._ (2021); Wu  _et al._ (2021); Arute  _et
al._ (2019); Braumüller  _et al._ (2022); Labuhn  _et al._ (2016); Lienhard
_et al._ (2018); Guardado-Sanchez  _et al._ (2018); Ebadi  _et al._ (2021);
Yan  _et al._ (2019); Roushan  _et al._ (2017); Chen  _et al._ (2021); Zhu
_et al._ (2022), and randomness induced by experimental imperfections.
Consequently, the PRH-QAOA-inspired VQE proposed in our work opens up an
avenue for the VQE experimentally generating states closer to the target
ground state than using the conventional protocol. Moreover, it is also shown
that the perfect fidelity GHZ state can be generated by the PRH-QAOA with
shallow-depth quantum circuit. This protocol is experimentally accessible
because the affordable circuit depth is limited for lack of efficient error
correction technique for NISQ computers. We also briefly discuss the effect of
decoherence on the PRH-QAOA generating the GHZ state in Appendix E. Taking the
non-local interactions in Rydberg-dressed quantum simulators Zeiher  _et al._
(2017) into consideration, we believe our work provides an alternative way of
generating high-fidelity GHZ states in neutral atoms.

Our work points to several further directions: (i) exploring how the
translation-invariant symmetry of target Hamiltonians or quantum circuits can
influence the performance of VQEs based on other ansatz, such as the hardware
efficient ansatzes Kandala  _et al._ (2017); Wecker  _et al._ (2015); Peruzzo
_et al._ (2014); McClean  _et al._ (2016); _et al._ (2018); Bravo-Prieto  _et
al._ (2020), (ii) exploring the performance of QAOA-inspired VQEs for other
complex target Hamiltonians, such as trapped-ion systems with long-range
interactions satisfying a power-law decay Kokail  _et al._ (2019); Pagano  _et
al._ (2020); Zhu  _et al._ (2020); Monroe  _et al._ (2021), and (iii) finding
more efficient ansatzes, taking the PRH into consideration, to design quantum
circuits of VQEs, in combination with other QAOA-like subsequences Sack and
Serbyn (2021); Brady  _et al._ (2021); Yu  _et al._ (2022); Zhu  _et al._
(2022); Herrman  _et al._ (2022), for example, the counterdiabatic QAOA Yao
_et al._ (2021); Chandarana  _et al._ (2022); Wurtz and Love (2022).

###### Acknowledgements.

We acknowledge the discussions with Roeland Wiersema, Kunpeng Wang, and Cheng
Sheng. This work was supported by the National Key Research and Development
Program of China (Grant No. 2021YFA1402001), the National Science Foundation
of China (NSFC) (Grant Nos. T2121001, 11934018, 11904018), Strategic Priority
Research Program of Chinese Academy of Sciences (Grant No. XDB28000000),
Beijing Natural Science Foundation (Grant No. Z200009), and Scientific
Instrument Developing Project of Chinese Academy of Sciences (Grant No.
YJKYYQ20200041).

##  Appendix A Results for 1D Heisenberg models

Figure 8: (a) The infidelities $1-f$ obtained after optimization as a function
of the depth $p$ for the Heisenberg model with both PBC and OBC, using the
conventional QAOA-inspired VQE. (b) The infidelities $1-f$ obtained after
optimization as a function of the depth $p$ for the Heisenberg model with OBC,
using the VQE based on conventional QAOA and PRH-QAOA. (c) The optimized
parameter $g^{\text{opt}}_{i,i+1}$ as a function of the spatial index $i$
obtained from the PRH-QAOA with a depth $p=6$. (d) is similar to (c) but for
the optimized parameter $\Delta^{\text{opt}}_{i}$.

