# Variational generation of spin squeezing on one-dimensional quantum devices
with nearest-neighbor interactions

Zheng-Hang Sun  Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences and CAS Center of Excellence for
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China  Yong-Yi Wang  Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Physical Sciences and CAS Center of
Excellence for Topological Quantum Computation, University of Chinese Academy
of Sciences, Beijing 100190, China  Yu-Ran Zhang<EMAIL_ADDRESS>School of Physics and Optoelectronics, South China University of Technology,
Guangzhou 510640, China  Theoretical Quantum Physics Laboratory, Cluster for
Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan  Center for
Quantum Computing, RIKEN, Wakoshi, Saitama 351-0198, Japan  Franco Nori
<EMAIL_ADDRESS>Theoretical Quantum Physics Laboratory, Cluster for Pioneering
Research, RIKEN, Wakoshi, Saitama 351-0198, Japan  Center for Quantum
Computing, RIKEN, Wakoshi, Saitama 351-0198, Japan  Physics Department,
University of Michigan, Ann Arbor, MI 48109-1040, USA  Heng Fan
<EMAIL_ADDRESS>Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences and CAS Center of Excellence for
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China  Songshan Lake Materials Laboratory, Dongguan 523808,
Guangdong, China  Beijing Academy of Quantum Information Sciences, Beijing
100193, China  Hefei National Laboratory, Hefei 230088, China

###### Abstract

Efficient preparation of spin-squeezed states is important for quantum-
enhanced metrology. Current protocols for generating strong spin squeezing
rely on either high dimensionality or long-range interactions. A key challenge
is how to generate considerable spin squeezing in one-dimensional systems with
only nearest-neighbor interactions. Here, we develop variational spin-
squeezing algorithms to solve this problem. We consider both digital and
analog quantum circuits for these variational algorithms. After the closed
optimization loop of the variational spin-squeezing algorithms, the generated
squeezing can be comparable to the strongest squeezing created from two-axis
twisting. By analyzing the experimental imperfections, the variational spin-
squeezing algorithms proposed in this work are feasible in recent developed
noisy intermediate-scale quantum computers.

###### pacs:

Valid PACS appear here

##  I INTRODUCTION

Spin squeezing plays a central role in the field of quantum metrology and the
characterization of quantum entanglement Pezzè  _et al._ (2018); _et al._
(2011). Generating highly spin-squeezed states is of fundamental interest
because of its promising applications in high-precision measurements
Giovannetti  _et al._ (2011); Gross  _et al._ (2010); Riedel  _et al._ (2010);
Wolfgramm  _et al._ (2010), such as the search for dark matter axions Backes
_et al._ (2021). Different from the Greenberger-Horne-Zeilinger state, which
is very fragile to noise Huelga  _et al._ (1997), spin-squeezed states are
more robust against amplitude-damping noise  _et al._ (2011); Wang  _et al._
(2010). A paradigmatic protocol for generating spin-squeezed states is one-
axis twisting (OAT) Kitagawa and Ueda (1993), which can be straightforwardly
realized in several experimental platforms with long-range interactions,
including atomic Bose-Einstein condensates Gross  _et al._ (2010); Riedel  _et
al._ (2010); Sebby-Strabley  _et al._ (2007); Muessel  _et al._ (2015);
Sørensen  _et al._ (2001); Strobel  _et al._ (2014), trapped ions Bohnet  _et
al._ (2016), Rydberg-dressed atoms Eckner  _et al._ (2023), and
superconducting circuits with all-to-all connectivity Song  _et al._ (2019);
Xu  _et al._ (2020, 2022).

Despite the enormous progress in generating squeezing via OAT, there are two
key challenges in this field. The first one is that OAT relies on the
infinite-range interactions, while only a few experimental platforms can
achieve sufficiently long-range or infinite-range interactions. Consequently,
exploring the possibility for generating highly spin-squeezed states in
systems with short-range interactions is highly desirable. The second
challenge is the realization of two-axis twisting (TAT), generating stronger
squeezing than OAT Kitagawa and Ueda (1993). Even though several schemes for
achieving TAT have been proposed Liu  _et al._ (2011); Huang  _et al._ (2015);
Hernández Yanes  _et al._ (2022); Wang  _et al._ (2017); Borregaard  _et al._
(2017), its experimental realization remains absent due to the limited
controllability and flexibility of artificial quantum systems.

Recent studies have shown that spin squeezing comparable to the best one
generated from OAT can be achieved in two- and three-dimensional short-range
systems Perlin  _et al._ (2020); Bornet _et al._ (2023); Block _et al._
(2023). It indicates that higher-dimensional systems have the potential to
yield stronger squeezing Gil  _et al._ (2014); Kaubruegger  _et al._ (2019).
Nevertheless, there remains a more challenging problem, i.e., whether _one-
dimensional_ (1D) quantum devices with only _nearest-neighbor_ interactions,
as one of the most experimentally achievable systems in quantum simulation,
can generate highly spin-squeezed states that are comparable to those from
OAT, or even from TAT.

In the past few years, tremendous developments on variational quantum
algorithms Cerezo  _et al._ (2021); Kokail  _et al._ (2019); McClean  _et al._
(2016); _et al._ (2018); Bharti  _et al._ (2022) opened new avenues for
improving the performance of quantum sensors Kaubruegger  _et al._ (2019); Huo
_et al._ (2022); Zheng  _et al._ (2022); Santra  _et al._ (2022); Kaubruegger
_et al._ (2021); Marciniak  _et al._ (2022), providing a method for overcoming
the aforementioned challenges. Here, by adopting the squeezing parameter
Gessner  _et al._ (2019) (lower value of the parameter corresponding to a
better spin-squeezed state) as the objective function, we propose variational
spin-squeezing algorithms on 1D quantum devices with nearest-neighbor
interactions to achieve not only OAT but also TAT.

Figure 1:  (a) A digital quantum computer consisting of a loop of
superconducting qubits connected with couplers. (b) Schematic of the
parameterized quantum circuit (PQC) based on the alternating layered ansatz
with $N=12$ qubits and the number of layers $p=2$. Here, the two-qubit gate
notation refers to the Fermionic Simulation (FSIM) gate defined by Eq. (8).
The initial state is chosen as $|\psi_{0}\rangle=|00...0\rangle$, where
$|0\rangle$ is the eigenstate of $\hat{\sigma}^{z}$ with the eigenvalue $-1$.
The variational parameters are denoted as $\vec{x}$. One layer of the PQC
contains two layers of entangler $\hat{U}_{E,1}$ and $\hat{U}_{E,2}$, each of
which is followed by a layer of single-qubit rotations. After measuring the
final state of the PQC, the value of objective function $\xi^{2}(\vec{x})$ can
be obtained and then is fed into the classical computer, where the
optimization algorithm is running, finding the next set of variational
parameters $\vec{x}_{\text{new}}$. Then, $\vec{x}_{\text{new}}$ is fed into
the quantum computer, completing one loop of the algorithm. Finally, by
iteratively running several loops, one can obtain the optimized objective
function $\xi^{2}_{\text{opt}}=\xi^{2}(\vec{x}_{\text{opt}})$. (c) The Husimi
$Q$ function for the optimized spin-squeezed states generated by the 12-qubit
PQC with $p=3$.

The performance of variational quantum algorithms greatly depends on the
choice of ansatz for designing the parameterized quantum circuit (PQC) Bharti
_et al._ (2022). Here, we first focus on a PQC based on the alternating
layered ansatz (ALA), which can be realized in several platforms of digital
quantum computers, ranging from near-term superconducting circuits Wu  _et
al._ (2021); Arute  _et al._ (2019); Neill  _et al._ (2021); Zhang  _et al._
(2022); Frey and Rachel (2022); Kandala  _et al._ (2017) [also see Fig. 1(a)
and (b)] to neutral Rydberg atoms Bluvstein  _et al._ (2022); Graham  _et al._
(2022). Flexible digital quantum computers allow us to implement gate-model-
based circuits using different ansatzes. We choose the ALA because of its
deterministic expressibility Nakaji and Yamamoto (2021) and ameliorated
barren-plateau phenomena Cerezo  _et al._ (2021). Then, we consider a
hardware-efficient PQC that is suitable for analog quantum computers, where
all qubits in the circuit can be entangled via the global unitary evolution of
a tailored Hamiltonian. We numerically study the performance of the
aforementioned two variational spin-squeezing algorithms. Remarkably, we find
that a spin-squeezed state with a squeezing parameter comparable to the lowest
one obtained from TAT can be generated from both the shallow-depth optimized
digital and analog PQCs. We also analyze the influence of experimental
imperfections, and explore the methods to reduce errors in the quantum circuit
of the variational spin-squeezing algorithms.

