# The principle of learning sign rules by neural networks in qubit lattice
models

Jin Cao  Department of Physics and Center for Advanced Quantum Studies,
Beijing Normal University, Beijing 100875, China  Shijie Hu
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100193, China  Department of Physics, Beijing Normal University, Beijing
100875, China  Zhiping Yin  Department of Physics and Center for Advanced
Quantum Studies, Beijing Normal University, Beijing 100875, China  Ke Xia
Department of Physics, Southeast University, Nanjing 211189, China

###### Abstract

A neural network is a powerful tool that can uncover hidden laws beyond human
intuition. However, it often appears as a black box due to its complicated
nonlinear structures. By drawing upon the Gutzwiller mean-field theory, we can
showcase a principle of sign rules for ordered states in qubit lattice models.
We introduce a shallow feed-forward neural network with a single hidden neuron
to present these sign rules. We conduct systematical benchmarks in various
models, including the generalized Ising, spin-$1/2$ XY, (frustrated)
Heisenberg rings, triangular XY antiferromagnet on a torus, and the Fermi-
Hubbard ring at an arbitrary filling. These benchmarks show that all the
leading-order sign rule characteristics can be visualized in classical forms,
such as pitch angles. Besides, quantum fluctuations can result in an imperfect
accuracy rate quantitatively.

Decoding hidden information from the ground-state wave function is essential
for understanding the properties of quantum closed systems at zero
temperature, including orders, correlations, and even intricate entanglement
features, etc Eisert  _et al._ (2010); Chertkov and Clark (2018); Wang  _et
al._ (2019); Irkhin and Skryabin (2019). For a real Hamiltonian, the sign
structure of elements in the real wave function can be summarized as a sign
rule within a selected representation Grover and Fisher (2015). For example,
the Perron-Frobenius theorem is applied to a class of Hamiltonians with non-
positive off-diagonal elements Perron (1907); Frobenius (1909). The Marshall-
Peierls rule (MPR) is another example applicable to antiferromagnetic spin
models on bipartite lattices Marshall (1955); Lieb  _et al._ (1961); Lieb and
Mattis (1962). These sign rules have been believed to be connected to various
physical phenomena, such as the volume law for the Rényi entanglement
entropies Grover and Fisher (2015), spatial periodicity of states Zeng and
Parkinson (1995); Bursill  _et al._ (1995), phase transitions Retzlaff  _et
al._ (1993); Richter  _et al._ (1994); Cai and Liu (2018); Westerhout  _et
al._ (2020), and so on.

Similar to the matrix product state (MPS) successfully applied to (quasi) one-
dimensional (1D) lattice models White (1992); Peschel  _et al._ (1999);
Schollwöck (2011); Orús (2014), the neural network quantum state (NNQS) and
fast-developing machine learning (ML) techniques provide a new approach for
multi-scale compression of the wave function, which has been widely used in
one and higher dimensional quantum many-body systems Carleo and Troyer (2017);
Carleo  _et al._ (2019); Jia  _et al._ (2019); Vivas  _et al._ (2022). By
using the empirical activation function cosine in the hidden layer of NNQS,
the complicated sign rules in qubit lattice models can be learned from the
wave functions Cai and Liu (2018), and subsequent studies have drawn
significant attention in recent years Choo  _et al._ (2019); Westerhout  _et
al._ (2020); Szabó and Castelnovo (2020); Bukov  _et al._ (2021). These
studies have shown that it is a practical advantage to enhance the
representation precision for complex sign rules Cai and Liu (2018); Choo  _et
al._ (2019); Westerhout  _et al._ (2020); Szabó and Castelnovo (2020); Bukov
_et al._ (2021). This can be achieved by adding more hidden layers/neurons in
NNQS or designing new architectures. Meanwhile, there is a growing concern
about interpreting the meaning of highly nonlinear structures in neural
networks Roscher  _et al._ (2020); He  _et al._ (2020); Fan  _et al._ (2021)
and finding links to existing physical insights Raissi  _et al._ (2019); Yuan
and Weng (2021); Cai  _et al._ (2021), which strongly motivates our work.

In this work, we establish a Gutzwiller mean-field (GWMF) principle of the
sign rules for ordered ground states in qubit lattice models. The leading-
order term can be well understood using a single-hidden-neuron feed-forward
neural network (shn-FNN). Our findings, tested on various spin and fermion
models, suggest that the leading-order sign rules have clear physical
interpretations tightly related to orders in spins or charges. The structure
of the paper is organized as follows: In Sec. I, we present the GWMF picture
of the sign rules for ordered states in qubit lattice models. In Sec. II, we
introduce shn-FNN in detail, which matches the GWMF picture and can be easily
interpreted. In Sec. III, we demonstrate the technical details of data set
preparation and shn-FNN training. In Sec. IV, we apply shn-FNN to extract the
leading-order sign rules in spin and Fermi-Hubbard models. We also discuss the
influence of frustration and global symmetries. At last, we summarize
conclusions and make a discussion briefly in Sec. V.

##  I Gutzwiller mean-field theory

A qubit is commonly used to represent a quantum state $\rvert n\rangle$ in
various fields of condensed matter physics. Examples include a spin-$1/2$ in
quantum magnets Vasiliev  _et al._ (2018), a single fermion state in ultra-
cold atomic systems Gardiner and Zoller (2017), a two-level atom in quantum
cavities Meher and Sivakumar (2022), and so on Makhlin  _et al._ (2001); Luo
(2008); Kjaergaard  _et al._ (2020). The binary value $n$ corresponds to an
empty or occupied fermion level, or a spin-$1/2$ polarizing $\uparrow$ or
$\downarrow$ in the z-axis. For a lattice with $L$ qubits, the basis can be
expressed as $\rvert\mathbf{n}\rangle=\otimes^{L}_{l=1}\rvert n_{l}\rangle$,
where $\rvert n_{l}\rangle$ represents the local basis at site-$l$, and the
quantum indices $n_{l}=0$, $1$ form a vector $\mathbf{n}=(n_{1},\ \dots,\
n_{L})$.

Without loss of generality, let us consider a spin-$1/2$ as an example. A spin
operator $\hat{\mathbf{S}}_{l}=(\hat{S}^{x}_{l}$, $\hat{S}^{y}_{l}$,
$\hat{S}^{z}_{l})$ defined at site-$l$ has three components in the $x$, $y$
and $z$-axes, respectively. For any state, there are only two free real
variables out of two complex coefficients in front of the basis
$\rvert\sigma_{l}\rangle$ of the $\hat{S}^{z}_{l}$-representation. The index
$\sigma_{l}=\uparrow$, $\downarrow$, corresponding to values of $\pm 1/2$.
These variables are governed by a pair of site-dependent angles $\theta_{l}$
and $\phi_{l}$ appearing in a spin-coherent state Penc and Läuchli (2010)

$\displaystyle\rvert\Omega_{l}\rangle=c^{\uparrow}_{l}\left\rvert\uparrow\right\rangle+c^{\downarrow}_{l}\left\rvert\downarrow\right\rangle\
,$ (1)

where the coefficients are given by

$\displaystyle c^{\uparrow}_{l}=\cos(\theta_{l}/2)\quad\text{and}\quad
c^{\downarrow}_{l}=\sin(\theta_{l}/2)e^{i\phi_{l}}\ .$ (2)

As convention, $\theta_{l}\in[0$, $\pi]$ and $\phi_{l}\in[0$, $2\pi)$. In such
a state,
$\mathbf{S}_{l}=\langle\hat{\mathbf{S}}_{l}\rangle=\mathbf{\Omega}_{l}/2$
behaves as half of the unit vector in three-dimensional coordinates, that is,

$\displaystyle\mathbf{\Omega}_{l}=(\sin\theta_{l}\cos\phi_{l},\
\sin\theta_{l}\sin\phi_{l},\ \cos\theta_{l})\ .$ (3)

The phase factor for the basis $\rvert\sigma_{l}\rangle$ with a non-vanishing
amplitude only depends on $\phi_{l}$, since both $\cos(\theta_{l}/2)$ and
$\sin(\theta_{l}/2)$ are positive. Besides, a phase angle $h_{l}$ can modulate
the phase factor in front of the spin-coherent state (1), i.e.,
$e^{ih_{l}}\rvert\Omega_{l}\rangle$.

In the GWMF theory Gutzwiller (1963, 1965), the wave function of the ground
state $\rvert\psi^{\text{gw}}\rangle$ is a product of bases for $L$
spin-$1/2$s, i.e.,

$\displaystyle\rvert\psi^{\text{gw}}\rangle=\bigotimes^{L}_{l=1}\rvert\Omega_{l}\rangle=\sum_{\\{\bm{\sigma}\\}}\bar{a}^{\text{gw}}_{\bm{\sigma}}p^{\text{gw}}_{\bm{\sigma}}\rvert\bm{\sigma}\rangle\
,$ (4)

where

$\displaystyle\bar{a}^{\text{gw}}_{\bm{\sigma}}=\prod_{l}\lvert
c^{\sigma_{l}}_{l}\rvert\quad\text{and}\quad
p^{\text{gw}}_{\bm{\sigma}}=\exp\left[i\left(\bm{\phi}\cdot\mathbf{n}+\sum_{l=1}^{L}h_{l}\right)\right]\quad$
(5)

represent the positive amplitude and the phase factor, respectively. Here, we
define the angle vector as $\bm{\phi}=(\phi_{1},\ \cdots,\ \phi_{L})$, and the
index vector $\bm{\sigma}=(\sigma_{1},\ \cdots,\ \sigma_{L})$ satisfies the
corresponding relation $\mathbf{n}=1/2-\bm{\sigma}$. For a quantum model
featuring a real Hamiltonian discussed in this work, the complex conjugate of
the GWMF wave function $\rvert\psi^{\text{gw}}\rangle^{*}$ also indicates a
state sharing the ground-state energy. Thus, the modified GWMF wave function
$e^{ih_{0}}|\psi^{\text{gw}}\rangle=|\psi^{\text{gw}}_{r}\rangle+i|\psi^{\text{gw}}_{i}\rangle$
for the ground state can be decomposed into the real part
$|\psi^{\text{gw}}_{r}\rangle$ and imaginary part
$|\psi^{\text{gw}}_{i}\rangle$, given a specific global phase angle $h_{0}$.
Both parts are real-valued and have an extra orthogonalization relation of
$\langle\psi^{\text{gw}}_{r}|\psi^{\text{gw}}_{i}\rangle=\langle\psi^{\text{gw}}_{i}|\psi^{\text{gw}}_{r}\rangle=0$.
Concretely, two parts are expressed as