The Hamiltonian of the 1D Heisenberg model with PBC is given by

$\displaystyle\hat{H}=\sum_{i=1}^{N}[\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}+\hat{\sigma}^{y}_{i}\hat{\sigma}^{y}_{i+1}+\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{i+1}].$
(19)

For the model with OBC, the superscript of the summation in equation (19) is
replaced by $N-1$. The system size is $N=12$. To prepare the ground states of
the 1D Heisenberg model with both OBC and PBC, we design the quantum circuit
of the QAOA-inspired VQE by employing equation (19) with corresponding
boundary conditions being the resource Hamiltonian. According to the VQE based
on the conventional QAOA in reference Wiersema  _et al._ (2020), the resource
Hamiltonian (19) is divided into four parts, i.e.,
$\hat{H}^{\text{even}}_{XY}+\hat{H}^{\text{even}}_{Z}+\hat{H}^{\text{odd}}_{XY}+\hat{H}^{\text{odd}}_{Z}$,
where
$\hat{H}^{\text{even}}_{XY}=\sum_{i=1}^{N/2}[\hat{\sigma}^{x}_{2i-1}\hat{\sigma}^{x}_{2i}+\hat{\sigma}^{y}_{2i-1}\hat{\sigma}^{y}_{2i}]$,
$\hat{H}^{\text{odd}}_{XY}=\sum_{i=1}^{N/2}[\hat{\sigma}^{x}_{2i}\hat{\sigma}^{x}_{2i+1}+\hat{\sigma}^{y}_{2i}\hat{\sigma}^{y}_{2i+1}]$,
$\hat{H}^{\text{even}}_{Z}=\sum_{i=1}^{N/2}\hat{\sigma}^{z}_{2i-1}\hat{\sigma}^{z}_{2i}$,
and
$\hat{H}^{\text{odd}}_{Z}=\sum_{i=1}^{N/2}\hat{\sigma}^{z}_{2i}\hat{\sigma}^{z}_{2i+1}$.
According to equation (1), the quantum circuit can be written as

$\displaystyle
U(\textbf{x})=\prod_{n=1}^{p}[U_{1}(x_{1}^{(n)})U_{2}(x_{2}^{(n)})U_{3}(x_{3}^{(n)})U_{4}(x_{4}^{(n)})]$
(20)

with $U_{1}(x_{1}^{(n)})=\exp(-ix_{1}^{(n)}\hat{H}^{\text{even}}_{Z})$,
$U_{2}(x_{2}^{(n)})=\exp(-ix_{2}^{(n)}\hat{H}^{\text{even}}_{XY})$,
$U_{3}(x_{3}^{(n)})=\exp(-ix_{3}^{(n)}\hat{H}^{\text{odd}}_{Z})$, and
$U_{4}(x_{4}^{(n)})=\exp(-ix_{4}^{(n)}\hat{H}^{\text{odd}}_{XY})$. The initial
state of the VQE is chosen as the ground state of both
$\hat{H}^{\text{even}}_{XY}$ and $\hat{H}^{\text{even}}_{Z}$, i.e.,
$|\psi_{0}\rangle=\bigotimes_{i=1}^{N/2}(|0\rangle_{2i-1}|1\rangle_{2i}-|1\rangle_{2i-1}|0\rangle_{2i})/\sqrt{2}$.

In reference Wiersema  _et al._ (2020), the quantum circuit (20) has been
employed to prepare the ground state of the modified Haldane-Shastry
Hamiltonian
$H^{(T)}=\sum_{n<m}^{N}\frac{1}{d_{nm}^{2}}[\hat{\sigma}^{z}_{n}\hat{\sigma}^{z}_{m}-\hat{\sigma}^{x}_{n}\hat{\sigma}^{x}_{m}-\hat{\sigma}^{y}_{n}\hat{\sigma}^{y}_{m}]$
with a complex long-range interactions
$d_{nm}=\frac{N}{\pi}|\sin[\frac{\pi}{N}(n-m)]|$. Here, the target Hamiltonian
$H^{(T)}$ is different from the resource Hamiltonian (19) where the
interactions are nearest-neighbor.

As shown in figure 8(a), the boundary condition of the 1D Heisenberg model
still influences the performance of the conventional QAOA-inspired VQE,
similar to the results in figure 1. To more efficiently prepare the ground
state of the 1D Heisenberg model with OBC, we also consider the quantum
circuit of the PRH-QAOA-inspired VQE according to equation (5). The PRH is