Figure 2:  (a) The optimized linear Ramsey squeezing parameter
$\xi^{2}_{L,\text{opt}}$ as a function of the depth $p$ of the PQC for
different numbers of qubits $N=10$, $12$, $14$, $16$, $18$, and $20$. The
inset of (a) shows the $\xi^{2}_{L,\textrm{opt}}$ for the deeper depth of the
PQC $p$ with $N=10$ and $N=12$. The dashed and dotted horizontal lines
represent the value of the fundamental limit of the linear Ramsey squeezing
parameter with $N=10$ and $N=12$, respectively. (b) The ratio of
$\xi^{2}_{L,\text{opt}}$ to the lowest squeezing parameter
$\xi^{2}_{L,\text{TAT}}$ obtained from TAT, i.e.,
$\xi^{2}_{L,\text{opt}}/\xi^{2}_{L,\text{TAT}}$, as a function of the depth
$p$. For each system size $N$, only the data with $p\leq p^{*}$ are plotted,
where $p^{*}$ denotes the smallest depth required for achieving
$\xi^{2}_{L,\text{opt}}/\xi^{2}_{L,\text{TAT}}<1$. The inset of (b) shows the
$p^{*}$ as a function of $N$, where the dashed line is the linear fitting. (c)
The optimized squeezing parameter $\xi^{2}_{L,\text{opt}}$ as a function of
the depth $p$ for different designs of the PQC with $N=12$. (d) The optimized
nonlinear Ramsey squeezing parameter $\xi^{2}_{NL,\text{opt}}$ as a function
of the depth $p$ for $N=10$ qubits. The dashed and dotted lines in (d)
represent the lowest nonlinear Ramsey squeezing parameter obtained from OAT
and TAT, respectively. Note that (a) and (c) have a vertical logarithmic axis.

##  II Variational spin-squeezing algorithm with digital quantum circuits

The workflow of variational spin-squeezing algorithms consists of four parts
(see Fig. 1): (i) the objective function; (ii) the PQC; (iii) measurement of
the squeezing parameter; and (iv) the classical optimizer. Here, for an
$N$-qubit system, the spin squeezing parameter, chosen as the objective
function, is defined as Gessner  _et al._ (2019)

$\displaystyle\xi^{2}[\hat{\rho},\hat{\mathbf{S}}]=\frac{N}{\lambda_{\text{max}}(\mathbb{M}[\hat{\rho},\hat{\mathbf{S}}])},$
(1)

where $\hat{\mathbf{S}}$ is a family of accessible operators, and
$\lambda_{\text{max}}$ is the maximum eigenvalue of the matrix
$\mathbb{M}[\rho,\hat{\mathbf{S}}]$. For the linear Ramsey squeezing parameter
$\xi^{2}_{L}$,
$\hat{\mathbf{S}}=\hat{\mathbf{S}}_{L}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$
($\hat{J}_{\alpha}=\frac{1}{2}\sum_{i=1}^{N}\hat{\sigma}_{\alpha}^{i}$,
$\alpha\in\\{x,y,z\\}$), and

$\displaystyle\mathbb{M}[\hat{\rho},\hat{\mathbf{S}}]=\mathbb{C}^{T}[\hat{\rho},\hat{\mathbf{S}}]\
\mathbb{V}^{-1}[\hat{\rho},\hat{\mathbf{S}}]\
\mathbb{C}[\hat{\rho},\hat{\mathbf{S}}],$ (2)

with

$\displaystyle\mathbb{V}_{ij}[\hat{\rho},\hat{\mathbf{S}}]=\langle\\{\hat{\mathbf{S}}_{i},\hat{\mathbf{S}}_{j}\\}\rangle/2-\langle\hat{\mathbf{S}}_{i}\rangle\langle\hat{\mathbf{S}}_{j}\rangle$
(3)

and

$\displaystyle\mathbb{C}[\hat{\rho},\hat{\mathbf{S}}]=-i\langle[\hat{\mathbf{S}}_{i},\hat{\mathbf{S}}_{j}]\rangle.$
(4)

The parameter $\xi^{2}_{L}$ can be easily measured in near-term quantum
computers Xu  _et al._ (2022), as discussed below in Appendix A. The
definition of the squeezing parameter (1) can be generalized to higher-order
nonlinear squeezing parameter, which characterizes the entanglement of non-
Gaussian states.

For digital quantum computers, we design the PQC enlightened by the ALA. As
shown in Fig. 1(b), the circuit is comprised of arbitrary single-qubit
rotations with the angles
$\vec{\omega}^{(k)}\equiv(\omega_{1}^{(k)},\omega_{2}^{(k)},\omega_{3}^{(k)})$,
i.e.,

$\hat{U}_{j}(\vec{\omega}^{(k)})=e^{-i\hat{\sigma}_{j}^{z}\omega_{1}^{(k)}}e^{-i\hat{\sigma}_{j}^{x}\omega_{2}^{(k)}}e^{-i\hat{\sigma}_{j}^{z}\omega_{3}^{(k)}},$
(5)

and two kinds of layers of Fermionic Simulation (FSIM) gates

$\hat{U}_{E,1}=\bigotimes_{i=1}^{N/2}\text{FSIM}(\theta,\phi)_{2i-1,2i}$ (6)

and

$\hat{U}_{E,2}=\bigotimes_{i=1}^{N/2}\text{FSIM}(\theta,\phi)_{2i,2i+1}$ (7)

being the entanglers, where $\text{FSIM}(\theta,\phi)_{j,j+1}$ refers to the
FSIM gate applied on the qubit pair consisting of the $j$-th and $(j+1)$-th
qubit with

$\text{FSIM}(\theta,\phi)={\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\cos\theta&-i\sin\theta&0\\\ 0&-i\sin\theta&\cos\theta&0\\\
0&0&0&e^{-i\phi}\end{array}\right)}.$ (8)

The FSIM gate in (8) plays a key role in the simulation of electronic
structures Kivlichan  _et al._ (2018); Neill  _et al._ (2021), and can provide
sufficient entanglement for demonstrating quantum advantage Wu  _et al._
(2021); Arute  _et al._ (2019). More importantly, a continuous set of high-
fidelity FSIM gates with different control angles $\theta$ and $\phi$ can be
experimentally realized Foxen  _et al._ (2020).

The adjustable couplers in Fig. 1(a) enable us to individually implement the
layer of entangler $\hat{U}_{E,1}$ and $\hat{U}_{E,2}$. Here, for simplicity,
we choose the same value of the angles $(\theta,\phi)$ in FSIM gates Eq. (8)
for different layers of entanglers $\hat{U}_{E,1}$ and $\hat{U}_{E,2}$, i.e.,
all the FSIM gates in the PQC shown in Fig. 1(b) are the same. For the layers
of single-qubit rotations, we adopt the protocol where the even (odd) qubits
have the same angle $\overrightarrow{\alpha}^{(P)}$
($\overrightarrow{\beta}^{(P)}$). The reason for choosing this protocol is
analyzed in Appendix B from the perspective of the expressibility Sim  _et
al._ (2019) of PQCs and the spatial symmetry.

Then, the final state after the $p$-layer PQC based on the ALA on the initial
state $|\psi_{0}\rangle=|00...0\rangle$ is written as

$\displaystyle|\psi(\vec{x})\rangle=\prod_{l=1}^{p}[\hat{U}_{s}^{(2l)}\hat{U}_{E,2}\hat{U}_{s}^{(2l-1)}\hat{U}_{E,1}]\hat{U}_{s}^{(0)}|\psi_{0}\rangle$
(9)

with

$\displaystyle\hat{U}_{s}^{(k)}\equiv\prod_{j\in\text{odd}}\hat{U}_{j}(\vec{\alpha}^{(k)})\prod_{j\in\text{even}}\hat{U}_{j}(\vec{\beta}^{(k)})$
(10)

being a layer of single-qubit rotations (5). Here,

$\displaystyle\hat{U}_{s}^{(0)}=\prod_{j\in\text{odd}}e^{-i\hat{\sigma}_{j}^{z}\alpha_{1}^{(0)}}e^{-i\hat{\sigma}_{j}^{x}\alpha_{2}^{(0)}}\prod_{j\in\text{even}}e^{-i\hat{\sigma}_{j}^{z}\beta_{1}^{(0)}}e^{-i\hat{\sigma}_{j}^{x}\beta_{2}^{(0)}}$
(11)

and the initial single-qubit rotation without the first $Z$ rotation in Eq.
(5), due to the chosen initial state $|\psi_{0}\rangle$. In the PQC with $p$
layers, in addition to four variational parameters in the initialization,
there are $12p$ variational parameters for the single-qubit rotations, and
only two variational parameters for the entanglers.

After introducing the PQC, we can implement the optimization algorithms, such
as the BFGS method Bharti  _et al._ (2022), to search for the optimum
parameters in the PQC for the state with the lowest squeezing parameter. We
present details of the optimization by using the BFGS method in Appendix C.
Note that the squeezing can be visualized via the Husimi $Q$ function
$Q(\Theta,\Phi)\propto|\langle\psi(\vec{x})|\Theta,\Phi\rangle|^{2}$ with
$|\Theta,\Phi\rangle\equiv\otimes_{j=1}^{N}(\cos\frac{\Theta}{2}|0\rangle_{j}+\sin\frac{\Theta}{2}e^{-i\Phi}|1\rangle_{j})$.
Figure 1(c) plots the $Q$ function of the optimized state obtained from the
variational spin-squeezing algorithm workflow with a depth of PQC $p=3$ and a
qubit number $N=12$. The squeezing parameter of the optimized spin-squeezed
state is $\xi_{\text{opt}}^{2}\simeq 0.2$, which is comparable to the lowest
value of the squeezing parameter for TAT, i.e., $\xi_{\text{TAT}}^{2}\simeq
0.24$ (see Appendix D for the definitions of OAT and TAT and related numerical
results).