$\displaystyle\begin{split}\rvert\psi^{\text{gw}}_{r}\rangle&=\sum_{\\{\bm{\sigma}\\}}\bar{a}^{\text{gw}}_{\bm{\sigma}}\cos(\bm{\phi}\cdot\mathbf{n}+\tilde{h})\rvert\bm{\sigma}\rangle\
,\\\
\rvert\psi^{\text{gw}}_{i}\rangle&=\sum_{\\{\bm{\sigma}\\}}\bar{a}^{\text{gw}}_{\bm{\sigma}}\sin(\bm{\phi}\cdot\mathbf{n}+\tilde{h})\rvert\bm{\sigma}\rangle\
,\end{split}$ (6)

where the phase angle $\tilde{h}=\sum^{L}_{l=1}h_{l}+h_{0}$ is given.
Regardless of whether the imaginary part $|\psi^{\text{gw}}_{i}\rangle$ is
null or two parts share the same energy, we can always obtain a real ground-
state wave function. It is worth noticing that the mean fields in the GWMF
theory prefer selecting one of the degenerate manifolds if they exist, which
artificially breaks the corresponding symmetry. So, the above-mentioned
rotation, adjusted by the phase angle $h_{0}$, would introduce an energy split
between the real part $|\psi^{\text{gw}}_{r}\rangle$ and the imaginary part
$|\psi^{\text{gw}}_{i}\rangle$. In our theory, we ignore this effect and
suppose their degeneracy still survives.

To assume that the real part
$\rvert\psi^{\text{gw}}_{r}\rangle=\sum_{\\{\bm{\sigma}\\}}c^{\text{gw}}_{\bm{\sigma}}\rvert\bm{\sigma}\rangle$
can be expanded in the representation of bases $\rvert\bm{\sigma}\rangle$, the
real expansion coefficient
$c^{\text{gw}}_{\bm{\sigma}}=\bar{a}^{\text{gw}}_{\bm{\sigma}}\cos(\bm{\phi}\cdot\mathbf{n}+\tilde{h})=s^{\text{gw}}_{\bm{\sigma}}a^{\text{gw}}_{\bm{\sigma}}$
comprises of an amplitude $a^{\text{gw}}_{\bm{\sigma}}=\lvert
c^{\text{gw}}_{\bm{\sigma}}\rvert$ and a sign
$s^{\text{gw}}_{\bm{\sigma}}=\text{Sgn}(c^{\text{gw}}_{\bm{\sigma}})$
following a rule:

$\displaystyle
s^{\text{gw}}_{\bm{\sigma}}=\text{Sgn}[\cos(\bm{\phi}\cdot\mathbf{n}+\tilde{h})]\
.$ (7)

The rule is called the leading-order version, removing short-range
fluctuations completely. The phase angle $\tilde{h}$ is determined by other
necessarily preserved global symmetries for a specified eigenstate, e.g.,
translational and inversion symmetries. And “Sgn” denotes the standard sign
function. When examining the sign rule for the nonzero imaginary part
$\rvert\psi^{\text{gw}}_{i}\rangle$, an extra $\pi/2$ needs to be added to the
phase angle $\tilde{h}$, which equivalently replaces the cosine function with
the sine function in Eq. (7). In the alternative notation-$\mathbf{n}$ of
utilizing bases $\rvert\mathbf{n}\rangle$, the sign rule remains the same,
i.e., $s^{\text{gw}}_{\mathbf{n}}\equiv s^{\text{gw}}_{\bm{\sigma}}$. For the
convenience of the following discussions, we only use the
notation-$\mathbf{n}$.

In the GWMF scenario, spins in the ordered state are typically visualized as
classical vectors that follow a regular profile $\\{\mathbf{\Omega}_{l}\\}$ in
space. Our derivation shows that the leading-order sign rule, which depends on
angles $\phi_{l}$, is closely related to the spin-order profile
$\\{\mathbf{\Omega}_{l}\\}$. The above conclusion is still valid for general
qubit lattice models. In Sec. II, we will demonstrate that the leading-order
sign rule can, in principle, be learned by shn-FNN.

##  II Single-hidden-neuron feed-forward neural network

The feed-forward neural network (FNN) is a powerful tool for approximating
continuous functions Cybenko (1989); Hornik  _et al._ (1989) and sorting
samples by discrete values of characters Goodfellow  _et al._ (2016). For
instance, it can be applied to the classification of the double-valued sign
$s_{\mathbf{n}}$ for arbitrary basis $\rvert\mathbf{n}\rangle$, when the
expansion coefficient $c_{\mathbf{n}}$ in the ground-state wave function
$\rvert\psi\rangle=\sum_{\mathbf{n}}c_{\mathbf{n}}\rvert\mathbf{n}\rangle$ for
a real Hamiltonian consists of an amplitude $a_{\mathbf{n}}$ and a sign
$s_{\mathbf{n}}$. However, as FNN becomes deeper, its complexity grows, making
it difficult to understand the sign rule and its connection to meaningful
physics insights. To address this issue, we introduce shn-FNN, similar to
previous shallow FNNs He  _et al._ (2020); Yuan and Weng (2021) but distinct
from recently developed operations in a compact latent space Iten  _et al._
(2020); Wang  _et al._ (2021).

Figure 1: Shn-FNN is utilized to acquire the leading-order sign rule of a
quantum state for $L$ qubits. The network consists of three layers: the input
layer (black squares) with $L$ neurons, the hidden layer (blue hexagons) with
$1$ neuron, and the output layer (red circles) with $2$ neurons. These layers
are connected by two weight vectors $\bm{w}$ (blue lines) and $\bm{w}_{O}$
(red lines). The hidden and output layers are activated using a cosine
function and a softmax function, respectively. The sign determined by shn-FNN
is positive if $y_{O,1}>y_{O,2}$ and negative otherwise.

The shn-FNN consists of an input, hidden, and output layer, as illustrated in
Fig. 1. The configuration $\mathbf{n}$ is assigned to the input layer of shn-
FNN by simply setting a $L$-dimensional vector $\bm{y}_{I}=\mathbf{n}$. The
hidden layer contains one neuron that produces a one-dimensional vector
$y_{H}$. The output layer consists of two neurons that form a one-hot vector
$\bm{y}_{O}=(y_{O,1},\ y_{O,2})$. These three layers are connected by two
weight vectors $\bm{w}$ and $\bm{w}_{O}$.

The activation function cosine is empirically chosen for the hidden layer so
that

$\displaystyle y_{H}=\cos(\bm{w}\cdot\mathbf{n})\ .$ (8)

The vector $\bm{y}_{O}$ in the output layer is determined by applying the
softmax function, i.e.,

$\displaystyle\bm{y}_{O}=\text{softmax}(y_{H}\bm{w}_{O})\ ,$ (9)

where the weight vector $\bm{w}_{O}=(w_{O,1}$, $w_{O,2})$ is fixed. The
function softmax executes normalization by an exponential function to obtain
probabilities, which is usually used in classification tasks Goodfellow  _et
al._ (2016). Specifically, the function softmax gives two neurons $y_{O,1}$
and $y_{O,2}$ in the output layer, given by

$\displaystyle\begin{split}y_{O,1}&=\frac{e^{w_{O,1}y_{H}}}{e^{w_{O,1}y_{H}}+e^{w_{O,2}y_{H}}}\
,\\\ y_{O,2}&=\frac{e^{w_{O,2}y_{H}}}{e^{w_{O,1}y_{H}}+e^{w_{O,2}y_{H}}}\
.\end{split}$ (10)

In this work, we choose $\bm{w}_{O}=(1,-1)$, so that $e^{w_{O,1}y_{H}}$ and
$e^{w_{O,2}y_{H}}$ are distinguishable. Thus, the desired sign
$s^{\text{fnn}}_{\mathbf{n}}$ is determined as follows:

$\displaystyle
s^{\text{fnn}}_{\mathbf{n}}=\left\\{\begin{array}[]{lc}+1&\quad\text{for}\quad
y_{O,1}>y_{O,2}\\\ \\\ -1&\quad\text{for}\quad y_{O,1}\leq
y_{O,2}\end{array}\right.\ ,$ (14)

which equivalently indicates the sign of $y_{H}$, i.e.,
$s^{\text{fnn}}_{\mathbf{n}}\equiv\text{Sgn}(y_{H})$ in principle. Without
one-hot representation, it has been proven that FNN performs worse since a
categorization task turns into a regression task compulsively Goodfellow  _et
al._ (2016).

In summary, the shn-FNN representation for the sign rule corresponds to a
function

$\displaystyle
s^{\text{fnn}}_{\mathbf{n}}=\text{Sgn}\left[\cos(\bm{w}\cdot\mathbf{n})\right]\
.$ (15)

Compared to the leading-order sign rule (7) that is derived from the GWMF
theory, we usually get

$\displaystyle w_{l}=\phi_{l}+\tilde{h}/N\quad\text{and}\quad
N=\sum^{L}_{l=1}n_{l}$ (16)

for models with constant $N$. Thus, Eq. (15) is also called the leading-order
sign rule.