$\displaystyle\hat{H}^{(R)}(\textbf{y})=\sum_{i=1}^{N-1}[g_{i,i+1}(\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}+\hat{\sigma}^{y}_{i}\hat{\sigma}^{y}_{i+1})+\Delta_{i,i+1}\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{i+1}],$
(21)

which is decomposed into four parts, i.e.,
$\hat{H}^{(R)}(\textbf{y})=\hat{H}^{(R)}_{1}(\textbf{y}_{1})+\hat{H}^{(R)}_{2}(\textbf{y}_{2})+\hat{H}^{(R)}_{3}(\textbf{y}_{3})+\hat{H}^{(R)}_{4}(\textbf{y}_{4})$
with
$\hat{H}^{(R)}_{1}(\textbf{y}_{1})=\sum_{i=1}^{N/2}g_{2i-1,2i}[\hat{\sigma}^{x}_{2i-1}\hat{\sigma}^{x}_{2i}+\hat{\sigma}^{y}_{2i-1}\hat{\sigma}^{y}_{2i}]$,
$\hat{H}^{(R)}_{2}(\textbf{y}_{2})=\sum_{i=1}^{N/2}\Delta_{2i-1,2i}\hat{\sigma}^{z}_{2i-1}\hat{\sigma}^{z}_{2i}$,
$\hat{H}^{(R)}_{3}(\textbf{y}_{3})=\sum_{i=1}^{N/2}g_{2i,2i+1}[\hat{\sigma}^{x}_{2i}\hat{\sigma}^{x}_{2i+1}+\hat{\sigma}^{y}_{2i}\hat{\sigma}^{y}_{2i+1}]$,
and
$\hat{H}^{(R)}_{4}(\textbf{y}_{4})=\sum_{i=1}^{N/2}\Delta_{2i,2i+1}\hat{\sigma}^{z}_{2i}\hat{\sigma}^{z}_{2i+1}$.
In comparison with the conventional QAOA-inspired VQE, the additional
parameters in the resource Hamiltonian are
$\textbf{y}=\textbf{y}_{1}\bigcup\textbf{y}_{2}\bigcup\textbf{y}_{3}\bigcup\textbf{y}_{4}$,
with $\textbf{y}_{1}=(g_{1,2},g_{3,4},\ldots,g_{11,12})$,
$\textbf{y}_{2}=(\Delta_{1,2},\Delta_{3,4},\ldots,\Delta_{11,12})$,
$\textbf{y}_{3}=(g_{2,3},g_{4,5},\ldots,g_{10,11})$, and
$\textbf{y}_{4}=(\Delta_{2,3},\Delta_{4,5},\ldots,\Delta_{10,11})$. The
results of the PRH-QAOA-inspired VQE are plotted in figures 8(b)-(d). Similar
to the results in figures 2(a)-(d), the parameters of the resource Hamiltonian
y obtained from optimization still satisfy an inversion symmetry [see figures
8(c) and (d)].

##  Appendix B Additional numerics of Ising models

Figure 9: The energy difference $E-E_{0}$ obtained after optimization as a
function of the depth of the VQE $p$ for the 1D ferromagnetic TFIM with OBC,
where $E_{0}$ is the ground state energy of target Hamiltonians. The inset
shows the energy difference $E-E_{0}$ as a function of $p$ for the TFIM with
PBC.
Figure 10: (a) The infidelities $1-f$ obtained after optimization using the
conventional QAOA-inspired VQE, as a function of the depth $p$ for the 2D
ferromagnetic TFIM with both PBC and OBC, and a transverse-field strength
$\lambda=2.9$. (b) The infidelities $1-f$ obtained after optimization using
the VQE based on both the conventional QAOA and PRH-QAOA, as a function of the
depth $p$ for the 2D ferromagnetic TFIM with OBC. (c) is similar to (a), but
for a transverse-field strength $\lambda=3.2$. (d) is similar to (b), but for
a transverse-field strength $\lambda=3.2$.
Figure 11: (a) The infidelities $1-f$ obtained after optimization as a
function of the depth $p$ for the 2D ferromagnetic TFIM with the triangular
geometry. (b) The values of optimized parameters $J_{i,j}^{\text{opt}}$
(represented by dashed lines, dotted lines, and solid lines with different
colors) and $h_{i}^{\text{opt}}$ (inside the circles) with the depth $p=6$.
(c) The ratio $R$ defined by equation (8) as a function of $p$. The inset of
(a) shows a schematic of a triangular 2D lattice with $N=10$ spins
(represented by circles), and the ferromagnetic Ising interactions between
spins (represented by solid lines).