We numerically simulate the variational spin-squeezing algorithm workflow with
the PQC based on the ALA, using the BFGS as the optimization algorithm for
total $12p+6$ variational parameters, and study the optimized linear Ramsey
squeezing parameter $\xi^{2}_{L,\text{opt}}$ as a function of the depth of the
PQC $p$. Here, we adopt the PQC with periodic boundary conditions (PBCs), as
displayed in Fig. 1(a) and (b).

Figure 2(a) shows the results with the system size up to $N=20$. It is shown
that for $p=1$ and $2$, the results for different $N$ give the same value of
$\xi^{2}_{L,\text{opt}}$. However, with increasing $p$, the values of
$\xi^{2}_{L,\text{opt}}$ diverge for different $N$. Actually,
$\xi^{2}_{L,\text{opt}}$ will approach the fundamental limit of the linear
Ramsey squeezing parameter $\xi^{2}_{\text{lim}}=2/(N+2)$ [see the inset of
Fig. 2(a) for the results of $\xi^{2}_{L,\text{opt}}$ with a deeper $p$
achieving $\xi^{2}_{\text{lim}}$]. Figure 2(b) displays
$\xi^{2}_{L,\text{opt}}/\xi^{2}_{L,\text{TAT}}$ as a function of $p$. We can
define the $p^{*}$ as the most shallow depth achieving
$\xi^{2}_{L,\text{opt}}/\xi^{2}_{L,\text{TAT}}<1$. We plot $p^{*}$ as a
function of $N$ in the inset of Fig. 2(b), showing a slow (approximately
linearly) growth of $p^{*}$ as $N$ increases.

Note that OAT or TAT, with infinite-range interactions (see Appendix D), can
also be realized by directly decomposing the infinite-range circuit into
nearest-neighbor ones, which is known as the _line swap strategy_ Weidenfeller
_et al._ (2022); Harrigan  _et al._ (2021). By using this strategy, a
$N$-qubit infinite-range circuit can be decomposed in $N$-layers of nearest-
neighbor two-qubit gates. Although the depth of the circuit based on the line
swap strategy also linearly scales with the qubit number $N$, to achieve OAT
or TAT via the variational spin-squeezing algorithms, the required depth of
the PQC is shallower [see the inset of Fig. 2(b)].

We then discuss whether a highly spin-squeezed state can be generated via the
variational algorithm without the PBC of the PQC. We first consider a simple
case where the PQC is the same as Fig. 1(b) but with open boundary conditions
(OBCs), i.e., the two-qubit gate $\text{FSIM}(\theta,\phi)_{1,N}$ is absent.
The results are displayed in Fig. 2(c), showing that the
$\xi^{2}_{L,\text{opt}}$ obtained from the PQC with OBC is worse than that for
the PQC with PBC.

Next, we consider a protocol, where the single-qubit rotations depends on the
index of qubits and the layers (see Appendix B for a schematic representation
of the PQC with site-dependent rotations). As shown in Fig. 2(c), with the
OBC, although one can obtain a lower value of $\xi^{2}_{L,\text{opt}}$ via the
protocol with site-dependent rotations, the variational algorithm with an
open-boundary PQC cannot generate an optimized spin-squeezed state with a
squeezing parameter comparable with the case of PBC, suggesting that the PBC
plays a key role in designing the PQCs for efficient variational spin-
squeezing algorithms. This can be interpreted by the fact that both the OAT
and TAT Hamiltonian, i.e., $\hat{H}_{\text{OAT}}\propto\hat{J}^{2}_{z}$ and
$\hat{H}_{\text{TAT}}\propto\hat{J}^{2}_{x}-\hat{J}^{2}_{y}$, have the PBC
with cyclic permutation symmetry. Thus, it is expected that the PQCs for
efficient variational spin-squeezing algorithms should fulfill the symmetry.

In addition, we consider the PQC, where the angles $(\theta,\phi)$ of the FSIM
gates (8) in each layer of entanglers are different, i.e., the protocol with
layer-dependent entanglers. We present the schematic of the PQC in Appendix B.
As displayed in Fig. 2(c), with PBC, the optimized squeezing parameters
$\xi^{2}_{L,\text{opt}}$ for the protocol with layer-dependent entanglers are
only sightly smaller than those for the conventional protocol. However, the
protocol with layer-dependent entanglers requires more experimental cost than
the protocol shown in Fig. 1(b) since the calibration of two-qubit FSIM gates
with different angles $(\theta,\phi)$ is more demanding than single-qubit
gates Barends  _et al._ (2019). Consequently, the PQC shown in Fig. 1(b) is
more experimentally feasible. In Appendix E, in order to guide subsequent
experimental investigations, we present the optimized angles
$(\theta_{\text{opt}},\phi_{\text{opt}})$ of the FSIM gates (8) for the
variational spin-squeezing algorithms with the PQC shown in Fig. 1(b).

The linear squeezing parameter can be generalized to the second-order non-
linear squeezing parameter by adopting
$\hat{\mathbf{S}}=\hat{\mathbf{S}}_{NL}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{J}_{x}^{2},\hat{J}_{y}^{2},\hat{J}_{z}^{2},\hat{J}_{xy}^{2},\hat{J}_{yz}^{2},\hat{J}_{zx}^{2})$
in Eq. (1), which can characterize the entanglement of non-Gaussian states
Gessner  _et al._ (2019). Here we consider the non-linear squeezing parameter
$\xi^{2}_{NL}$ as the objective function and employed the variational spin-
squeezing algorithm workflow with the PQC in Fig. 1(b) to obtain the optimized
non-linear squeezing parameter $\xi^{2}_{NL,\text{opt}}$.

As displayed in Fig. 2(d), for $N=10$, $\xi^{2}_{NL,\text{opt}}$ can achieve
the lowest value of $\xi^{2}_{NL}$ obtained from OAT and TAT with the depths
$p=4$ and $7$, respectively. In comparison with the linear squeezing
parameter, a deeper depth of the PQC is required for achieving the lowest
$\xi^{2}_{NL}$ of TAT. It has been verified that there is a hierarchy

$\displaystyle\xi^{-2}_{L}\leq\xi^{-2}_{NL}\leq F/N$ (12)

with $F$ being the quantum Fisher information Gessner  _et al._ (2019).
Consequently, the inverse of the second-order squeezing parameter
$\xi^{-2}_{NL}$ can estimate the _lower bound_ of $F/N$. For $p=7$,
$\xi^{-2}_{NL,\text{opt}}\simeq 6.87$. Because the violation of the inequality
$F/N\leq\kappa$ signals $(\kappa+1)$-partite entanglement Pezzè  _et al._
(2018), the optimized state with $p=7$ _at least_ has $7$-partite
entanglement.

Figure 3:  (a) Schematic of the PQC for the analog-HEA protocol with the
number of layers $p=2$. (b) Semilogarithmic plot of the optimized linear
Ramsey squeezing parameter $\xi^{2}_{L,\text{opt}}$ versus the depth $p$ of
the PQC for $N=10$. The dotted horizontal line is the saturated value of
$\xi^{2}_{L,\text{opt}}$ for the analog-HEA protocol, i.e.,
$\xi^{2}_{L,\text{opt}}\simeq 0.2$. The dashed horizontal line represents the
fundamental limit of the squeezing parameter with $N=10$, i.e.,
$\xi^{2}_{\text{lim}}\simeq 0.167$. (c) The $\xi^{2}_{L,\text{opt}}$ as a
function of the number of qubits $N$ with the depth of PQCs $p=5$. Here, ALA
refers to alternating layered ansatz.
Figure 4: The optimized linear Ramsey squeezing parameter
$\xi^{2}_{L,\textrm{opt}}$ obtained from the variational spin-squeezing
algorithm based on the PQC shown in Fig. 3(a) with depth $p=p^{*}$, for
different numbers $N$ of qubits, in comparison with the
$\xi^{2}_{\text{TAT}}$.
Figure 5: (a) Schematic of the PQC for the analog-HEA protocol consisting of
both the time evolution of the XY model $\hat{U}_{G}(t_{j})$ and the Ising
model $\hat{U}_{Z}(T_{j})$ (the analog-HEA-Ising protocol). Here, the depth of
the PQC is $p=2$. (b) The optimized linear Ramsey squeezing parameter
$\xi^{2}_{L,\text{opt}}$ versus the depth $p$ of the PQC for $N=10$. Here,
both the analog-HEA-Ising protocol and the analog-HEA protocol are considered.