##  III Data sets and training

Data sets. We use the exact diagonalization (ED) method to obtain the ground-
state wave function. In the wave function, the sign $s_{\mathbf{n}}$ and the
configuration $\mathbf{n}$ for a specified basis $\rvert\mathbf{n}\rangle$
constitute a sample in the data set $\mathbf{T}$. Each sign $s_{\mathbf{n}}$
is encoded as a one-hot vector $\bm{y}^{(\mathbf{n})}=(y^{(\mathbf{n})}_{1}$,
$y^{(\mathbf{n})}_{2})$, which can only take two valid combinations:

$\displaystyle\bm{y}^{(\mathbf{n})}=\left\\{\begin{array}[]{lc}(1,\
0)&\quad\text{for}\quad s_{\mathbf{n}}>0\\\ \\\ (0,\ 1)&\quad\text{for}\quad
s_{\mathbf{n}}<0\end{array}\right.\ .$ (20)

After arranging the samples in descending order of amplitude $a_{\mathbf{n}}$,
we discard those with $a_{\mathbf{n}}<10^{-15}$ to avoid any artificial
effects caused by the limited numeric precision. The remaining
$\mathcal{N}_{s}$ samples in the data set $\mathbf{T}$ are divided into a
training set $\mathbf{T}_{\text{train}}$ and a testing set
$\mathbf{T}_{\text{test}}$ in a $4:1$ ratio Goodfellow  _et al._ (2016). Thus,
the number of samples in the testing set $\mathbf{T}_{\text{test}}$ is given
by $\mathcal{N}_{\text{test}}=\mathcal{N}_{s}/5$.

Training. During the training scheme-$\rm{I}$, we employ the back-propagation
(BP) algorithm Rumelhart  _et al._ (1986) to optimize the variables in the
weight vector $\bm{w}$ while adaptively adjusting the learning rate using the
Adam algorithm P and Ba (2014). The process aims to minimize the cross
entropy, defined as

$\displaystyle{\cal
S}_{\times}=-\sum_{\\{\mathbf{n}\\}}\left(y^{(\mathbf{n})}_{1}\ln
y^{(\mathbf{n})}_{O,1}+y^{(\mathbf{n})}_{2}\ln y^{(\mathbf{n})}_{O,2}\right)\
,$ (21)

which sums over samples in the entire training set. Here, the one-hot vector
$\bm{y}^{(\mathbf{n})}_{O}=(y^{(\mathbf{n})}_{O,1}$, $y^{(\mathbf{n})}_{O,2})$
is the output of shn-FNN as the vector $\mathbf{n}$ is input.

We employ the mini-batch method based on the stochastic gradient descent (SGD)
Goodfellow  _et al._ (2016); Wilson and Martinez (2003) to reduce the huge
computational costs. Instead of using the entire training set
$\mathbf{T}_{\text{train}}$ directly, we randomly select
$\mathcal{N}_{\text{step}}=100$ samples to calculate the gradients of the
weights $w_{l}$ at each training step. In such a case, Eq. (21) only sums over
selected $\mathcal{N}_{\text{step}}$ samples. This method performs well in
accuracy and speed, and the random selection helps prevent the “over-fitting”
problem. In our program, we implement shn-FNN and the Adam optimization using
the ML library “TensorFlow” Abadi  _et al._ (2016).

To evaluate the performance of shn-FNN, we introduce the accuracy rate (AR),
which is calculated as the ratio of the number of successfully classified
samples $\mathcal{N}^{c}_{s}$ to the total number of samples $\mathcal{N}_{s}$
in the data set $\mathbf{T}$, i.e.,
$\text{AR}=\mathcal{N}^{c}_{s}/\mathcal{N}_{s}$. To monitor the optimization
process, we define two additional accuracy rates. At each training step, we
calculate the number of samples successfully classified
$\mathcal{N}^{c}_{\text{step}}$. Then, we evaluate the optimization by
computing the accuracy rate
$\text{AR}_{\text{step}}=\mathcal{N}^{c}_{\text{step}}/\mathcal{N}_{\text{step}}$.
To provide a comprehensive evaluation, we utilize the testing set
$\mathbf{T}_{\text{test}}$ and define the accuracy rate
$\text{AR}_{\text{test}}=\mathcal{N}^{c}_{\text{test}}/\mathcal{N}_{\text{test}}$,
where $\mathcal{N}^{c}_{\text{test}}$ represents the number of correctly
classified samples in the testing set $\mathbf{T}_{\text{test}}$.

After each training step, we assess the convergence criterion $\epsilon$,
which measures the absolute value of the difference between the accuracy rates
$\text{AR}_{\text{test}}$ obtained in the current step and the previous step.
We halt the training process once the convergent criterion $\epsilon$ falls
below a threshold $\epsilon_{0}=10^{-3}$.

Figure 2: Following scheme-$\rm{I}$, we train shn-FNN to acquire the leading-
order sign rule (15) of the ground-state wave function for a generalized
ferromagnetic ($J<0$) Ising model (22) of $L=16$ spins in a ring. The spin
polarization is along the x-axis, i.e., $\mathbf{\Omega}=\hat{x}$. Throughout
the training process, we monitor the cross entropy $\mathcal{S}_{\times}$
(green) as well as three different accuracy rates: $\text{AR}_{\text{step}}$
(black), $\text{AR}_{\text{test}}$ (red) and AR (blue).

To exemplify how shn-FNN learns the sign rule, we study the ground state of a
generalized Ising ring described by the Hamiltonian

$\displaystyle\hat{H}_{\text{Ising}}(J,\
\mathbf{\Omega})=J\sum^{L}_{l=1}(\hat{\mathbf{S}}_{l}\cdot\mathbf{\Omega})(\hat{\mathbf{S}}_{l+1}\cdot\mathbf{\Omega})\
,$ (22)

where $J$ represents the strength of the coupling, and $\mathbf{\Omega}$
determines the orientation of the spin polarization.

For the case of ferromagnetic coupling ($J<0$) and spin polarization along the
x-axis ($\mathbf{\Omega}=\hat{x}$), we obtain the ground-state wave function
for $L=16$ spins, and then prepare the data sets $\mathbf{T}$,
$\mathbf{T}_{\text{train}}$ and $\mathbf{T}_{\text{test}}$, as stated above.
We initialize the weight vectors $\bm{w}$ and $\bm{w}_{O}$ in shn-FNN and
start the training process. As illustrated in Fig. 2, the cross entropy
$\mathcal{S}_{\times}$ rapidly decreases around the $170$-th training step.
After approximately $200$ training steps, the cross entropy
$\mathcal{S}_{\times}$ tends towards stability, and the three accuracy rates
$\text{AR}_{\text{step}}$, $\text{AR}_{\text{train}}$ and AR consistently
reach a maximum value $1$ or $100\%$. We terminate the training process when
the convergence criterion $\epsilon<\epsilon_{0}$ is met. In addition to
scheme-$\rm{I}$, we will introduce scheme-$\rm{II}$ in Sec. IV.2.1, tailored
for frustrated spin models and the Fermi-Hubbard ring later.

##  IV Qubit lattice models

Using shn-FNN, we analyze the leading-order sign rules for various ordered
ground states in qubit lattice models, including non-frustrated spin models in
Sec. IV.1, frustrated spin models in Sec. IV.2, and interacting fermions in
Sec. IV.3.

###  IV.1 Non-frustrated spin models

Figure 3: (a.1-d.1) The accuracy rate AR of the sign rule (15) as a function
of the angle $\varphi$ and phase angle $\tilde{h}$. In (a.1) and (b.1), the
angle $\phi_{l}=\varphi$, while in (c.1) and (d.1) $\phi_{l}=\varphi l$.
(a.2-d.2) Following scheme-$\rm I$, the accuracy rate AR changes with the
training step number during the training process. After full training, the
optimized shn-FNN with $\text{AR}=1$ gives weights $w_{l}$. These weights are
then converted to $\phi_{l}$ and $\tilde{h}$ using Eq. (16), which follow
simple rules: $\phi_{l}=\varphi$ in (a.2) and (b.2), while $\phi_{l}=\varphi
l$ in (c.2) and (d.2). Consequently, we mark coordinates $\varphi$ and
$\tilde{h}$ in (a.1-d.1) by black open triangles for representing the
optimized weights. Here, we investigate ground states in (a.1, a.2) a
generalized ferromagnetic Ising ring with $\mathbf{\Omega}=\hat{x}$, (b.1,
b,2) a ferromagnetic spin-$1/2$ XY ring, (c.1, c.2) a twisted ferromagnetic
spin-$1/2$ XY ring (even parity), and (d.1, d.2) an antiferromagnetic
spin-$1/2$ XY ring. The system size is $L=16$.

####  IV.1.1 A generalized ferromagnetic Ising ring

For the case of $J<0$ and $\mathbf{\Omega}=\hat{x}$ in the model (22), the
optimized shn-FNN with $\text{AR}=1$, as illustrated in Fig. 3(a.2), suggests
that the weights $w_{l}=0$ in the sign rule (15). The physical interpretation
of $w_{l}$ can be understood from the connection between Eq. (15) and the
leading-order sign rule (7). Using Eq. (16), $w_{l}=0$ can be converted to a
combination of the angles $\phi_{l}=0$ and the phase angle $\tilde{h}=0$ in
the sign rule (7). This demonstrates the presence of ferromagnetic order along
the x-axis, according to the spin-coherent state representation (2). To assume
that $\phi_{l}=\varphi$ in Eq. (16), we can visualize AR for the sign rule
(15) in the ($\varphi$, $\tilde{h}$) plane, as shown in Fig. 3(a.1). The
optimized weights $w_{l}=0$ are positioned at the coordinates of a maximum,
i.e., $\varphi=0$ and $\tilde{h}=0$ (black open triangle).

####  IV.1.2 A ferromagnetic spin-$1/2$ XY ring

For a spin-$1/2$ XY ring with $L=16$ sites, the Hamiltonian is given by

$\displaystyle\hat{H}^{P}_{\text{xy}}(J)=\frac{1}{2}\left(\sum^{L}_{l=1}J\hat{S}^{+}_{l}\hat{S}^{-}_{l+1}+\textrm{h.c.}\right)\
,$ (23)

where $\hat{S}^{\pm}_{l}$ represent the flipping-up and flipping-down
operators for a spin-$1/2$. The coupling strengths $J$ in the $x$ and $y$-axes
are equal in this ring.