In this section, we present several additional numerical results. First, above
studies mainly focus on the fidelity as the cost function. We now employ the
energy as the cost function. The target Hamiltonian $\hat{H}^{(T)}$ is chosen
as the critical 1D TFIM ($\lambda=1$) with both PBC and OBC. The system size
is $N=12$. For the conventional QAOA-inspired VQE, the quantum circuit is
equation (2), and the cost function is the energy
$E(\textbf{x})=\langle\psi(\textbf{x})|\hat{H}^{(T)}|\psi(\textbf{x})\rangle$.
For the PRH-QAOA-inspired VQE, the quantum circuit is equation (5), and the
cost function is the energy
$E(\textbf{x},\textbf{y})=\langle\psi(\textbf{x},\textbf{y})|\hat{H}^{(T)}|\psi(\textbf{x},\textbf{y})\rangle$
with the additional parameters y in equations (6) and (7). In figure 9, we
plot the results of $E-E_{0}$ as a function of the depth $p$, where $E$ is the
parametrized $E(\textbf{x})$ or $E(\textbf{x},\textbf{y})$ obtained after
optimization, and $E_{0}$ is the ground state energy of $\hat{H}^{(T)}$. The
behaviors of the energy as the cost function are similar to those with the
fidelity as the cost function [see figure 1(a) and figure 2(e)].

Second, in figures 1(b) and 3, for the 2D ferromagnetic TFIM with the geometry
of square lattice and system size $N=3\times 3$, the strength of transverse
field is chosen as $\lambda=3.05$. Here, we consider other values of the
transverse-field strength, including $\lambda=2.9$ and $3.2$. The results are
plotted in figure 10, and the behaviors of $1-f$ are similar to those with
$\lambda=3.05$ [see figure 1(b) and figure 3(a)].

Finally, we consider another 2D ferromagnetic TFIM different from the TFIM
with the geometry of square lattice. The target Hamiltonian reads
$\hat{H}^{(T)}=-\sum_{\langle
i,j\rangle}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}-\lambda\sum_{i=1}^{N}\hat{\sigma}_{i}^{x}$
with $\langle i,j\rangle$ referring to the nearest-neighbor sites $i$ and $j$
on the triangular lattice [see the inset of figure 11(a)] and $\lambda=2$
being the transverse-field strength. It is noted that the Ising model with
such triangular geometry has been experimentally realized in Rydberg atoms
Lienhard  _et al._ (2018). The results are plotted in figure 11. It is seen
that the PRH-QAOA can improve the performance of the VQE for 2D target
Hamiltonians with triangular geometry, which may enlightens further works
aiming to improve the performance of VQEs for frustrated quantum systems
Kattemölle and van Wezel (2021); Bosse and Montanaro (2022); Uvarov  _et al._
(2020).

##  Appendix C Quantum circuit representation of the PRH-QAOA protocols for
generating GHZ states

In this section, we show the quantum circuit representation of the PRH-QAOA
protocols with the 1D OBC geometry (see figure 12) and the cross-shape
geometry (see figure 13). The PRH-QAOA protocols consists of the unitary
evolution under Ising Hamiltonians
$U_{2}(x_{2}^{(n)},\mathbf{J})=\exp(ix_{2}^{(n)}\sum_{i=1}^{N}J_{i,i+1}\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z})$,
i.e., the Ising dynamics, and single-qubit rotations
$U_{1}(x_{1}^{(n)},\mathbf{h})=\exp(ix_{1}^{(n)}\sum_{i=1}^{N}h_{i}\hat{\sigma}_{i}^{x})$.
For digital-gate-model quantum circuit, the Ising dynamics
$U_{2}(x_{2}^{(n)},\mathbf{J})$ can be realized in two layers of two-qubit
gates defined in figure 12(b), whose decomposition is given by the quantum
circuit in figure 12(d). The single-qubit gate in figure 12(c) directly
realizes $U_{1}(x_{1}^{(n)},\mathbf{h})$.