##  III Variational spin-squeezing algorithm with analog quantum circuits

For the PQC as shown in Fig. 1(b), between two layers of single-qubit
rotations, the entanglers $\hat{U}_{E,1}$ and $\hat{U}_{E,2}$ only make local
qubit pairs entangled, as a typical digital quantum circuit. In contrast, for
the hardware-efficient ansatz (HEA) introduced in Ref. Kandala  _et al._
(2017), all qubits are entangled between two layers of single-qubit rotations,
as shown in Fig. 3(a). Here, instead of using a gate-model-based way to
construct the global entangler, i.e., $\hat{U}_{G}=\hat{U}_{E,1}\hat{U}_{E,2}$
Kandala  _et al._ (2017), we focus on the analog quantum process, where the
global entangler is realized by letting all qubits evolve under a tailored
Hamiltonian $\hat{H}_{T}$.

According to the results in Fig. 2(c), a PQC with PBC and layer-dependent
entanglers has a better performance. Thus, for the design of analog quantum
circuits based on the HEA, i.e., the analog-HEA protocol, we still adopt PBC
and the layer-dependent global entangler
$\hat{U}_{G}(t_{j})=\exp(-i\hat{H}_{T}t_{j})$ with variational parameters
$t_{j}$ ($j=1,2,...,p$ for a $p$-layer PQC). We consider 1D analog
superconducting circuits with nearest-neighbor interactions Kockum and Nori
(2019); Chen  _et al._ (2021); Zhu  _et al._ (2022); Yan  _et al._ (2019);
Roushan  _et al._ (2017); Kannan  _et al._ (2020); You  _et al._ (2007); Gu
_et al._ (2017) as an example, and the tailored Hamiltonian can be written as
(see Appendix F for details of the experimental realization)

$\hat{H}_{T}=\sum_{i=1}^{N}(\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+1}^{x}+\hat{\sigma}_{i}^{y}\hat{\sigma}_{i+1}^{y}),$
(13)

which is known as the 1D XY model. Overall, the final parameterized state
obtained from the $p$-layer PQC of the analog-HEA protocol is

$\displaystyle|\psi(\vec{x})\rangle=\prod_{l=1}^{p}[\hat{U}_{s}^{(l)}\hat{U}_{G}(t_{l})]\hat{U}_{s}^{(0)}|\psi_{0}\rangle,$
(14)

where the initial state $|\psi_{0}\rangle$ and the single-qubit rotations
$\hat{U}_{s}^{(l)}$ and $\hat{U}_{s}^{(0)}$ are the same as those in Eq. (9).
More precisely, the PQC
$\hat{U}(\vec{x})=\prod_{l=1}^{p}[\hat{U}_{s}^{(l)}\hat{U}_{G}(t_{l})]\hat{U}_{s}^{(0)}$
is an analog-digital circuit, because the layers of single-qubit rotations
$\hat{U}_{s}^{(l)}$ can be regarded as digital blocks Parra-Rodriguez  _et
al._ (2020).

With the linear Ramsey squeezing parameter $\xi^{2}_{L}$ being the objective
function, after the optimization, we can obtain the lowest squeezing parameter
$\xi^{2}_{L,\text{opt}}$ for the analog-HEA protocol. As shown in Fig. 3(b),
when increasing the depth $p$, the $\xi^{2}_{L,\text{opt}}$ obtained from the
ALA tends to the fundamental limit $\xi^{2}_{\text{lim}}$, while
$\xi^{2}_{L,\text{opt}}$ obtained from the analog-HEA protocol saturates to a
larger value, indicating that the variational spin-squeezing algorithm based
on the ALA outperforms with the analog-HEA protocol.

As shown in the inset of Fig. 2(b), TAT can be achieved by using the
variational spin-squeezing algorithm based on the ALA with the depth
$p^{*}=2$, $3$, $3$, $4$, $4$, and $5$ for $N=10$, $12$, $14$, $16$, $18$, and
$20$. Here, in Fig. 4, we plot the $\xi^{2}_{L,\textrm{opt}}$ obtained from
the analog-HEA protocol with $p=p^{*}$ for different $N$. It is seen that the
shallow-depth analog-HEA protocol can still generate a spin-squeezed state
better than that obtained from the TAT, which provides the possibility for
generating strong squeezing on near-term analog quantum computers.

Figure 6:  (a) The optimized linear Ramsey squeezing parameter
$\xi^{2}_{L,\text{opt}}$ obtained from the variational spin-squeezing
algorithm based on the ALA as a function of the number $N$ of qubits with
depth $p=5$ and different coherent errors $r$. (b) For $N=10$ and $r=0.005$,
the squeezing parameter $\xi^{2}_{L}$ versus iterations during the
optimization, for both the conventional ALA protocol and the ALA with site-
dependent rotations. (c) For $N=10$ and $p=5$, the optimized squeezing
parameter $\xi^{2}_{L,\text{opt}}$ with different coherent errors $r$. Here,
we consider both the conventional ALA protocol and the ALA with site-dependent
rotations. Note that (a) has a vertical logarithmic axis, and (b) has both
logarithmic axes.

The aforementioned analog-HEA protocol only considers the unitary evolution
under the 1D XY model (13), because it can be directly realized in analog
superconducting circuits Kockum and Nori (2019); Chen  _et al._ (2021); Zhu
_et al._ (2022); Yan  _et al._ (2019); Roushan  _et al._ (2017); Kannan  _et
al._ (2020); You  _et al._ (2007); Gu  _et al._ (2017) . For the ALA protocol
with the PQC shown in Fig. 1(b), the entanglers consist of the FSIM gates (8),
which can be rewritten as

$\displaystyle\text{FSIM}(\theta,\phi)=\exp[-i\frac{\theta}{2}(\hat{\sigma}^{x}\otimes\hat{\sigma}^{x}+\hat{\sigma}^{y}\otimes\hat{\sigma}^{y})]$
(15)
$\displaystyle\times\exp[-i\frac{\phi}{4}(\hat{I}-\hat{\sigma}^{z})\otimes(\hat{I}-\hat{\sigma}^{z})]$

($\hat{I}$ is the two-dimensional identity matrix). It is seen that both the
XY-type interactions
$\exp[-i\frac{\theta}{2}(\hat{\sigma}^{x}\otimes\hat{\sigma}^{x}+\hat{\sigma}^{y}\otimes\hat{\sigma}^{y})]$
and Ising interactions
$\exp(-i\frac{\phi}{4}\hat{\sigma}^{z}\otimes\hat{\sigma}^{z})$ exist in the
FSIM gates. However, for the analog-HEA protocol with the PQC in Fig. 3(a),
the Ising interactions are absent.

Next, we explore the variational spin-squeezing algorithm based on the PQC
shown in Fig. 5(a), with an additional entangler

$\displaystyle\hat{U}_{z}(T_{j})=\exp[-iT_{j}(\sum_{k=1}^{N}\hat{\sigma}^{z}_{k}\hat{\sigma}^{z}_{k+1})]$
(16)

as the global Ising interaction, i.e., the analog-HEA-Ising protocol. The
final parameterized state obtained from the $p$-layer PQC shown in Fig. 5(a)
is

$\displaystyle|\psi(\vec{x})\rangle=\Pi_{l=1}^{p}[\hat{U}_{s}^{(l)}\hat{U}_{G}(t_{l})\hat{U}_{z}(T_{l})]\hat{U}_{s}^{(0)}|\psi_{0}\rangle,$
(17)

where $|\psi_{0}\rangle$, $\hat{U}_{s}^{(0)}$, and $\hat{U}_{s}^{(l)}$ are the
same as those in Eq. (14).

In Fig. 5(b), we display the optimized squeezing parameter
$\xi^{2}_{L,\text{opt}}$ for the analog-HEA-Ising protocol, in comparison with
the analog-HEA protocol shown in Fig. 3(a). It is seen that the
$\xi^{2}_{L,\text{opt}}$ obtained from the analog-HEA-Ising protocol can
achieve lower values, which tend to the fundamental limit
$\xi^{2}_{\text{lim}}\simeq 0.167$ with increasing $p$. The results shown in
Fig. 5(b) indicate that the Ising interactions are indispensable for the
design of variational spin-squeezing algorithms, which can achieve the
fundamental limit of the linear Ramsey squeezing parameter.

##  IV Experimental imperfections

We now analyze the influence of experimental imperfections on variational
spin-squeezing algorithms. We focus on the algorithm based on the ALA, which
is shown in Fig. 1(b), since it outperforms the analog-HEA protocol with $p=5$
of the PQC [see Fig. 3(c)]. Here we study the influence of the coherent errors
in the FSIM gate (8). In experiments, the coherent errors can be induced by
the uncertainty of $\theta$ and $\phi$ in Eq. (8), and the additional phases
$\Delta_{+}$, $\Delta_{-}$, and $\Delta_{-,\text{off}}$ Mi  _et al._ (2021).
One can see Appendix G for the formulation of the FSIM gate with coherent
errors, i.e.,
$\text{FSIM}_{\text{exp.}}(\theta_{*},\phi_{*},\Delta_{+},\Delta_{-},\Delta_{-,\text{off}})$.
The coherent error can be quantified by

$\displaystyle
r=1-\frac{1}{16}|\text{Tr}(\text{FSIM}_{\text{exp.}}^{\dagger}\cdot\text{FSIM}_{\text{target}})|^{2},$
(18)

where the target FSIM gate is
$\text{FSIM}_{\text{target}}=\text{FSIM}(\theta_{\text{opt}},\phi_{\text{opt}})$
with the optimized parameters $(\theta_{\text{opt}},\phi_{\text{opt}})$
obtained from the variational algorithm without the coherent error, and
$\text{FSIM}_{\text{exp.}}$ represents the FSIM gate with the coherent error.