For the case of $J<0$ shown in Fig. 3(b.2), the accuracy rate AR for shn-FNN
reaches a perfect-classification limit of $1$ after approximately $150$
training steps. In the optimized shn-FNN, the weights are given by
$w_{l}=\pi/8$, which are equivalent to $\phi_{l}=\pi/8$ and $\tilde{h}=0$
according to Eq. (16). The result remains consistent with an in-plane
ferromagnetic order, where all spins are confined to the xy-plane and aligned
in the same polarization direction. Therefore, we still set $\phi_{l}=\varphi$
in Eq. (16), and plot the AR distribution in the ($\varphi$, $\tilde{h}$)
plane in Fig. 3(b.1). We can easily find that the weights $w_{l}=\pi/8$
correspond to the coordinates $\varphi=\pi/8$ and $\tilde{h}=0$ (black open
triangle). Since $w_{l}=\pi/8$ is uniform in space, the resulting sign rule
(15) can be summarized as the Perron-Frobenius theorem Perron (1907);
Frobenius (1909) by removing a global phase angle $\pi$, when $L=16$.

####  IV.1.3 A twisted ferromagnetic spin-$1/2$ XY ring

For a spin-$1/2$ XY ring with the ferromagnetic coupling strength $J<0$ under
the twisted boundary condition (TBC), an antiferromagnetic bond connects
site-$1$ and site-$L$ in the Hamiltonian

$\displaystyle\hat{H}^{T}_{\text{xy}}(J)=\frac{1}{2}\left(\sum^{L-1}_{l=1}J\hat{S}^{+}_{l}\hat{S}^{-}_{l+1}-J\hat{S}^{+}_{L}\hat{S}^{-}_{1}+\textrm{h.c.}\right)\
.\quad$ (24)

To achieve convergence of the accuracy rate $\text{AR}=1$, we train the shn-
FNN for $L=16$, as shown in Fig. 3(c.2). The optimized weights obtained from
this training are given by

$\displaystyle w_{l}=\frac{\pi}{L}l-\frac{(L+1)\pi}{2L}$ (25)

in the sign rule (15) for the ground-state wave function. We find Eq. (25)
corresponds to a combination of the angles $\phi_{l}=\pi l/16$ and the phase
angle $\tilde{h}=-17\pi/4$ according to Eq. (16). This suggests a gradually
varying spin profile in space with a pitch angle of $\pi/16$. In a similar
manner, by setting $\phi_{l}=\varphi l$ in Eq. (16), we parameterize the sign
rule (15) with two parameters $\varphi$ and $\tilde{h}$. The AR distribution
in the ($\varphi$, $\tilde{h}$) plane is plotted in Fig. 3(c.1), where the
coordinates $\varphi=\pi/16$ and $\tilde{h}=-17\pi/4$ are marked by a black
open triangle.

The above result can be better understood through the following analysis.
Under a rotation defined by the operators

$\displaystyle\hat{\mathcal{U}}_{\varphi}=\prod^{L}_{l=1}\hat{\mathcal{R}}_{l}(l\varphi)\quad\text{and}\quad\hat{\mathcal{R}}_{l}(q)=e^{iq\hat{S}^{z}_{l}}\
,\quad$ (26)

the twisting effect from the antiferromagnetic bond is absorbed into a gauge
field $\tilde{J}=\exp(i\varphi)$ in a new Hamiltonian
${\hat{H}}^{P}_{\text{xy}}(J\tilde{J})$, where $\varphi=\pi/L$. Meanwhile,

$\displaystyle\hat{\mathcal{U}}^{\dagger}_{\varphi}{\hat{S}}^{\pm}_{l}\hat{\mathcal{U}}_{\varphi}={\hat{S}}^{\pm}_{l}\exp(\mp
il\varphi)\ .$ (27)

Based on the arguement in App. A, it is found that the even-parity ground-
state wave function $\rvert\psi^{P}_{\text{xy}}(J\tilde{J})\rangle$ for the
Hamiltonian ${\hat{H}}^{P}_{\text{xy}}(J\tilde{J})$ has positive signs, i.e.,
$s_{\mathbf{n}}>0$. As a result, the ground-state wave function
$\rvert\psi^{T}_{\text{xy}}(J)\rangle$ for the Hamiltonian
${\hat{H}}^{T}_{\text{xy}}(J)$ carries a nonzero complex phase factor due to
the rotation $\hat{\mathcal{U}}_{\varphi}$. Specifically, it can be written as

$\displaystyle\rvert\psi^{T}_{\text{xy}}(J)\rangle=\hat{\mathcal{U}}_{\varphi}\rvert\psi^{P}_{\text{xy}}(J\tilde{J})\rangle\
.$ (28)

Thus, the real part of the wave function is given by

$\displaystyle|\psi^{T}_{\text{xy},r}(J)\rangle=\cos(\bm{w}\cdot\mathbf{n})\rvert\psi^{P}_{\text{xy}}(J\tilde{J})\rangle\
,$ (29)

which is inversion-symmetric concerning the chain center. It is worth noting
that although Eq. (29) implies the same sign rule (15) obtained from the GWMF
theory, this analysis is rigorous for the twisted spin-$1/2$ XY ring.

####  IV.1.4 An antiferromagnetic spin-$1/2$ XY ring

For the case of antiferromagnetic coupling $J>0$, the optimized shn-FNN
suggests $w_{l}=\pi l-\pi/8$ in the sign rule (15), as shown in Fig. 3(d.1).
We find that $\bm{w}$ corresponds to a combination of the angles $\phi_{l}=\pi
l$ and the phase angle $\tilde{h}=-\pi$, indicating the presence of
N$\acute{e}$el order. Remarkably, the sign rule defined by $\bm{w}$ is
equivalent to MPR, where $s_{\mathbf{n}}=(-1)^{N_{O}}$, and the quantity

$\displaystyle N_{O}=\sum_{l\in\text{odd}}\langle{\hat{S}}^{z}_{l}+1/2\rangle$
(30)

sums over all odd sites. Here, we parameterize the sign rule (15) with two
parameters $\varphi$ and $\tilde{h}$, by setting $\phi_{l}=\varphi l$ in Eq.
(16). Fig. 3(d.1) illustrates the AR distribution in the ($\varphi$,
$\tilde{h}$) plane, with the corresponding coordinates of the optimized
$\bm{w}$ represented by a black open triangle, specifically $\varphi=\pi$ and
$\tilde{h}=-\pi$. This sign rule is well understood because the ground-state
wave function for $J>0$ is connected to the one for $J<0$ by a $\pi$-rotation
operation

$\displaystyle\hat{\mathcal{U}}_{\pi}=\prod^{L}_{l=1}\hat{\mathcal{R}}_{l}(l\pi)=\prod_{l\in\text{odd}}\hat{\mathcal{R}}_{l}(\pi)\
.$ (31)

Hence, it is evident that the sign rules in the XY ring with perfect AR, shown
in Sec. IV.1.2, IV.1.3, and IV.1.4, adhere to a standardized format of weights
Eq. (16) by setting $\phi_{l}=\varphi l$ with specific pitch angles
$\varphi=0$, $\pi/L$ and $\pi$. This pitch angle $\varphi$ is related to the
profile of spins rotating in space and can be acquired by training shn-FNN.

####  IV.1.5 An antiferromagnetic Heisenberg ring

In a pure antiferromagnetic Heisenberg ring (AFHR) with equal nearest-
neighboring antiferromagnetic couplings $J_{1}$ in the $x$, $y$, and $z$-axes,
the spins at odd sites align anti-parallel to the spins at even sites
according to GWMF. Even though the precise ground state behaves as the
Tomonago-Luttinger liquid (TLL) Tomonaga (1950); Luttinger (1963); Haldane
(1981), the optimized shn-FNN suggests MPR, which is consistent with previous
studies Marshall (1955); Richter  _et al._ (1994). We discuss the sign rule
uniformly in the $J_{1}$-$J_{2}$ AFHR in Sec. IV.2.1.

###  IV.2 Frustrated spin models

Figure 4: Training shn-FNN to learn the leading-order sign rule (15) of the
ground-state wave function for a $J_{1}$-$J_{2}$ antiferromagnetic Heisenberg
model (32) of $L=16$ spins in a ring. The chosen ratio is $\alpha=0.9$. (a) In
scheme-$\rm{I}$, all $\mathcal{N}_{s}$ samples in the data set $\mathbf{T}$
are utilized for training shn-FNN. The training process is monitored by the
cross entropy $\mathcal{S}_{\times}$ (green) as well as three distinct
accuracy rates: $\text{AR}_{\text{step}}$ (black), $\text{AR}_{\text{test}}$
(red) and AR (blue). (b) The samples are sorted in descending order of
magnitude $a_{\mathbf{n}}$. They are then regrouped into two data sets:
$\mathbf{T}_{\text{obey}}$ that follows the sign rule proposed by the
optimized shn-FNN obtained from training scheme-$\rm{I}$ (red), and
$\mathbf{T}_{\text{disobey}}$ that violates the sign rule (blue). (c) In
scheme-$\rm{II}$, the first $\widetilde{\mathcal{N}}_{s}\leq\mathcal{N}_{s}$
samples in the data set $\mathbf{T}$ are used for training shn-FNN. The
accuracy rate $\widetilde{\text{AR}}$ is plotted as a function of the
selection rate $\beta=\widetilde{\mathcal{N}}_{s}/\mathcal{N}_{s}$. (d) The
spatial distribution of the weight vector elements $w_{l}$ in the sign rule
(15) from training scheme-$\rm{I}$ (blue) and scheme-$\rm{II}$ (red) are
shown.

