Ying Yang(杨莹)¹, Huaixin Cao(曹怀信)^,21School of Mathematics and Information Technology, Yuncheng University, Yuncheng 044000, China
²School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, China

Received:2021-05-16Revised:2021-07-3Accepted:2021-07-5Online:2021-08-31

Fund supported:

* Supported by the National Natural Science Foundation of China.12001480 Supported by the National Natural Science Foundation of China.11871318 Applied Basic Research Program of Shanxi Province.201901D211461 Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi.2020L0554 Excellent Doctoral Research Project of Shanxi Province.QZX-2020001 PhD Start-up Project of Yuncheng University.YQ-2019021

Abstract
The quantum many-body problem (QMBP) has become a hot topic in high-energy physics and condensed-matter physics. With an exponential increase in the dimensions of Hilbert space, it becomes very challenging to solve the QMBP, even with the most powerful computers. With the rapid development of machine learning, artificial neural networks provide a powerful tool that can represent or approximate quantum many-body states. In this paper, we aim to explicitly construct the neural network representations of hypergraph states. We construct the neural network representations for any k-uniform hypergraph state and any hypergraph state, respectively, without stochastic optimization of the network parameters. Our method constructively shows that all hypergraph states can be represented precisely by the appropriate neural networks introduced in [Science 355 (2017) 602] and formulated in [Sci. China-Phys. Mech. Astron. 63 (2020) 210312].
Keywords： hypergraph state;neural network quantum state;representation


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Ying Yang(杨莹), Huaixin Cao(曹怀信). Representations of hypergraph states with neural networks^*. Communications in Theoretical Physics, 2021, 73(10): 105103- doi:10.1088/1572-9494/ac1101

1. Introduction

In quantum physics, fully understanding and characterising a complex system with a large number of interacting particles is an extremely challenging problem. Solutions within the standard framework of quantum mechanics generally require knowledge of the full quantum many-body wave function. Thus, the problem becomes one of how to solve the many-body Schrödinger equation of a large dimension system. This is just the so-called quantum many-body problem (QMBP) in quantum physics, which has become a hot topic in high-energy physics and condensed-matter physics. When the dimension of the Hilbert space describing the system is exponentially large, it becomes very challenging to solve the QMBP, even with the most powerful computers.

Many methods have been used to overcome this exponential difficulty and solve the QMBP, including the tensor network method (TNM) [1–3] and quantum Monte Carlo simulation (QMCS) [4]. However, the TNM has difficulty dealing with high-dimensional systems [5] or systems with massive entanglement [6]; the QMCS suffers from the sign problem [7]. Thus, some new methods of finding the QMBP are required.

The approximation capabilities of artificial neural networks (ANNWs) have been investigated by many authors, including Cybenko [8], Funahashi [9], Hornik [10, 11], Kolmogorov [12], and Roux [13]. It is known that ANNWs can be used in many fields, including the representation of complex correlations in multiple-variable functions or probability distributions [13], the study of artificial intelligence through the popularity of deep learning methods [14], and so on [15–19].

Undoubtedly, the interaction between machine learning and quantum physics will benefit both fields [20, 21]. For instance, in light of the idea of machine learning, Carleo and Troyer [22] found an interesting connection between the variational approach in the QMBP and learning methods based on neural network representations. They used a restricted Boltzmann machine (RBM) to describe a many-body wave function and obtained an efficient variational representation by optimizing those variational parameters with powerful learning methods. Yang et al [23] researched an approximation of an unknown ground state of a given Hamiltonian using neural network quantum states. Numerical evidence suggests that an RBM optimized by the reinforcement learning method can provide a good solution to several QMBPs [24–31].

However, the solutions obtained are approximate but not exact. To find the exact solution of a QMBP using an ANNW, the authors of [32] introduced neural network quantum states (NNQSs) with general input observables from the mathematical point of view, and found some N-qubit states that can be represented by normalized NNQS, such as all separable pure states, Bell states, and Greenberger-Horne-Zeilinger (GHZ) states. Gao et al [33] showed that every graph state has an RBM representation (RBMR) and gave a simple construction of the RBMR for a graph state.