As shown in Fig. 6(a), with coherent errors, the squeezing generated by the
variational algorithm becomes weaker. When the coherent errors in the
entanglers are considered, the FSIM gates of different qubit pairs are not
identical (see Appendix G), and the partial permutation symmetry in the PQC as
shown in Fig. 1(b), where all the FSIM gates are the same, is absent for the
PQC with coherent errors. Consequently, according to the results of the PQC
with site-dependent rotations [see Fig. 2(c)], one can expect a lower
$\xi^{2}_{L,\text{opt}}$ by employing the PQC with site-dependent rotations
than its conventional version. As shown in Fig. 6(b), the optimization
trajectory for the saturation of $\xi^{2}_{L}$ to its optimized value
$\xi^{2}_{L,\text{opt}}$ is longer for the protocol with site-dependent
rotations compared to the conventional protocol, due to its larger number of
variational parameters. Moreover, the influence of the coherent errors on the
generated squeezing is suppressed by using the protocol with site-dependent
rotations [see Fig. 6(c)].

##  V DISCUSSIONS

We have developed variational spin-squeezing algorithms for 1D quantum devices
with nearest-neighbor interactions. We designed the PQC enlightened by the
ALA, suitable for digital quantum computers, as well as the PQC based on the
HEA, which can be naturally realized in analog superconducting circuits. We
demonstrated that variational spin-squeezing algorithms with both digital and
analog versions of the PQC can generate the spin-squeezed states comparable to
the best spin-squeezed state obtained from TAT. Our work sheds light on the
variational optimization of quantum sensors Kaubruegger  _et al._ (2021);
Marciniak  _et al._ (2022); Degen  _et al._ (2017) for 1D quantum devices with
nearest-neighbor interactions.

The variational spin-squeezing algorithms proposed in this work pave the way
for experimentally achieving the strongest squeezing generated from TAT. The
efficient numerical simulation of the PQC shown in Fig. 1(b) with PBC for
large system sizes is an intractable problem, because the matrix-product-
state-based algorithm is less efficient for PBC Schollwöck (2005).
Consequently, testing variational spin-squeezing algorithms on actual
experimental platforms is desirable, because it can extend the results shown
in the inset of Fig. 2(b) to larger system sizes and demonstrate the
scalability of variational spin-squeezing algorithms.

For the analog version of the variational spin-squeezing algorithm, the global
entangler is implemented by the unitary dynamics under the tailored
Hamiltonian that describes an array of superconducting qubits with nearest-
neighbor capacitive couplings. A further direction is to study the performance
of the variational spin-squeezing algorithm based on other experimental
platforms such as the Rydberg-dressed atoms Henriet  _et al._ (2020); Browaeys
and Lahaye (2020); Zeiher  _et al._ (2017); Borish  _et al._ (2020) and cavity
quantum electrodynamics systems Ballester  _et al._ (2012); _et al._ (2020);
Macrì  _et al._ (2020).

###### Acknowledgements.

We acknowledge the discussions with Markus Heyl and Xiao Yuan. This work was
supported by National Natural Science Foundation of China (Grants Nos.
92265207, T2121001, 11934018), Innovation Program for Quantum Science and
Technology (Grant No. 2-6), Beijing Natural Science Foundation (Grant No.
Z200009), and Scientific Instrument Developing Project of Chinese Academy of
Sciences (Grant No. YJKYYQ20200041), Nippon Telegraph and Telephone
Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via
the Quantum Leap Flagship Program (Q-LEAP), and the Moonshot R&D Grant Number
JPMJMS2061], the Asian Office of Aerospace Research and Development (AOARD)
(via Grant No. FA2386-20-1-4069), and the the office of Naval Research (ONR).

##  Appendix A Measurement of spin squeezing parameters

Figure 7: (a) A PQC with global single-qubit gates and PBC. (b) A PQC which is
similar to that shown in the Fig 1(b) but with the system size $N=10$. (c) A
PQC with site-dependent single-qubit gates and PBC. Here, the two-qubit gate
notation refers to the FSIM gate defined by Eq. (8).
Figure 8: (a) For the PQCs shown in Fig. 7, the expressibility as a function
of the depth of the PQC $p$. (b) The optimized linear Ramsey squeezing
parameter $\xi^{2}_{L,\text{opt}}$, obtained from the variational spin-
squeezing algorithms based on the the PQCs shown in Fig. 7, as a function of
$p$.
Figure 9: (a) A PQC which is similar to that shown in the Fig 7(b), but with
the open boundary condition (OBC). (b) A PQC which is similar to that shown in
the Fig 7(b), but with layer-dependent entanglers. (c) A PQC with site-
dependent single-qubit gates and OBC. Here, the two-qubit gate notation refers
to the FSIM gate defined by Eq. (8).

We first present the measurement scheme of the linear spin squeezing parameter
$\xi_{L}^{2}$ with a set of collective spin operators
$\hat{\mathbf{S}}_{L}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$. According to Eq.
(1), the parameter $\xi_{L}^{2}$ can be calculated with the matrix

$\displaystyle\mathbb{C}[\hat{\mathbf{S}}_{L}]=\begin{pmatrix}0&\langle\hat{J}_{z}\rangle&-\langle\hat{J}_{y}\rangle\\\
-\langle\hat{J}_{z}\rangle&0&\langle\hat{J}_{x}\rangle\\\
\langle\hat{J}_{y}\rangle&-\langle\hat{J}_{x}\rangle&0\end{pmatrix},$ (19)

and

$\displaystyle\mathbb{V}[\hat{\mathbf{S}}_{L}]=\begin{pmatrix}(\Delta\hat{J}_{x})^{2}&\text{cov}(\hat{J}_{x},\hat{J}_{y})&\text{cov}(\hat{J}_{x},\hat{J}_{z})\\\
\text{cov}(\hat{J}_{x},\hat{J}_{y})&(\Delta\hat{J}_{y})^{2}&\text{cov}(\hat{J}_{y},\hat{J}_{z})\\\
\text{cov}(\hat{J}_{x},\hat{J}_{z})&\text{cov}(\hat{J}_{y},\hat{J}_{z})&(\Delta\hat{J}_{z})^{2}\end{pmatrix}.$
(20)

For terms $\langle\hat{J}_{\gamma}\rangle$ and
$(\Delta\hat{J}_{\gamma})^{2}=\langle\hat{J}_{\gamma}^{2}\rangle-(\langle\hat{J}_{\gamma}\rangle)^{2}$
$(\gamma\in\\{x,y,z\\})$, the measurement requires the single-shot readout
along the axis $\hat{J}_{\gamma}$. Moreover, according to

$\displaystyle\text{cov}(\hat{J}_{\alpha},\hat{J}_{\beta})=\langle\hat{J}_{\alpha\beta}^{2}\rangle-\frac{\langle\hat{J}_{\alpha}^{2}\rangle+\langle\hat{J}_{\beta}^{2}\rangle}{2}-\langle\hat{J}_{\alpha}\rangle\langle\hat{J}_{\beta}\rangle$
(21)

with $\alpha,\beta\in\\{x,y,z\\}$ ($\alpha\neq\beta$) and
$\hat{J}_{\alpha\beta}=(\hat{J}_{\alpha}+\hat{J}_{\beta})\sqrt{2}$, the
measurement of the covariances $\text{cov}(\hat{J}_{x},\hat{J}_{y})$,
$\text{cov}(\hat{J}_{x},\hat{J}_{z})$ and
$\text{cov}(\hat{J}_{y},\hat{J}_{z})$ requires a single-shot readout along the
additional three axes $\hat{J}_{xy}$, $\hat{J}_{xz}$ and $\hat{J}_{yz}$. In
short, to obtain the linear Ramsey spin squeezing parameter $\xi_{L}^{2}$, one
should perform the single-shot readout measurements on each qubit along the
following axes:
$\\{\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{J}_{xy},\hat{J}_{yz},\hat{J}_{zx}\\}$.