####  IV.2.1 A $J_{1}$-$J_{2}$ antiferromagnetic Heisenberg ring

When the antiferromagnetic next-nearest-neighboring (NNN) Heisenberg coupling
$J_{2}>0$ is introduced, we investigate the behavior of the frustrated
spin-$1/2$ $J_{1}$-$J_{2}$ AFHR. The Hamiltonian for this system is given by

$\displaystyle\hat{H}_{J_{1}\text{-}J_{2}}=\sum_{l=1}^{L}(J_{1}\hat{\mathbf{S}}_{l}\cdot\hat{\mathbf{S}}_{l+1}+J_{2}\hat{\mathbf{S}}_{l}\cdot\hat{\mathbf{S}}_{l+2})\
,$ (32)

where $\alpha=J_{2}/J_{1}$ is a dimensionless ratio.

Using the techniques suggested in Sec. III and following training
scheme-$\rm{I}$ exemplified in Fig. 2, we initially train shn-FNN with all
$\mathcal{N}_{s}$ samples in the data set $\mathbf{T}$. For example, when the
ratio $\alpha=0.9$, the optimization of shn-FNN searches for the minima of the
cross entropy $\mathcal{S}_{\times}$ with extremely low efficiency. As
illustrated in Fig. 4(a), three accuracy rates oscillate irregularly near the
worst-performance limit of $0.5$. After approximately $3000$ training steps,
all accuracy rates suddenly increase and reach another plateau. During this
phase, the cross entropy $\mathcal{S}_{\times}$ exhibits random oscillation
and fails to offer a meaningful gradient direction for updating the weight
vector $\bm{w}$ in shn-FNN. Once the accurate rate AR reaches approximately
$0.63$, the weight vector $\bm{w}$ in the optimized shn-FNN (blue circles), as
shown in Fig. 4(d), becomes difficult to interpret.

To address this issue, we sort samples in the descending order of amplitude
$a_{\mathbf{n}}$, as shown in Fig. 4(b), and we observe that correct
classification in the data set $\mathbf{T}_{\text{obey}}$ and wrong
classification in the data set $\mathbf{T}_{\text{disobey}}$ are irregularly
mixed. Instead, we adopt training scheme-$\rm{II}$, where we use the first
$\widetilde{\mathcal{N}}_{s}\leq\mathcal{N}_{s}$ samples in the data set
$\mathbf{T}$ to train shn-FNN. In Fig. 4(c), as we reduce the selection rate
$\beta=\widetilde{\mathcal{N}}_{s}/\mathcal{N}_{s}$, the accuracy rate
$\widetilde{\text{AR}}=\widetilde{\mathcal{N}}^{c}_{s}/\widetilde{\mathcal{N}}_{s}$
approaches the perfect-classification limit of $1$. With the optimized shn-
FNN, $\widetilde{\mathcal{N}}^{c}_{s}$ out of $\widetilde{\mathcal{N}}_{s}$
samples are correctly classified. The resulting weight vector $\bm{w}$, shown
in Fig. 4(d), exhibits a straight line in the sign rule (15), which will be
used to demonstrate physical insight later.

Figure 5: (a.1-a.5) The structure factor $\mathcal{S}_{k}$ as a function of
the momentum $k$ for a $J_{1}$-$J_{2}$ antiferromagnetic Heisenberg ring (32)
with $L=16$. Five different values of the ratio $\alpha=J_{2}/J_{1}$ are
considered: (a.1) $0.35$, (a.2) $0.52$, (a.3) $0.58$, (a.4) $0.73$, and (a.5)
$1.2$. (b) The optimized shn-FNN is trained using scheme-II, and the resulting
weights Eq. (16) with $\phi_{l}=\varphi l$ gives the pitch angle $\varphi$.
(c) Meanwhile, the corresponding accuracy rate AR (black) and the correct
weight $\rho$ (red) are also shown. The plots demonstrate a series of level
crossings at $\alpha_{\text{MG}}=0.5$, $\alpha_{c1}\approx 0.53$,
$\alpha_{c2}\approx 0.64$, and $\alpha_{\text{DC}}\approx 0.99$.

After conducting a systematical analysis of the ground states for $L=16$
sites, we have discovered that the optimized shn-FNN proposes the sign rules
with weights Eq. (16) by setting $\phi_{l}=\varphi l$, where the phase angle
$\varphi=2p\pi/L$, as shown in Fig. 5(b). The integer number $p$ ranges from
$L/2$ to $L/4$, which allows us to divide the broad regime $\alpha\in[0,\
1.2]$ into five intervals. Within each interval, the value of $\varphi$ plays
the role of the commensurate/incommensurate pitch angle White and Affleck
(1996); Soos  _et al._ (2016). In Fig. 5(a.1-a.5), we observe that the double
peaks in the structure factor

$\displaystyle{\cal
S}_{k}=\frac{1}{L^{2}}\sum_{l,l^{\prime}}e^{ik(l-l^{\prime})}\langle\hat{\mathbf{S}}_{l}\cdot\hat{\mathbf{S}}_{l^{\prime}}\rangle$
(33)

are located at the momenta $k=\pm\varphi$. Due to the interplay of
interactions, the accuracy rate $\text{AR}<1$ and the correct weight

$\displaystyle\rho=\sum_{\mathbf{n}\in\mathbf{T}_{\text{obey}}}|a_{\mathbf{n}}|^{2}<1$
(34)

are shown in Fig. 5(c), where the data set $\mathbf{T}_{\text{obey}}$ includes
samples obeying the leading-order sign rule (15). Besides, we find that the
correct weight $\rho$ in the whole parameter regime is close to $1$, which
means that most of the samples with the largest amplitudes obey the leading-
order sign rule, so the proposed scheme-$\rm{II}$ works well.

The investigation of the sign rule provides consistent physical insights. At
the Majumdar-Ghosh (MG) point $\alpha_{\text{MG}}=0.5$, and for $\varphi=\pi$
within the range $\alpha\in[0,\ \alpha_{\text{MG}}]$, the leading-order sign
rule is in accordance with MPR. As the ratio $\alpha$ approaches infinity, one
of the decoupled chains, composed of odd or even sites, individually follows
MPR. However, away from that limit, a relatively tiny positive $J_{1}$
promotes a stable commensurate spin order with a pitch angle $\varphi=\pi/2$.
When the ratio $\alpha<\alpha_{\text{DC}}\approx 0.99$, commensurability is
disrupted due to the emergence of triplet defects White and Affleck (1996);
Soos  _et al._ (2016). Between $\alpha_{\text{MG}}$ and $\alpha_{\text{DC}}$,
the ground state undergoes an incommensurate crossover White and Affleck
(1996); Soos  _et al._ (2016), which is indicated by the varying pitch angle
$\varphi$ in the weights of the leading-order sign rule (15), as shown in Fig.
5(b).

Besides, the ground state maintains the translation symmetry with a conserved
momentum of either $0$ or $\pi$, depending on the integer number $p$, so that
$\phi_{l}=\varphi l$ in the leading-order sign rule (7). Moreover, the center
inversion symmetry of the chain imposes a constraint of $\tilde{h}=p\pi/2$.
Consequently, in the equivalent sign rule (15), we choose the activation
function sine/cosine for odd/even values of $p$.

Figure 6: The sign-fidelity susceptibility density $\chi_{f}$ for the ground
state is examined in the region $\alpha\in[0$, $\alpha_{\text{MG}})$ for
different system sizes $L=8$, $12$, $16$, $20$, $22$ and $24$ in the
$J_{1}$-$J_{2}$ antiferromagnetic Heisenberg ring (32).

To quantitatively assess the violation of MPR when $\alpha\in[0,\
\alpha_{\text{MG}})$, we introduce a sign-fidelity

$\displaystyle f=\langle\psi^{\text{MPR}}|\psi\rangle\ .$ (35)

Here, we define the MPR state $|\psi^{\text{MPR}}\rangle$ as

$\displaystyle\rvert\psi^{\text{MPR}}\rangle=\sum_{\\{\mathbf{n}\\}}s^{\text{MPR}}_{\mathbf{n}}a_{\mathbf{n}}\rvert\mathbf{n}\rangle\
,$ (36)

where the sign $s^{\text{MPR}}_{\mathbf{n}}$ fully satisfies MPR. Thus, we get
$f=2\rho-1$ Retzlaff  _et al._ (1993); Richter  _et al._ (1994); Zeng and
Parkinson (1995). In the vicinity of a continuous transition point, the
minimum sign-fidelity $f$ or correct weight $\rho$ is expected to be achieved,
indicating the most complicated sign rule Cai and Liu (2018); Westerhout  _et
al._ (2020). Like the orthogonalization catastrophe for free fermions Anderson
(1967), fidelity follows a pow-law function of the system size $L$. In
principle, the relevant sign-fidelity susceptibility density, given by

$\displaystyle\chi_{f}=-(\ln f)/L\ ,$ (37)

is capable of identifying the places of continuous transition points Gu
(2010). However, the maximum of $\chi_{f}$ is located at
$\alpha_{\text{peak}}\approx 0.43>\alpha_{\text{BKT}}\approx 0.241$ in the
dimerized (DM) region Wang  _et al._ (2009), where $\chi_{f}$ approaches a
$L$-independent function as shown in Fig. 6. It is possibly caused by the
anomalous behavior of the exponential closure of gaps at the famous
Berzinskii-Kosterlitz-Thouless (BKT) transition point $\alpha_{\text{BKT}}$
Cincio  _et al._ (2019).

####  IV.2.2 A spin-$1/2$ triangular XY antiferromagnet on a torus

Shn-FNN can learn the leading-order sign rules for the ground-state wave
function of $2$D quantum models, such as the XY model on triangular lattices
with a size of $L_{x}\times L_{y}$ sites, as shown in Fig. 7(a). The
corresponding Hamiltonian for the model reads

$\displaystyle{\hat{H}}_{\triangle}=\frac{1}{2}\sum_{\langle
l,l^{\prime}\rangle}(\hat{S}^{+}_{l}\hat{S}^{-}_{l^{\prime}}+\textrm{h.c.})\
,$ (38)

and $\langle\rangle$ sums over all nearest-neighboring sites $l$ and
$l^{\prime}$. In the XC geometry, the lattice site labeled as
$l=l_{y}L_{x}+l_{x}+1$, is identified by binary indices $(l_{x},l_{y})$ with
$l_{x}=0$, $\cdots$, $L_{x}-1$ and $l_{y}=0$, $\cdots$, $L_{y}-1$. The
displacement for the site is given by $\mathbf{r}_{l}$.