Lu et al [19] theoretically proved that every hypergraph state can be represented by an RBM with a $\{0,1\}$ input and obtained the RBMRs of 2- and 3-uniform hypergraph states, which are not the NNQS introduced by [22] and formulated in [32]. What we care about is whether we can construct a neural network representation of any hypergraph state using the $\{1,-1\}$-input NNQS considered in [22] and [32].

In this paper, we will aim to explicitly construct the neural network representations of arbitrary hypergraph states. In section 2, some notations and conclusions for NNQS with general input observables will be recalled and some related properties will be proved. In sections 3 and 4, the neural network representations for any k-uniform hypergraph state and any hypergraph state will be constructed, respectively, without stochastic optimization of the network parameters.

2. Neural network quantum states

Let us start with a brief introduction to some notations in the neural network architecture introduced by [22] and formulated in [32].

Let ${Q}_{1},{Q}_{2},\ldots ,{Q}_{N}$ be N quantum systems with state spaces ${{ \mathcal H }}_{1},{{ \mathcal H }}_{2},\ldots ,{{ \mathcal H }}_{N}$ with dimensions of ${d}_{1},{d}_{2},\ldots ,{d}_{N}$, respectively. We consider the composite system Q of ${Q}_{1},{Q}_{2},\ldots ,{Q}_{N}$ with the state space ${ \mathcal H }:= {{ \mathcal H }}_{1}\otimes {{ \mathcal H }}_{2}\otimes \ldots \otimes {{ \mathcal H }}_{N}$.

Let ${S}_{1},{S}_{2},\ldots ,{S}_{N}$ be non-degenerate observables of the systems ${Q}_{1},{Q}_{2},\ldots ,{Q}_{N}$, respectively. Then $S={S}_{1}\otimes {S}_{2}\otimes \ldots \otimes {S}_{N}$ is an observable of the composite system Q. We use ${\{| {\psi }_{{k}_{j}}\rangle \}}_{{k}_{j}=0}^{{d}_{j}-1}$ to denote the eigenbasis of S_j corresponding to the eigenvalues ${\{{\lambda }_{{k}_{j}}\}}_{{k}_{j}=0}^{{d}_{j}-1}$. Thus,(1)$\begin{eqnarray}{S}_{j}| {\psi }_{{k}_{j}}\rangle ={\lambda }_{{k}_{j}}| {\psi }_{{k}_{j}}\rangle ({k}_{j}=0,1,\ldots ,{d}_{j}-1).\end{eqnarray}$It is easy to check that the eigenvalues and corresponding eigenbasis of $S={S}_{1}\otimes {S}_{2}\otimes \cdots \otimes {S}_{N}$ are ${\lambda }_{{k}_{1}}{\lambda }_{{k}_{2}}\ldots {\lambda }_{{k}_{N}},$(2)$\begin{eqnarray}| {\psi }_{{k}_{1}}\rangle \otimes | {\psi }_{{k}_{2}}\rangle \otimes \ldots \otimes | {\psi }_{{k}_{N}}\rangle ({k}_{j}=0,1,\ldots ,{d}_{j}-1),\end{eqnarray}$respectively. We write$\begin{eqnarray*}\begin{array}{rcl}V(S) & = & \left\{{{\rm{\Lambda }}}_{{k}_{1}{k}_{2}\ldots {k}_{N}}\equiv {\left({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}}\right)}^{{\rm{T}}}:\right.\\ {k}_{j} & = & 0,1,\left.\ldots ,{d}_{j}-1\right\},\end{array}\end{eqnarray*}$which is called an input space. For parameters$\begin{eqnarray*}a={\left({a}_{1},{a}_{2},\ldots ,{a}_{N}\right)}^{{\rm{T}}}\in {{\mathbb{C}}}^{N},b={\left({b}_{1},{b}_{2},\ldots ,{b}_{M}\right)}^{{\rm{T}}}\in {{\mathbb{C}}}^{M},\end{eqnarray*}$$\begin{eqnarray*}W=[{W}_{{ij}}]\in {{\mathbb{C}}}^{M\,\times \,N},\end{eqnarray*}$we write ${\rm{\Omega }}=(a,b,W)$ and(3)$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ =\sum _{{h}_{i}=\pm 1}\exp \left(\sum _{j=1}^{N}{a}_{j}{\lambda }_{{k}_{j}}+\sum _{i=1}^{M}{b}_{i}{h}_{i}+\sum _{i=1}^{M}\sum _{j=1}^{N}{W}_{{ij}}{h}_{i}{\lambda }_{{k}_{j}}\right).\end{array}\end{eqnarray}$We then obtain a complex-valued function ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$ of the input variable ${{\rm{\Lambda }}}_{{k}_{1}{k}_{2}\ldots {k}_{N}}$. We call it a neural network quantum wave function (NNQWF). It may be identical to zero. In what follows, we assume that this is not the case, that is, we assume that ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\ne 0$ for some input variable ${{\rm{\Lambda }}}_{{k}_{1}{k}_{2}\ldots {k}_{N}}.$ We then define(4)$\begin{eqnarray}\begin{array}{rcl}| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle & = & \sum _{{{\rm{\Lambda }}}_{{k}_{1}{k}_{2}\ldots {k}_{N}}\in V(S)}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ & & \times | {\psi }_{{k}_{1}}\rangle \otimes | {\psi }_{{k}_{2}}\rangle \otimes \cdots \otimes | {\psi }_{{k}_{N}}\rangle ,\end{array}\end{eqnarray}$which is a nonzero vector (not necessarily normalized) of the Hilbert space ${ \mathcal H }$. We call it a neural network quantum state (NNQS) induced by the parameter ${\rm{\Omega }}=(a,b,W)$ and the input observable $S={S}_{1}\otimes {S}_{2}\otimes \cdots \otimes {S}_{N}$ (figure 1).