Then, we present the measurement scheme of the nonlinear spin squeezing
parameter $\xi_{NL}^{2}$ with a set of collective spin operators
$\hat{\mathbf{S}}_{NL}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{J}_{x}^{2},\hat{J}_{y}^{2},\hat{J}_{z}^{2},\hat{J}_{xy}^{2},\hat{J}_{yz}^{2},\hat{J}_{zx}^{2})$.
The expression of the matrices $\mathbb{V}[\hat{\mathbf{S}}_{NL}]$ and
$\mathbb{C}[\hat{\mathbf{S}}_{NL}]$ are more complex. According to the
analytical results in Ref. Xu  _et al._ (2022), to measure the parameter
$\xi_{NL}^{2}$, one requires the single-shot readout measurements of the
following operators

$\displaystyle\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{J}_{xy},\hat{J}_{yz},\hat{J}_{zx},\hat{J}_{x\overline{y}},\hat{J}_{y\overline{z}},\hat{J}_{z\overline{x}}$
(22)

with
$\hat{J}_{\alpha\overline{\beta}}=(\hat{J}_{\alpha}-\hat{J}_{\beta})/\sqrt{2}$,
and $\hat{J}_{xy^{\prime}},\hat{J}_{yz^{\prime}},\hat{J}_{zx^{\prime}}$, with
$\hat{J}_{\alpha\beta^{\prime}}=(\hat{J}_{\alpha}+\sqrt{3}\hat{J}_{\beta})/\sqrt{2}$,
and
$\hat{J}_{x\overline{y}^{\prime}},\hat{J}_{y\overline{z}^{\prime}},\hat{J}_{z\overline{x}^{\prime}}$
with
$\hat{J}_{\alpha\overline{\beta}^{\prime}}=(\hat{J}_{\alpha}-\sqrt{3}\hat{J}_{\beta})/\sqrt{2}$,
and

$\displaystyle\hat{J}_{xyz}=\frac{\hat{J}_{x}+\hat{J}_{y}+\hat{J}_{z}}{\sqrt{3}},\hat{J}_{\overline{x}yz}=\frac{-\hat{J}_{x}+\hat{J}_{y}+\hat{J}_{z}}{\sqrt{3}},$
(23)
$\displaystyle\hat{J}_{x\overline{y}z}=\frac{\hat{J}_{x}-\hat{J}_{y}+\hat{J}_{z}}{\sqrt{3}},\hat{J}_{xy\overline{z}}=\frac{\hat{J}_{x}+\hat{J}_{y}-\hat{J}_{z}}{\sqrt{3}}.$
(24)

##  Appendix B Expressibility and performance of parameterized quantum
circuits with different designs

In this section, we study the expressibility of the parameterized quantum
circuits (PQCs) employed to perform variational spin squeezing algorithms. We
first introduce the method of calculating the expressibility defined in Ref.
Sim  _et al._ (2019). The PQC can be described as $\hat{U}(\vec{x})$, where
$\vec{x}$ is the variational parameter vector. The first step for calculating
the expressibility involves uniformly sampling 20,000 (here for $N=10$) sets
of the variational parameter. The second step requires computing the overlap
$F=|\langle\psi_{0}|\hat{U}(\vec{x}_{2})\hat{U}(\vec{x}_{1})|\psi_{0}\rangle|^{2}$,
with $\vec{x}_{1}$ and $\vec{x}_{2}$ denoting two different sets of the
variational parameter. The third step is obtaining the probability
distribution of $F$, i.e., $P(F)$. The final step is calculating the
expressibility defined as

Expr. $\displaystyle=$ $\displaystyle D_{\text{KL}}(P(F)||P_{\text{Haar}}(F))$
$\displaystyle=$
$\displaystyle\sum_{F}P(F)\ln[\frac{P(F)}{P_{\text{Haar}}(F)}],$

which is actually the Kullback-Liebler divergence between the distribution
$P(F)$ and the distribution corresponding to the Haar random states
$P_{\text{Haar}}(F)=(2^{N}-1)(1-F)^{2^{N}-2}$. The expressibility can quantify
the ability of a PQC to capture a rich class of quantum states. For a given
PQC, a smaller value of the Expr. defined by Eq. (B) indicates a higher
expressibility.

We now introduce some PQCs with periodic boundary conditions (PBCs) and
different designs of single-qubit rotations. In Fig. 7(a), we show the PQC
with global single-qubit gates, i.e., all qubits have the same single-qubit
rotations in each layer. The $k$-th layer of the single-qubit rotations in the
PQC as shown in Fig. 7(a) can be represented as

$\displaystyle\hat{U}_{s}^{(k)}(\vec{\alpha}^{(k)})=\Pi_{j=1}^{N}[e^{-i\hat{\sigma}_{j}^{z}\alpha_{1}^{(k)}}e^{-i\hat{\sigma}_{j}^{x}\alpha_{2}^{(k)}}e^{-i\hat{\sigma}_{j}^{z}\alpha_{3}^{(k)}}],$
(26)

with the site-independent angles of rotations
$\vec{\alpha}^{(k)}=(\alpha_{1}^{(k)},\alpha_{2}^{(k)},\alpha_{3}^{(k)})$.

In Fig. 7(b), we display the PQC with the same design of Fig. 1(b) in contrast
to other designs of the PQCs. In Fig. 7(c), we plot the PQC with site-
dependent single-qubit gates, i.e., all qubits could have different single-
qubit rotations in each layers. The $k$-th layer of the single-qubit rotations
in the PQC as shown in Fig. 7(c) can be represented as
$\hat{U}_{s}^{(k)}=\Pi_{j=1}^{N}[\hat{U}^{(k)}_{j}(\vec{\alpha}^{(k)}_{j})]$
with

$\displaystyle\hat{U}^{(k)}_{j}(\vec{\alpha}^{(k)}_{j})=e^{-i\hat{\sigma}_{j}^{z}\alpha_{1,j}^{(k)}}e^{-i\hat{\sigma}_{j}^{x}\alpha_{2,j}^{(k)}}e^{-i\hat{\sigma}_{j}^{z}\alpha_{3,j}^{(k)}},$
(27)

and the variational parameters
$\vec{\alpha}^{(k)}_{j}=(\alpha_{1,j}^{(k)},\alpha_{2,j}^{(k)},\alpha_{3,j}^{(k)})$
being the angles of rotations for the $j$-th qubit.

We then calculate the expressibility of the PQCs in Fig. 7, and plot the
results in Fig. 8(a). It is seen that the expressibility of the PQC in Fig.
7(b) is higher than that of the PQC with global single-qubit rotations [see
Fig. 7(a)], and the PQC with site-dependent single-qubit rotations [see Fig.
7(c)] has a much higher expressibility.

In Fig. 8(b)., we plot the optimized linear Ramsey squeezing parameter
$\xi^{2}_{L,\text{opt}}$ obtained from the variational algorithms based on
three different PQCs. For the variational spin-squeezing algorithm based on
the PQC in Fig. 7(b), the $\xi^{2}_{L,\text{opt}}$ is smaller than that for
PQC in Fig. 7(a), which indicates that the performance of variational spin-
squeezing algorithms is related to the expressibility of the chosen PQC, and a
high-expressibility PQC is required for the efficient generation of spin-
squeezed states.

Nevertheless, as shown in Fig. 8, although the expressibility of the PQC
displayed in Fig. 7(b) is lower than of the PQC in Fig. 7(c), the obtained
values of $\xi^{2}_{L,\text{opt}}$ are more or less the same. Thus, the
performance of variational spin-squeezing algorithms does not solely rely on
the expressibility of the chosen PQC.

Actually, the spatial symmetry also plays an important role in the design of
PQC. The major objective of the variational spin-squeezing algorithm is
generating squeezing comparable to the maximum squeezing created from the two-
axis twisting (TAT). The unitary evolution of the TAT has a cyclic permutation
symmetry (see the following section for more details). Meanwhile, the PQC as
shown in Fig. 7(b) can be regarded as a constrained version of the PQC in Fig.
7(c), with partial cyclic permutation symmetry imposed. Consequently, the PQC
shown in Fig. 7(b) and the unitary evolution of the TAT share a symmetry. The
symmetry can reduce the number of parameters in the PQC, and we mainly study
the variational spin-squeezing algorithm based on the form of PQC as shown in
Fig. 7(b).

In Fig. 2(d), we compare the performance of variational spin-squeezing
algorithms based on different PQCs. To more clearly represent the additional
PQCs in Fig. 2(d), we plot the schematics of the PQCs in Fig 9. The PQC shown
in Fig 9(a) and (c) are the open-boundary-condition version of the PQC shown
in Fig 7(a) and (c), respectively. The PQC as shown in Fig 9(b) is similar to
the PQC shown in Fig 7(b), but with layer-dependent entanglers. Specifically,
the entanglers in the $p$-th layer of the PQC are
$\hat{U}_{E,1}^{(p)}(\theta_{1}^{(p)},\phi_{1}^{(p)})=\otimes_{i=1}^{N/2}\text{FSIM}(\theta_{1}^{(p)},\phi_{1}^{(p)})_{2i-1,2i}$
and
$\hat{U}_{E,2}^{(p)}(\theta_{2}^{(p)},\phi_{2}^{(p)})=\otimes_{i=1}^{N/2}\text{FSIM}(\theta_{2}^{(p)},\phi_{2}^{(p)})_{2i,2i+1}$,
with the variational parameters
$(\theta_{1}^{(p)},\phi_{1}^{(p)},\theta_{2}^{(p)},\phi_{2}^{(p)})$ dependent
on the number of layers.