Figure 7: (a) The spin-$1/2$ XY antiferromagnet (38) with the XC geometry on a
torus. $a_{1}$ and $a_{2}$ denote two primitive vectors. All sites are labeled
with a single index $l$. (b) For the geometry $3\times 4$, we employ
scheme-$\rm{II}$ to train shn-FNN for the ground-state wave function. The
spatial distribution of weights $w_{l}$ in the optimized shn-FNN is plotted.
Figure 8: Regarding the ground state of the spin-$1/2$ XY antiferromagnet on a
torus (38), we demonstrate (a) the accuracy rate AR (black) and (b) correct
weight $\rho$ (red) for the sign rule (15) with weights defined in Eq. (39).
Both are shown as a function of the geometry $L_{x}\times L_{y}$. It is worth
noting that the optimized shn-FNN indicates that $\tilde{h}=0$ for even
$(L_{y}/2)$, but $\tilde{h}=\pi/2$ for odd $(L_{y}/2)$.

To ensure an exact hit at relevant high-symmetry momentum points
$\mathbf{K}^{\pm}$ in the first Brillouin zone, the length $L_{x}$ is chosen
as a multiple of $3$. Following training scheme-$\rm{II}$, the weights
$w_{(l_{x},l_{y})}$ in the leading-order sign rule (15) are determined by the
optimized shn-FNN. Specifically, we get

$\displaystyle w_{(l_{x},l_{y})}=\frac{2\pi}{3}(l_{x}+[l_{y}])\ ,$ (39)

where $[l_{y}]=1$ if $l_{y}$ is even and $0$ otherwise, as illustrated in Fig.
7(b). This result matches the physical scenario of the coplanar $120^{\circ}$
order Bach  _et al._ (2021) observed in previous studies, where the angle
between spin polarization orientations at neighboring sites is always
$2\pi/3$.

Moreover, the ground state possesses point group symmetries of the torus.
These symmetry operations listed in table 1 carry eigenvalues of $+1$ or $-1$,
corresponding to the symmetric/even or antisymmetric/odd sector of the group
representation in mathematics.

Table 1: The symmetry operations are measured on the ground-state wave function for the spin-$1/2$ triangular XY antiferromagnet on a torus. The table includes the translation by a site ${\cal T}_{x}$ in the x-axis and ${\cal T}_{y}$ in the y-axis, mirror inversion ${\cal M}_{x}$ about the x-axis and ${\cal M}_{y}$ about the y-axis, center inversion ${\cal I}_{c}$. $L_{x}\times L_{y}$ | $\langle{\cal T}_{x}\rangle$ | $\langle{\cal T}_{y}\rangle$ | $\langle{\cal M}_{x}\rangle$ | $\langle{\cal M}_{y}\rangle$ | $\langle{\cal I}_{c}\rangle$
---|---|---|---|---|---
$3\times 4$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
$3\times 6$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$
$3\times 8$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
$6\times 4$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
$3\times 10$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$

Let us discuss the mirror inversion ${\cal M}_{y}$ about the y-axis. Under
${\cal M}_{y}$, the basis $|\mathbf{n}\rangle$ becomes
$|\mathbf{n}^{\prime}\rangle$ but the sign is unchanged, so we have

$\displaystyle\bm{w}\cdot\mathbf{n}^{\prime}=L_{y}\pi-\bm{w}\cdot\mathbf{n}\quad\text{modulo}\quad
2\pi\ .$ (40)

For even $(L_{y}/2)$, such as the $3\times 4$ lattice, the ${\cal
M}_{y}$-symmetric ground-state wave function leads to $\tilde{h}=0$ in the
leading-order sign rule (7), or equivalently the activation function cosine in
Eq. (15), which obeys

$\displaystyle\text{Sgn}[\cos(\bm{w}\cdot\mathbf{n})]=\text{Sgn}[\cos(\bm{w}\cdot\mathbf{n^{\prime}})]\
.$ (41)

In contrast, for odd $(L_{y}/2)$, e.g., the geometry of $3\times 6$ lattices,
the ${\cal M}_{y}$-antisymmetric ground-state wave function prefers
$\tilde{h}=\pi/2$ in Eq. (7) and the sine function in Eq. (15). This
difference is captured by shn-FNN in Fig. 8.

Furthermore, based on the mean-field picture of spinless Dirac fermions
coupled to Chern-Simons gauge fields Wang  _et al._ (2018); Sedrakyan  _et
al._ (2020), it has been shown that for different lattice geometries with
finite $L_{x}$ and $L_{y}$, non-condensed BCS pairs of spinons from high
symmetry points $\mathbf{K}^{\pm}$ would violate the leading-order sign rule,
where both AR and $\rho$ deviate from $1$. However, a more nuanced
understanding of the subtle relationship between lattice geometry and the
deviation from GWMF still needs to be included.

###  IV.3 A Fermi-Hubbard ring

The Fermi-Hubbard model is a simple model that describes the physics in
strongly correlated electron systems, which is closely connected to quantum
magnetism, metal-insulator transition, and the promising theory of high-
temperature superconductivity Henderson  _et al._ (1992); Essler  _et al._
(2005); Moriya (2012). In a ring, the Hamiltonian for two-species fermions can
be written as

$\displaystyle\hat{H}_{\text{F}}=\sum^{L}_{l=1}\left[-t\sum_{\sigma}(\hat{c}^{{\dagger}}_{l,\sigma}\hat{c}_{l+1,\sigma}+{\rm
h.c.})+U\hat{n}_{l,\uparrow}\hat{n}_{l,\downarrow}\right]\ ,\quad$ (42)

where $\hat{c}^{{\dagger}}_{l,\sigma}$, $\hat{c}_{l,\sigma}$ and
$\hat{n}_{l,\sigma}=\hat{c}^{{\dagger}}_{l,\sigma}\hat{c}_{l,\sigma}$
represent the creation, annihilation and particle number operators of fermion
at site-$l$ respectively, $\sigma=\uparrow$, $\downarrow$ denotes the spin
polarization, $t>0$ is the hopping amplitude between two nearest-neighboring
sites, and $U$ is the onsite coulomb repulsion.

Figure 9: (a) In the noninteracting limit $U/t=0$, we examine the double-even-
parity ground state for the Fermi-Hubbard ring (42). We train shn-FNN by
scheme-$\rm{II}$. In the optimized shn-FNN, we plot the weights $w_{l}$, which
match Eq. (47). (b) When $U/t=0.1$ and $N_{\downarrow}=2$, we plot the
accuracy rate AR of the sign rule
$s_{\mathbf{n}_{\uparrow},\mathbf{n}_{\downarrow}}=\text{Sgn}[\cos(\bm{w}\cdot\mathbf{n}+\vartheta)]$
as a function of the variational angle $\vartheta$ for the ground state. The
maximum AR value occurs at $\vartheta=0$. The system size is given by $L=16$.

In the Fock space, each basis is a product of the bases for two species, that
is,

$\displaystyle|\mathbf{n}_{\uparrow},\mathbf{n}_{\downarrow}\rangle=\left[\bigotimes^{L}_{l=1}|n_{l,\uparrow}\rangle\right]\left[\bigotimes^{L}_{l=1}|n_{l,\downarrow}\rangle\right]\
.$ (43)

Here, we define the vectors $\mathbf{n}_{\sigma}=(n_{1,\sigma},\ \cdots,\
n_{L,\sigma})$ for species-$\sigma$. Under the conventional Jordan-Wigner
transformation Derzhko (2001), the two-channel spin flipping operators
$\hat{S}^{\pm}_{l,\sigma}$ can be represented by fermion operators as follows:

$\displaystyle\begin{split}\hat{S}^{+}_{l,\sigma}&=\hat{c}^{\dagger}_{l,\sigma}e^{\phantom{-}i\pi\sum_{k<l}\hat{n}_{k,\sigma}}\
,\\\
\hat{S}^{-}_{l,\sigma}&=\hat{c}_{l,\sigma}e^{-i\pi\sum_{k<l}\hat{n}_{k,\sigma}}\
.\end{split}$ (44)

Thus, we get a two-leg spin-$1/2$ ladder

$\displaystyle\hat{H}_{\text{Ladder}}=\sum_{\sigma}\hat{H}_{\parallel,\sigma}+\hat{H}_{\perp}\
,$ (45)

where

$\displaystyle\hat{H}_{\parallel,\sigma}=\\!-t\left[\sum^{L-1}_{l=1}{\hat{S}}^{+}_{l,\sigma}{\hat{S}}^{-}_{l+1,\sigma}\\!+\\!(-1)^{\hat{N}_{\sigma}-1}{\hat{S}}^{+}_{L,\sigma}{\hat{S}}^{-}_{1,\sigma}\\!+\\!{\rm
h.c.}\right]\ ,$
$\displaystyle\hat{H}_{\perp}=U\sum^{L}_{l=1}(\hat{S}^{z}_{l,\uparrow}+1/2)(\hat{S}^{z}_{l,\downarrow}+1/2)\
.$ (46)

denote the transverse and longitudinal parts, respectively. The particle
number operator in total for species-$\sigma$ is given by
$\hat{N}_{\sigma}=\sum^{L}_{l=1}\hat{n}_{l,\sigma}$. We are interested in the
ground state for the case of $N_{\uparrow}+N_{\downarrow}=L$, and even
$N_{\sigma}=\langle\hat{N}_{\sigma}\rangle$.

Figure 10: (a) The accuracy rate AR of the sign rule (50) as a function of
$N_{\downarrow}$ when we choose $L=16$. (b) The same accuracy rate as a
function of $L$ when we choose $N_{\downarrow}=2$. Here, we concern the ground
state for the Fermi-Hubbard ring (42) with $U/t=0.1$.