Figure 1.

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Figure 1.Artificial neural network encoding an NNQS. It is a restricted Boltzmann machine architecture that features a set of N visible artificial neurons (blue disks) and a set of M hidden neurons (yellow disks). For each value ${{\rm{\Lambda }}}_{{k}_{1}{k}_{2}\ldots {k}_{N}}$ of the input observable S, the neural network computes the value of ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$.


The NNQWF can be reduced to(5)$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ =\prod _{j=1}^{N}{{\rm{e}}}^{{a}_{j}{\lambda }_{{k}_{j}}}\cdot \prod _{i=1}^{M}2\cosh \left({b}_{i}+\sum _{j=1}^{N}{W}_{{ij}}{\lambda }_{{k}_{j}}\right).\end{array}\end{eqnarray}$

There is a special class of NNQSs; when $S={\sigma }_{1}^{z}\otimes {\sigma }_{2}^{z}\otimes \cdots \otimes {\sigma }_{N}^{z},1\leqslant j\leqslant N,$ we have

(6)$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{{k}_{j}} & = & \left\{\begin{array}{ll}1, & {k}_{j}=0\\ -1, & {k}_{j}=1\end{array}\right.,\\ | {\psi }_{{k}_{j}}\rangle & = & \left\{\begin{array}{ll}| 0\rangle , & {k}_{j}=0\\ | 1\rangle , & {k}_{j}=1\end{array}\right.\end{array}\end{eqnarray}$and $V(S)=\{1,-1\}{}^{N}$.

In this case, the NNQS (4) becomes(7)$\begin{eqnarray}\begin{array}{rcl}| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle & = & \sum _{{{\rm{\Lambda }}}_{{k}_{1}{k}_{2}\ldots {k}_{N}}\in \{1,-1\}{}^{N}}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ & & \times | {\psi }_{{k}_{1}}\rangle \otimes | {\psi }_{{k}_{2}}\rangle \otimes \cdots \otimes | {\psi }_{{k}_{N}}\rangle .\end{array}\end{eqnarray}$This leads to the NNQS introduced in [22] and discussed in [34]. We call such an NNQS a spin-z NNQS.

From the definition of an NNQWF, we can easily obtain the following results.