Figure 10: Linear Ramsey squeezing parameter $\xi_{L}^{2}$ versus iterations
during the optimization. This is for the system size $N=12$ and the $p=5$ PQC
depth. The solid curves show the optimization with 10 randomly chosen initial
parameters $\vec{x}_{0}^{(N=12)}$. The dotted curve shows the optimization
with $\vec{x}_{0}^{(N=12)}=\vec{x}_{\text{opt}}^{(N=10)}$ (also see the
inset).
Figure 11: (a) Linear spin squeezing parameter $\xi^{2}_{L}$ as a function of
the evolution time $\tau$ for the OAT and TAT with system size $N=20$. The
dashed and dotted horizontal lines represent the minimum $\xi^{2}_{L}$
generated by the OAT and TAT, i.e., $\xi_{\text{OAT}}^{2}=0.1979$ and
$\xi_{\text{TAT}}^{2}=0.1623$, respectively. (b) The minimum $\xi^{2}_{L}$
generated by the OAT and TAT, i.e., the $\xi^{2}_{\text{OAT}}$ and
$\xi^{2}_{\text{TAT}}$ as a function of the system size $N$.
Figure 12: (a) For the OAT, the time evolution of both the linear spin
squeezing parameter $\xi^{-2}_{L}$ and the second-order non-linear squeezing
parameter $\xi^{-2}_{NL}$ for system size $N=10$. (b) For the TAT, the time
evolution of $\xi^{-2}_{NL}$ with system size $N=10$.

##  Appendix C Details of the optimization procedure of variational spin-
squeezing algorithms

In this section, we present details of the optimization procedure. We employ
the BFGS method, as a gradient-based approach, to optimize the squeezing
parameters. For the optimization of the function $f(\vec{x})$, with
$\vec{x}=(x_{1},x_{2},...,x_{m})$ being the variational parameters, the
gradients can be computed using forward finite differences, i.e.,
$\partial_{j}f(\vec{x})\simeq[f(\vec{x}+\epsilon\vec{e}_{j})-f(\vec{x})]/\epsilon$,
where $\vec{e}_{j}$ is the unit vector with 1 as its $j$-th element and 0
otherwise, and $\epsilon$ denotes the finite-difference step size. Here, we
chose $\epsilon=10^{-8}$, which has been found to effectively suppress
statistical fluctuations in our applications. For each step of the
optimization procedure, in order to obtain the gradient, the number of
repetitions for the calculation of $f(\vec{x})$ is $m$.

It has been recognized that the BFGS method looks for local minima. To avoid
the influence of local minima, for systems with $N=10$, we perform 1000
optimizations using randomly chosen initial parameters $\vec{x}_{0}$, and
extract the minimum from 1000 optimization trajectories. The optimized
variational parameter for $N=10$ is denoted as
$\vec{x}_{\text{opt}}^{(N=10)}$. For larger system size $N=12$, we find that
$\vec{x}_{\text{opt}}^{(N=10)}$ is an appropriate initial parameter for the
optimization procedure. Here, we consider the PQC shown in Fig. 1(b) with
depth $p=5$ as an example. As shown in Fig. 10, with the initial parameter
$\vec{x}_{0}^{(N=12)}=\vec{x}_{\text{opt}}^{(N=10)}$, the squeezing parameter
$\xi_{L}^{2}$ tends to a lower value during the optimization, in comparison
with 10 randomly chosen $\vec{x}_{0}^{(N=12)}$. Thus, we adopt the initial
parameter $\vec{x}_{0}^{(N)}=\vec{x}_{\text{opt}}^{(N-2)}$ to efficiently
perform the optimization with system sizes $N\geq 12$.

##  Appendix D One-axis twisting and two-axis twisting

In this section, we first briefly introduce the one-axis twisting (OAT) and
TAT. The OAT and TAT Hamiltonian can be written as
$\hat{H}_{\text{OAT}}=\hat{J}_{z}^{2}$ and
$\hat{H}_{\text{TAT}}=\hat{J}_{x}^{2}-\hat{J}_{y}^{2}$, respectively  _et al._
(2011); Kitagawa and Ueda (1993). The OAT is the dynamical process described
by the unitary evolution under $\hat{H}_{\text{OAT}}$, i.e.,
$\hat{U}_{\text{OAT}}(\tau)=\exp(-i\hat{H}_{\text{OAT}}\tau)$. For the OAT,
the initial state is chosen as
$|\psi_{0}\rangle_{\text{OAT}}=\otimes_{j=1}^{N}|+\rangle_{j}$, with
$|+\rangle$ being the eigenstate of $\hat{\sigma}^{x}$ with the eigenvalue
$+1$. The non-equilibrium state generated from the OAT is
$|\psi(\tau)\rangle_{\text{OAT}}=\hat{U}_{\text{OAT}}(\tau)|\psi_{0}\rangle_{\text{OAT}}$.
Similarly, for the TAT, the unitary evolution is
$\hat{U}_{\text{TAT}}(\tau)=\exp(-i\hat{H}_{\text{TAT}}\tau)$, and
$|\psi(\tau)\rangle_{\text{TAT}}=\hat{U}_{\text{TAT}}(\tau)|\psi_{0}\rangle_{\text{TAT}}$,
with the chosen initial state for the TAT being
$|\psi_{0}\rangle_{\text{TAT}}=\otimes_{j=1}^{N}|0\rangle_{j}$.

We then can calculate the linear spin squeezing parameter defined in Eq. (1)
with $\hat{\textbf{S}}_{L}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$, i.e.,
$\xi^{2}_{L}$, for the OAT states $|\psi(\tau)\rangle_{\text{OAT}}$ and the
TAT states $|\psi(\tau)\rangle_{\text{TAT}}$. In Fig. 11(a), we plot the
$\xi^{2}_{L}$ as a function of the evolved time $\tau$ for both the OAT and
TAT with system size $N=20$. We note that the minimum $\xi^{2}_{L}$, signaling
the best spin-squeezed state generated by the OAT and TAT, can be achieved at
the time $\tau_{\text{OAT}}^{*}$ and $\tau_{\text{TAT}}^{*}$, respectively. We
denote the minimum $\xi^{2}_{L}$ generated by the OAT and TAT as
$\xi^{2}_{\text{OAT}}$ and $\xi^{2}_{\text{TAT}}$, respectively. In Fig.
11(b), we plot the $\xi^{2}_{\text{OAT}}$ and $\xi^{2}_{\text{TAT}}$ with
different system sizes $N$, showing that the TAT can generate stronger
squeezing than the OAT.

Finally, for the OAT and TAT, we calculate the second-order non-linear
squeezing parameter defined in the Eq. (1) with
$\hat{\textbf{S}}_{NL}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z},\hat{J}_{x}^{2},\hat{J}_{y}^{2},\hat{J}_{z}^{2},\hat{J}_{xy}^{2},\hat{J}_{yz}^{2},\hat{J}_{zx}^{2})$.
In Fig. 12(a), we show the time evolution of $\xi^{-2}_{NL}$ in comparison
with the $\xi^{-2}_{NL}$ in the OAT, demonstrating that
$\xi^{-2}_{NL}\geq\xi^{-2}_{L}$. In Fig. 12(b), we display the dynamics of
$\xi^{-2}_{NL}$ in the TAT, showing that the TAT can also generate a state
with larger $\xi^{-2}_{NL}$ than the OAT.

Figure 13: The entanglement power of FSIM gates, i.e.,
$\text{EP}[\text{FSIM}(\theta,\phi)]$, as a function of $\theta$ and $\phi$.
The values of $\theta_{\text{opt}}$ and $\phi_{\text{opt}}$ in Table I are
marked.
Figure 14: Schematic of the experimental pulse sequence corresponding to the
PQC shown in Fig. 3(a). After initialization, all qubits are tuned to the
working point by a pulse lasting a time $t_{1}$, and the unitary evolution
$\hat{U}(t_{1})$ can be realized. Next, all qubits are tuned to the idle point
and single-qubit pulses are applied for single-qubit rotations.

##  Appendix E Details of the optimized FSIM gates

In Table I, we present the optimized angles
$(\theta_{\text{opt}},\phi_{\text{opt}})$ of FSIM gates (8) in the PQC shown
in Fig. 1(b) with different depth $p$ and system size $N$. For each $p$, one
can see a weak dependence of $\theta_{\text{opt}}$ and $\phi_{\text{opt}}$ on
$N$. However, the values of $\theta_{\text{opt}}$ and $\phi_{\text{opt}}$ are
sensitive to the depth of the PQC $p$. For $p=1$ and $2$,
$\theta_{\text{opt}}>\phi_{\text{opt}}$, while for $p\geq 3$,
$\theta_{\text{opt}}<\phi_{\text{opt}}$.