Figure 12: (a) The quantum circuit representation of the PRH-QAOA for
generating GHZ states. Here, the 12-qubit 1D quantum circuit with OBC
corresponds to the results shown in figures 5(a)-(c). The initial state is
$\otimes_{i=1}^{N}|+\rangle_{i}$ with $|+\rangle$ being the eigenstate of
$\sigma^{x}$ with the eigenvalue $+1$. As an example, the depth of the circuit
is $p=2$. The quantum circuit consists of the two-qubit gate $ZZ(\phi)$ shown
in (b) and the single-qubit rotation $X_{\theta}$ shown in (c). (d) The gate
$ZZ(\phi)$ can be decomposed into two CNOT gates and one single-qubit gate
$R_{z}(2\phi)=e^{-i\phi\sigma^{z}}$.
Figure 13: (a) The quantum circuit representation of the PRH-QAOA for
generating 9-qubit GHZ states with the geometry in the figure 5(f). The
initial state $\otimes_{i=1}^{N}|+\rangle_{i}$, the two-qubit gate $ZZ(\phi)$,
and the single-qubit rotation $X_{\theta}$ are the same as those shown in
figure 12. The gate $I$ refers to the identity gate. (b) The qubits
$Q_{1}$-$Q_{9}$ are labeled, according to the results shown in the figures
5(e) and (f).

##  Appendix D Additional numerics of generating the Greenberg-Horne-Zeilinger
state with square lattice geometry

Figure 14: The infidelities $1-f$ with $f$ being the GHZ state fidelity,
obtained after optimization, as a function of the depth $p$. The inset shows a
schematic of a regular 2D lattice with $N=16$ spins (represented by circles).
For the OBC case, only the interactions represented by solid lines are
considered. For the PBC case, both the interactions represented by solid and
dashed lines are considered.

As shown in figure 6(a), one can generate the 17-qubit GHZ state with perfect
fidelity using the 4-depth quantum circuit of the QAOA with the cross-shape
geometry shown in figure 6(c). We now consider the Ising model on 2D square
lattice with a larger system size $N=4\times 4$ as the resource Hamiltonian,
and to generate 16-qubit GHZ state, both the conventional QAOA and PRH-QAOA
are studied. The results are plotted in figure 14. It is seen that for the
depth of quantum circuit $p\leq 4$, all of the conventional QAOA with the both
OBC and PBC quantum circuits, as well as the PRH-QAOA cannot achieve the
perfect fidelity of the 16-qubit GHZ state. When $p>4$, the PRH-QAOA can
generate higher fidelity GHZ states than those generated by the convention
QAOA. Nevertheless, employing the Ising model on 2D square lattice as the
resource Hamiltonian, for generating GHZ states, the performance of QAOA is
less efficient than that employing the the cross-shape geometry shown in
figures 6(b) and (c).

Figure 15: The infidelities $1-f$ as a function of the damping strength
$\Gamma$. The dashed line is the linear fitting for $\log(1-f)$ and
$\log\Gamma$, whose slope is $1.992$.

##  Appendix E The effect of decoherence

In this section, we numerically study the effect of decoherence on the PRH-
QAOA for generating the GHZ state. In particular, we consider the quantum
circuit with the cross-shape geometry and the system size $N=9$. As shown in
figures 5(d)-(f), the depth of the circuit for achieving the perfect fidelity
is $p=2$. Here, we consider the damping as an example of the effect of
decoherence.

To simulate the Ising dynamics with damping effect, we consider the Lindblad
master equation
$\frac{\text{d}\hat{\rho}(t)}{\text{d}t}=-i[\hat{H}_{zz},\hat{\rho}(t)]+\frac{1}{2}\sum_{j=1}^{N}[2\hat{C}_{j}\hat{\rho}(t)\hat{C}_{j}^{\dagger}-\\{\hat{C}_{j}^{\dagger}\hat{C}_{j},\hat{\rho}(t)\\}]$,
where $\hat{H}_{zz}=-\sum_{\langle
i,j\rangle}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}$ describing ferromagnetic
Ising interactions with the cross-shape geometry shown in figure 5(f), and
$C_{j}=\Gamma\hat{\sigma}_{j}^{-}$ is the collapse operator for the damping
effect with a strength $\Gamma$.

We display the results in figure 15. It is seen that the infidelity has a
power-law increase with the damping strength $\Gamma$, i.e.,
$1-f\propto\Gamma^{\alpha}$ with $\alpha\simeq 1.992$. For recently developed
quantum processors, the strength of damping $\Gamma$ is near $10^{-2}$ Wu  _et
al._ (2021); Arute  _et al._ (2019); de Léséleuc  _et al._ (2018), and
according to the results shown in figure 15, the achievable infidelity of the
GHZ state is $1-f\sim 10^{-4}$.

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