When $U=0$, TBC is effectively applied to two decoupled chains in the spin-
ladder model, as both $N_{\uparrow}$ and $N_{\downarrow}$ are even. For each
species, when the parity of the ground state is even, the optimized shn-FNN
with perfect AR can identify the leading-order sign rule given by

$\displaystyle
s^{(e)}_{\mathbf{n}_{\sigma}}=\text{Sgn}[\cos(\bm{w}\cdot\mathbf{n}_{\sigma})]$
(47)

with the weights

$\displaystyle w_{l}=-\frac{(l-1)\pi}{L}+\frac{\pi}{2}-\frac{\pi}{2L}\ ,$ (48)

depicted in Fig. 9(a). For the degenerate ground state with odd parity, the
function cosine is replaced by the function sine, i.e.,

$\displaystyle
s^{(o)}_{\mathbf{n}_{\sigma}}=\text{Sgn}[\sin(\bm{w}\cdot\mathbf{n}_{\sigma})]\
.$ (49)

For small $U/t>0$ and any $N_{\sigma}$, the parity of the unique ground state
is always even. In the case of $U/t=0.1$, $N_{\downarrow}=2$ and $L=16$, as
illustrated in Fig. 9(b), the optimized shn-FNN suggests a sign rule

$\displaystyle
s_{\mathbf{n}_{\uparrow},\mathbf{n}_{\downarrow}}=\text{Sgn}[\cos(\bm{w}\cdot\mathbf{n})]\
,$ (50)

with the accuracy rate $\text{AR}\approx 0.97$, where the vector is defined as

$\displaystyle\mathbf{n}=\mathbf{n}_{\uparrow}+\mathbf{n}_{\downarrow}=(n_{1,\uparrow}+n_{1,\downarrow},\
\cdots,\ n_{L,\uparrow}+n_{L,\downarrow})\ .$ (51)

So, the resulting leading-order sign rule for the Fermi-Hubbard model remains
consistent with the sign rule (15).

The leading-order sign rule (50) is robust and less dependent on the filling
fraction and system size $L$. In the case of $U/t=0.1$ and $L=16$ (Fig. 10),
the accuracy rate AR is greater than $0.968$ for different $N_{\downarrow}$.
Additionally, as $L$ grows, the accuracy rate AR for $N_{\downarrow}=2$ gets
closer to $0.99$.

Figure 11: The correct weight $\rho$ of the sign rule (50) as a function of
$U/t$ in the Fermi-Hubbard ring (42). We concern the ground state with
$N_{\downarrow}=2$, and four different system sizes are investigated: $L=8$
(green), $12$ (blue), $16$ (black), and $20$ (red).

In the limit of large $U$, only single occupations can exist in the ground
state because of a considerable charge gap. As a result, spin fluctuations in
the reduced Hilbert space of either spin-up
$\hat{c}^{\dagger}_{l,\uparrow}\rvert 0\rangle$ or spin-down
$-\hat{c}^{\dagger}_{l,\downarrow}\rvert 0\rangle$ are described by the
effective antiferromagnetic Heisenberg ring. The ground state for the
effective model follows the MPR sign rule exactly. Returning to the fermion
bases, it is easy to prove that the weights $w_{l}$ are the same as ones in
the leading-order sign rule at $U=0$. We can observe that the corresponding
correct weight $\rho$ approaches $1$ when $U/t\geq 8$, as shown in Fig. 11.

According to the Bethe ansatz solution Lieb and Wu (1968), the Fermi liquid
only survives at $U=0$ in the thermodynamical limit (TDL). However, because of
a tiny charge gap close to $U=0$, fermions exhibit behavior like a Fermi
liquid in the ground state for system size limited to $L\leq 20$, much smaller
than the correlation length. Consequently, a quasi-critical point is indicated
by the minimum of the correct weight $\rho$, where the strong quantum
fluctuations would strongly violate the leading-order sign rules. As $L$ grows
in Fig. 11, the quasi-critical point gradually approaches $U=0$.

In an alternative definition of bases, that is,

$\displaystyle\rvert\mathbf{n}\rangle=\bigotimes^{L}_{l=1}\bigg{[}\rvert
n_{l,\uparrow}\rangle\rvert n_{l,\downarrow}\rangle\bigg{]}\ ,$ (52)

the Jordan Wigner transformation changes accordingly, and an additional
nonlinear appendix

$\displaystyle(-1)^{\sum^{L}_{l=2}n_{l,\uparrow}\sum^{l-1}_{k=1}n_{k,\downarrow}}$
(53)

exists in front of the predicted sign rules. However, this appendix can not be
expressed in shn-FNN.

##  V Summary and discussions

We have successfully developed a Gutzwiller mean-field theory of sign rules
for the ordered ground states in qubit lattice models, which perfectly matches
the sign predicted by a shallow FNN with a single hidden neuron, called shn-
FNN. By utilizing this principle, we provide a consistent explanation for the
excellent performance of activation functions in the neural network and offer
a vivid interpretation of the sign rule represented by FNN.

We systematically test our theory on various spin models and the Fermi-Hubbard
ring. For non-frustrated spin-$1/2$ models, such as a generalized Ising ring,
(twisted) XY rings, and an antiferromagnetic Heisenberg ring, the sign rules
for ground states with magnetic orders can be fully captured by shn-FNN, where
the accurate rate of the prediction can archive $1$ exactly. However, in the
case of frustrated models where interactions compete, the complexity of sign
rules for ground states is significantly enhanced, reducing prediction
accuracy. Nonetheless, the leading-order sign rules obtained by optimizing
shn-FNN still provide a visual scenario of orders in spins, with the
characteristic weight vector closely related to pitch angles. In the Fermi-
Hubbard ring, we can obtain a unified sign rule by selecting suitable bases.

GWMF may not be suitable for 1D models since quantum fluctuations tend to
destroy long-range orders. However, our current work presents a fresh
perspective by demonstrating that GWMF can effectively capture the leading-
order sign rule in the wave function, where fluctuations in amplitudes are
erased. Our theory is a simple starting point by removing short-range details
in ordered states. It would be intriguing to explore the information encoded
in high-order microscopic processes instead of focusing solely on the leading-
order ones. Of course, the theory for general lattice models also deserves
profound studies in the future.

##  VI Acknowlegement

We thank Tao Li, Rui Wang, Ji-Lu He, Wei Su, and Wei Pan for the grateful
discussion. S. H. acknowledges funding from the Ministry of Science and
Technology of China (Grant No. 2022YFA1402700) and the National Science
Foundation of China (Grants No. 12174020). Z. P. Y. acknowledges funding from
the National Science Foundation of China (Grants No. 12074041). S. H. and K.
X. further acknowledge support from Grant NSAF-U2230402. The computations were
performed on the Tianhe-2JK at the Beijing Computational Science Research
Center (CSRC) and the high-performance computing cluster of Beijing Normal
University in Zhuhai.

##  Appendix A Sign rule for the twisted ferromagnetic spin-$1/2$ XY rings

Here we prove that the even-parity ground-state wave function for the
Hamiltonian $\hat{H}^{P}_{\text{xy}}(J\tilde{J})$ with $J<0$, mentioned in
Sec. IV.1.3 of the main text, has positive signs $s_{\mathbf{n}}>0$.

Under the Jordan-Wigner transformation Derzhko (2001), the spin operators can
be represented by fermion operators as follows:

$\displaystyle\hat{S}^{+}_{l}=\hat{c}^{\dagger}_{l}e^{i\pi\sum_{k<l}\hat{n}_{k}}\
\text{and}\ \hat{S}^{-}_{l}=\hat{c}_{l}e^{-i\pi\sum_{k<l}\hat{n}_{k}}\ $ (54)

where $\hat{c}_{l}$, $\hat{c}^{\dagger}_{l}$, and
$\hat{n}_{l}=\hat{c}^{\dagger}_{l}\hat{c}_{l}$ represent the annihilation,
creation, and particle number operators of fermion, respectively. As a result,
the Hamiltonian $\hat{H}^{P}_{\text{xy}}(J\tilde{J})$ can be transformed into
the one for spinless free fermions defined as

$\displaystyle\hat{H}^{P/T}_{F}(J)=J\left(\sum^{L-1}_{l=1}\hat{c}^{\dagger}_{l}\hat{c}_{l+1}\pm\hat{c}^{\dagger}_{L}\hat{c}_{1}\right)+{\rm
h.c.}.$ (55)

The selection of periodic or twisted boundary conditions in the fermion model
depends on whether the particle number $N$ is odd or even.

For odd $N$, the single-particle levels are described by plane waves
$\exp(ik_{m}l)$ with discrete momenta $k_{m}=2m\pi/L$, where the integer $m$
ranges from $0$ to $L-1$. The energy $\epsilon_{m}$ for the $m$-th single-
particle level follows a formula $\epsilon_{m}=2J\cos(k_{m}+\varphi)$ with the
phase angle $\varphi=\pi/L$. At half-filling $N=L/2$, the ground state selects
single-particle levels with the integer $m\in(-L/4,\ L/4]$. Therefore, the
ground-state wave function for the Hamiltonian $\hat{H}^{P}_{F}(J\tilde{J})$,
seen as a determinant of selected plane waves, is the same as the one for the
other Hamiltonian $\hat{H}^{P}_{F}(J)$ at half-filling. Similarly, when $N$ is
even, the ground-state wave function for the Hamiltonian
$\hat{H}^{P}_{F}(J\tilde{J})$ is the same as the one for the Hamiltonian
$\hat{H}^{T}_{F}(J)$ as well. In conclusion, both Hamiltonian
$\hat{H}^{P/T}_{F}(J)$ for odd/even $N$ can be transformed back to the unique
Hamiltonian $\hat{H}^{P}_{\text{xy}}(J)$ through the inverse Jordan-Wigner
transformation, where the even-parity ground-state wave function always has
positive signs, according to the Perron-Frobenius theorem Perron (1907);
Frobenius (1909).