$p=1$ | $N=10$ | $N=12$ | $N=14$ | $N=16$ | $N=18$ | $N=20$
---|---|---|---|---|---|---
$\theta_{\text{opt}}$ | 0.7380 | 0.7380 | 0.7380 | 0.7380 | 0.7380 | 0.7380
$\phi_{\text{opt}}$ | 0.5256 | 0.5256 | 0.5256 | 0.5256 | 0.5256 | 0.5256
$p=2$ | $N=10$ | $N=12$ | $N=14$ | $N=16$ | $N=18$ | $N=20$
$\theta_{\text{opt}}$ | 0.7468 | 0.7474 | 0.7470 | 0.7470 | 0.7470 | 0.7470
$\phi_{\text{opt}}$ | 0.2542 | 0.2543 | 0.2544 | 0.2544 | 0.2544 | 0.2544
$p=3$ | $N=10$ | $N=12$ | $N=14$ | $N=16$ | $N=18$ | $N=20$
$\theta_{\text{opt}}$ | 1.3200 | 1.3242 | 1.3287 | 1.3287 | 1.3287 | 1.3288
$\phi_{\text{opt}}$ | 3.1449 | 3.1494 | 3.1408 | 3.1408 | 3.1408 | 3.1439
$p=4$ | $N=10$ | $N=12$ | $N=14$ | $N=16$ | $N=18$ | $N=20$
$\theta_{\text{opt}}$ | 1.8320 | 1.7906 | 1.7746 | 1.7693 | 1.7683 | 1.7683
$\phi_{\text{opt}}$ | 3.2124 | 3.1775 | 3.1589 | 3.1614 | 3.1614 | 3.1614
$p=5$ | $N=10$ | $N=12$ | $N=14$ | $N=16$ | $N=18$ | $N=20$
$\theta_{\text{opt}}$ | 1.3175 | 1.3482 | 1.3778 | 1.3928 | 1.3955 | 1.3955
$\phi_{\text{opt}}$ | 3.1379 | 3.0976 | 3.0768 | 3.1018 | 3.0865 | 3.0865
Table 1: Optimized values of $\theta$ and $\phi$ of the FSIM gate for
different values of $p$ and $N$.

To quantify the entanglement of the FSIM gates defined by Eq. (8), we consider
the entanglement power (EP) Zanardi  _et al._ (2000), which is express as

$\displaystyle\text{EP}(\hat{U})=\overline{E(\hat{U}|\psi_{1}\rangle\otimes|\psi_{2}\rangle)},$
(28)

where $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are single-qubit states
uniformly sampled on the Bloch sphere, the top bar denotes the average over to
all states $|\psi_{1}\rangle\otimes|\psi_{2}\rangle$, and
$E(|\Psi\rangle)=1-\text{Tr}(\hat{\rho}_{1})$
($\hat{\rho}_{1}=\text{Tr}_{2}|\Psi\rangle\langle\Psi|$) being the linear
entropy. The entanglement power of the FSIM gate $\text{FSIM}(\theta,\phi)$
can be analytically derived as $\text{EP}[\text{FSIM}(\theta,\phi)]=-\cos
2\theta(\cos 2\theta+\cos\phi)/18+1/6$. The maximum value of
$\text{EP}[\text{FSIM}(\theta,\phi)]$ is $2/9$. We plot
$\text{EP}[\text{FSIM}(\theta,\phi)]$ as a function of $\theta$ and $\phi$ in
Fig. 13. One can see that for all $p=1,2,...,5$, the optimized values
$(\theta_{\text{opt}},\phi_{\text{opt}})$ do not lie in the region with
maximum entanglement power. Consequently, the variational spin-squeezing
algorithm with the PQC shown in Fig. 1(b) does not require FSIM gates with
large entanglement power.

##  Appendix F Experimental realization of the global entanglers by using
analog superconducting circuits

In this section, we will present more details for the experimental realization
of the global entanglers under the time evolution of the Hamiltonian
$\hat{H}_{T}=\sum_{i=1}^{N}(\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+1}^{x}+\hat{\sigma}_{i}^{y}\hat{\sigma}_{i+1}^{y})$
by using superconducting circuits. Actually, a loop of superconducting qubits
coupled via a fixed capacitor can be described by the Bose-Hubbard model Chen
_et al._ (2021); Zhu  _et al._ (2022); Yan  _et al._ (2019); Roushan  _et al._
(2017)

$\displaystyle\hat{H}_{\textrm{BH}}=$ $\displaystyle J$
$\displaystyle\sum_{i=1}^{N}(\hat{a}_{i}^{\dagger}\hat{a}_{i+1}+\textrm{H.c.})$
$\displaystyle+$
$\displaystyle\frac{U}{2}\sum_{i=1}^{N}\hat{n}_{i}(\hat{n}_{i}-1)+\sum_{i=1}^{N}\mu_{i}\hat{n}_{i},$

where $\hat{n}_{i}=\hat{a}_{i}^{\dagger}\hat{a}_{i}$, $\hat{a}_{i}$
($\hat{a}_{i}^{\dagger}$) denotes the bosonic annihilation (creation)
operator, $J$ is the strength of the hopping interaction, $U$ is the nonlinear
interaction, and $\mu_{i}$ is the chemical potential.

In a typical superconducting quantum processor, the ratio $|U|/J$ is quite
large. As an example, $|U|/2\pi\simeq 230$ MHz, $J/2\pi\simeq 12$ MHz, and
$|U|/J\simeq 19$ for the quantum device in Ref. Zhu  _et al._ (2022) . In this
situation, the Bose-Hubbard model Eq. (F) is approximate to the hard-core
limit, where the bosonic operators $\hat{a}_{i}$ and $\hat{a}_{i}^{\dagger}$
are replaced by the spin raising and lowering operators, respectively. The
Hamiltonian with the hard-core limit (also known as the $XY$ model) can be
written as

$\displaystyle\hat{H}_{XY}=J\sum_{i=1}^{N}(\hat{\sigma}_{i}^{+}\hat{\sigma}_{i+1}^{-}+\textrm{H.c.})+\sum_{i=1}^{N}\mu_{i}\hat{\sigma}_{i}^{+}\hat{\sigma}_{i}^{-}.$
(30)

When we tune all qubits to the same working point (see Fig. 14), the chemical
potentials satisfy $\mu_{i}=0$, and the unitary evolution
$\hat{U}(t)=\exp(-i\hat{H}_{XY}\tau)=\exp[-i\sum_{j=1}^{N}(\hat{\sigma}_{j}^{x}\hat{\sigma}_{j+1}^{x}+\hat{\sigma}_{j}^{y}\hat{\sigma}_{j+1}^{y})t]$,
with $t=J\tau/2$ can be experimentally realized. An experimental pulse
sequence with control waveforms corresponding to the PQC shown in Fig. 3(a) is
depicted in Fig. 14.

##  Appendix G FSIM gates with coherent errors

The FSIM gate with coherent errors can be written as Mi  _et al._ (2021)

$\displaystyle\text{FSIM}_{\text{exp.}}$ $\displaystyle=$
$\displaystyle\text{FSIM}_{\text{exp.}}(\theta_{*},\phi_{*},\Delta_{+},\Delta_{-},\Delta_{-,\text{off}})$
(31) $\displaystyle=$ $\displaystyle\begin{pmatrix}1&0&0&0\\\
0&u_{22}&u_{23}&0\\\ 0&u_{32}&u_{33}&0\\\ 0&0&0&u_{44}\end{pmatrix},$ (32)

where

$\displaystyle u_{22}=\exp\\{i(\Delta_{+}+\Delta_{-})\\}\cos\theta_{*},$ (33)
$\displaystyle
u_{23}=-i\exp\\{i(\Delta_{+}-\Delta_{-,\text{off}})\\}\sin\theta_{*},$ (34)
$\displaystyle
u_{32}=-i\exp\\{i(\Delta_{+}+\Delta_{-,\text{off}})\\}\sin\theta_{*},$ (35)
$\displaystyle u_{33}=\exp\\{i(\Delta_{+}-\Delta_{-})\\}\cos\theta_{*},$ (36)
$\displaystyle u_{44}=\exp\\{i(2\Delta_{+}-\phi_{*})\\}.$ (37)

The influence of coherent errors can be analyzed by simulating the PQC with
the FSIM gates described by Eq. (32). Taking the PQC based on the ALA shown in
Fig. 1(b) with $N=12$ and the degree of coherent error $r=0.005$ as an
example, we can numerically generate a series of FSIM gates, i.e.,
$\text{FSIM}_{\text{exp.}}^{(n,m)}=\text{FSIM}_{\text{exp.}}(\theta_{*}^{(n,m)},\phi_{*}^{(n,m)},\Delta_{+}^{(n,m)},\Delta_{-}^{(n,m)},\Delta_{-,\text{off}}^{(n,m)})$
as the FSIM gate between the $n$-th and $m$-th qubit (here $m=n+1$, for
$n=1,2,...,11$, and $m=1$ for $n=12$), which satisfies $r=0.005$. It is noted
that the parameters
$(\theta_{*}^{(n,m)},\phi_{*}^{(n,m)},\Delta_{+}^{(n,m)},\Delta_{-}^{(n,m)},\Delta_{-,\text{off}}^{(n,m)})$
are not necessarily identical for different $n$. After generating the FSIM
gates with coherent errors, we can perform the optimization algorithm for the
parameters in the single-rotations, and obtain the optimized squeezing
parameter for the PQC with coherent errors.

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