## References

  * Eisert  _et al._ (2010) J. Eisert, M. Cramer, and M. B. Plenio, Reviews of Modern Physics 82, 277 (2010).
  * Chertkov and Clark (2018) E. Chertkov and B. K. Clark, Physical Review X 8, 031029 (2018).
  * Wang  _et al._ (2019) C. Wang, H. Zhai, and Y. Z. You, Science Bulletin 64, 1228 (2019).
  * Irkhin and Skryabin (2019) V. Y. Irkhin and Y. N. Skryabin, Physics of Metals and Metallography 120, 513 (2019).
  * Grover and Fisher (2015) T. Grover and M. P. A. Fisher, Physical Review A 92, 042308 (2015).
  * Perron (1907) O. Perron, Mathematische Annalen 64, 248 (1907).
  * Frobenius (1909) G. Frobenius,  _Matrices from positive elements. II_ , 1 (Walter de Gruyter & CO, Berlin, Germany, 1909) pp. 514–518.
  * Marshall (1955) W. Marshall, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 232, 48 (1955).
  * Lieb  _et al._ (1961) E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961).
  * Lieb and Mattis (1962) E. Lieb and D. Mattis, Journal of Mathematical Physics 3, 749 (1962).
  * Zeng and Parkinson (1995) C. Zeng and J. B. Parkinson, Phys. Rev. B 51, 11609 (1995).
  * Bursill  _et al._ (1995) R. Bursill, G. A. Gehring, D. J. J. Farnell, J. B. Parkinson, T. Xiang, and C. Zeng, Journal of Physics: Condensed Matter 7, 8605 (1995).
  * Retzlaff  _et al._ (1993) K. Retzlaff, J. Richter, and N. B. Ivanov, Zeitschrift für Physik B Condensed Matter 93, 21 (1993).
  * Richter  _et al._ (1994) J. Richter, N. B. Ivanov, and K. Retzlaff, Europhysics Letters 25, 545 (1994).
  * Cai and Liu (2018) Z. Cai and J. Liu, Phys. Rev. B 97, 035116 (2018).
  * Westerhout  _et al._ (2020) T. Westerhout, N. Astrakhantsev, K. S. Tikhonov, M. I. Katsnelson, and A. A. Bagrov, Nature Communications 11, 1593 (2020).
  * White (1992) S. R. White, Physical Review Letters 69, 2863 (1992).
  * Peschel  _et al._ (1999) I. Peschel, M. Kaulke, X. Wang, and K. Hallberg, eds.,  _Density-Matrix Renormalization_ (Springer Berlin Heidelberg, 1999).
  * Schollwöck (2011) U. Schollwöck, Annals of Physics 326, 96 (2011).
  * Orús (2014) R. Orús, Annals of Physics 349, 117 (2014).
  * Carleo and Troyer (2017) G. Carleo and M. Troyer, Science 355, 602 (2017).
  * Carleo  _et al._ (2019) G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Reviews of Modern Physics 91, 045002 (2019).
  * Jia  _et al._ (2019) Z. A. Jia, B. Yi, R. Zhai, Y. C. Wu, G. C. Guo, and G. P. Guo, Advanced Quantum Technologies 2, 1800077 (2019).
  * Vivas  _et al._ (2022) D. R. Vivas, J. Madroñero, V. Bucheli, L. O. Gómez, and J. H. Reina, arXiv preprint arXiv:2204.12966 (2022).
  * Choo  _et al._ (2019) K. Choo, T. Neupert, and G. Carleo, Physical Review B 100, 125124 (2019).
  * Szabó and Castelnovo (2020) A. Szabó and C. Castelnovo, Physical Review Research 2, 033075 (2020).
  * Bukov  _et al._ (2021) M. Bukov, M. Schmitt, and M. Dupont, SciPost Physics 10, 147 (2021).
  * Roscher  _et al._ (2020) R. Roscher, B. Bohn, M. F. Duarte, and J. Garcke, IEEE Access 8, 42200 (2020).
  * He  _et al._ (2020) C. He, M. Ma, and P. Wang, Neurocomputing 387, 346 (2020).
  * Fan  _et al._ (2021) F. L. Fan, J. J. Xiong, M. Z. Li, and G. Wang, IEEE Transactions on Radiation and Plasma Medical Sciences 5, 741 (2021).
  * Raissi  _et al._ (2019) M. Raissi, P. Perdikaris, and G. Karniadakis, Journal of Computational Physics 378, 686 (2019).
  * Yuan and Weng (2021) J. Yuan and Y. Weng, in  _2021 IEEE International Conference on Data Mining (ICDM)_ (IEEE, 2021).
  * Cai  _et al._ (2021) S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis, Acta Mechanica Sinica 37, 1727 (2021).
  * Vasiliev  _et al._ (2018) A. Vasiliev, O. Volkova, E. Zvereva, and M. Markina, npj Quantum Materials 3, 18 (2018).
  * Gardiner and Zoller (2017) C. Gardiner and P. Zoller,  _The Quantum World of Ultra-Cold Atoms and Light Book III: Ultra-Cold Atoms_ (World Scientific (Europe), 2017).
  * Meher and Sivakumar (2022) N. Meher and S. Sivakumar, arXiv:2204.01322 (2022).
  * Makhlin  _et al._ (2001) Y. Makhlin, G. Schön, and A. Shnirman, Review Modern Physics 73, 357 (2001).
  * Luo (2008) S. L. Luo, Physics Review A 77, 042303 (2008).
  * Kjaergaard  _et al._ (2020) M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annual Review of Condensed Matter Physics 11, 369 (2020).
  * Penc and Läuchli (2010) K. Penc and A. M. Läuchli, in  _Introduction to Frustrated Magnetism_ (Springer Berlin Heidelberg, 2010) pp. 331–362.
  * Gutzwiller (1963) M. C. Gutzwiller, Physical Review Letters 10, 159 (1963).
  * Gutzwiller (1965) M. C. Gutzwiller, Physical Review 137, A1726 (1965).
  * Cybenko (1989) G. Cybenko, Mathematics of Control, Signals and Systems 2, 303 (1989).
  * Hornik  _et al._ (1989) K. Hornik, M. Stinchcombe, and H. White, Neural Networks 2, 359 (1989).
  * Goodfellow  _et al._ (2016) I. Goodfellow, Y. Bengio, and A. Courville,  _Deep Learning_ (MIT Press, 2016) http://www.deeplearningbook.org.
  * Iten  _et al._ (2020) R. Iten, T. Metger, H. Wilming, L. del Rio, and R. Renner, Physical Review Letters 124, 010508 (2020).
  * Wang  _et al._ (2021) W. Wang, Z. Wang, Y. Zhang, B. Sun, and K. Xia, Physical Review Applied 16, 014005 (2021).
  * Rumelhart  _et al._ (1986) D. E. Rumelhart, G. E. Hinton, and R. J. Williams, in  _Parallel Distributed Processing_ , edited by D. E. Rumelhart and R. J. McClelland (MIT Press, Cambridge, Mass., 1986) Chap. 8.
  * P and Ba (2014) D. P, Kingma and J. Ba, arxiv:1412.6980 (2014).
  * Wilson and Martinez (2003) D. Wilson and T. R. Martinez, Neural Networks 16, 1429 (2003).
  * Abadi  _et al._ (2016) M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, M. Kudlur, J. Levenberg, R. Monga, S. Moore, D. G. Murray, B. Steiner, P. Tucker, V. Vasudevan, P. Warden, M. Wicke, Y. Yu, and X. Zheng, in  _12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16)_ (USENIX Association, Savannah, GA, 2016) pp. 265–283.
  * Tomonaga (1950) S. I. Tomonaga, Progress of Theoretical Physics 5, 544 (1950).
  * Luttinger (1963) J. M. Luttinger, Journal of Mathematical Physics 4, 1154 (1963).
  * Haldane (1981) F. D. M. Haldane, Journal of Physics C: Solid State Physics 14, 2585 (1981).
  * White and Affleck (1996) S. R. White and I. Affleck, Physical Review B 54, 9862 (1996).
  * Soos  _et al._ (2016) Z. G. Soos, A. Parvej, and M. Kumar, Journal of Physics: Condensed Matter 28, 175603 (2016).
  * Anderson (1967) P. W. Anderson, Physical Review Letters 18, 1049 (1967).
  * Gu (2010) S. J. Gu, International Journal of Modern Physics B 24, 4371 (2010).
  * Wang  _et al._ (2009) L. Wang, S.-J. Gu, and S. Chen, arXiv:0903.4242 (2009).
  * Cincio  _et al._ (2019) L. Cincio, M. M. Rams, J. Dziarmaga, and W. H. Zurek, Physical Review B 100, 081108(R) (2019).
  * Bach  _et al._ (2021) A. Bach, M. Cicalese, L. Kreutz, and G. Orlando, Calculus of Variations and Partial Differential Equations 60, 149 (2021).
  * Wang  _et al._ (2018) R. Wang, B. Wang, and T. A. Sedrakyan, Physical Review B 98, 064402 (2018).
  * Sedrakyan  _et al._ (2020) T. Sedrakyan, R. Moessner, and A. Kamenev, Physical Review B 102, 024430 (2020).
  * Henderson  _et al._ (1992) J. A. Henderson, J. Oitmaa, and M. C. B. Ashley, Phys. Rev. B 46, 6328 (1992).
  * Essler  _et al._ (2005) F. H. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin,  _The one-dimensional Hubbard model_ (Cambridge University Press, 2005).
  * Moriya (2012) T. Moriya,  _Electron Correlation and Magnetism in Narrow-Band Systems: Proceedings of the Third Taniguchi International Symposium, Mount Fuji, Japan, November 1–5, 1980_ , Vol. 29 (Springer Science & Business Media, 2012).
  * Derzhko (2001) O. Derzhko, arXiv:cond-mat/0101188 (2001).
  * Lieb and Wu (1968) E. H. Lieb and F. Y. Wu, Physical Review Letters 20, 1445 (1968).