Proposition 1. If a hidden-layer neuron ${h}_{M+1}$ is added into an RBM with NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$, then the NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}^{\prime} }({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$ of the resulting network reads$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{S,{\rm{\Omega }}^{\prime} }({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ \quad =\,{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\cdot {{\rm{\Psi }}}_{S,\widetilde{{\rm{\Omega }}}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}}),\end{array}\end{eqnarray*}$where$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{S,\widetilde{{\rm{\Omega }}}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ =\,\sum _{{h}_{M+1}=\pm 1}\exp \left(\sum _{j=1}^{N}{\tilde{a}}_{j}{\lambda }_{{k}_{j}}+\tilde{b}{h}_{M+1}+\sum _{j=1}^{N}{h}_{M+1}{\widetilde{W}}_{(M+1)j}{\lambda }_{{k}_{j}}\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}{\rm{\Omega }}=(a,b,W),{\rm{\Omega }}^{\prime} =(a^{\prime} ,b^{\prime} ,W^{\prime} ),\widetilde{{\rm{\Omega }}}=(\tilde{a},\tilde{b},\widetilde{W}),\end{eqnarray*}$$\begin{eqnarray*}a^{\prime} =a+\tilde{a},\tilde{b}={b}_{M+1},\end{eqnarray*}$$\begin{eqnarray*}\widetilde{W}=({\widetilde{W}}_{(M+1)1},{\widetilde{W}}_{(M+1)2},\cdots ,{\widetilde{W}}_{(M+1)N}),\end{eqnarray*}$$\begin{eqnarray*}b^{\prime} =\left(\begin{array}{c}b\\ \tilde{b}\\ \end{array}\right)\in {{\mathbb{C}}}^{M+1},\ W^{\prime} =\left(\begin{array}{c}W\\ \widetilde{W}\\ \end{array}\right)\in {{\mathbb{C}}}^{(M+1)\times N}.\end{eqnarray*}$

This result can be illustrated by figure 2.

Figure 2.

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Figure 2.The network that results from adding a hidden-layer neuron ${h}_{M+1}$ into a network with visible layers ${S}_{1},{S}_{2},\ldots ,{S}_{N}$ and hidden layers ${h}_{1},{h}_{2},\ldots ,{h}_{M}$.


Proposition 2. Supposing that ${{\rm{\Psi }}}_{S,{\rm{\Omega }}^{\prime} }({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$ and ${{\rm{\Psi }}}_{S,{\rm{\Omega }}^{\prime\prime} }({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$ are two spin-z NNQWFs with the same input observable$\begin{eqnarray*}S={\sigma }_{1}^{z}\otimes {\sigma }_{2}^{z}\otimes \cdots \otimes {\sigma }_{N}^{z},\end{eqnarray*}$and the individual parameters$\begin{eqnarray*}{\rm{\Omega }}^{\prime} =(a^{\prime} ,b^{\prime} ,W^{\prime} ),{\rm{\Omega }}^{\prime\prime} =(a^{\prime\prime} ,b^{\prime\prime} ,W^{\prime\prime} ),\end{eqnarray*}$respectively. Then$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{S,{\rm{\Omega }}^{\prime} }({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\cdot {{\rm{\Psi }}}_{S,{\rm{\Omega }}^{\prime\prime} }({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})\\ \quad ={{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}}),\end{array}\end{eqnarray*}$where$\begin{eqnarray*}{\rm{\Omega }}=(a,b,W),a=a^{\prime} +a^{\prime\prime} ,\end{eqnarray*}$$\begin{eqnarray*}b=\left(\begin{array}{c}b^{\prime} \\ b^{\prime\prime} \\ \end{array}\right)\in {{\mathbb{C}}}^{M^{\prime} +M^{\prime\prime} },\ W=\left(\begin{array}{c}W^{\prime} \\ W^{\prime\prime} \\ \end{array}\right)\in {{\mathbb{C}}}^{(M^{\prime} +M^{\prime\prime} )\times N}.\end{eqnarray*}$

3. Neural network representations of k-uniform hypergraph states

Generally, for a given pure state $| \psi \rangle $, if an NNQS $| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $ and a normalized constant z exist, such that $| \psi \rangle =z| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $, then we say that $| \psi \rangle $ can be represented by the NNQS $| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $. The authors of [32] found some N-qubit states that can be represented by a normalized NNQS, such as all separable pure states, Bell states, and GHZ states. It was proved in [33] and [19] that all graph states and all hypergraph states can be represented by an RBM with a $\{0,1\}$ input.

In this section, we aim to construct a neural network representation of any k-uniform hypergraph state by using {1, −1}-input NNQS given by [22], rather than $\{0,1\}$-input NNQS.

To do this, let us start by briefly recalling the definition of the k-uniform hypergraph state, which is an extension of the concept of graph state. A k-uniform hypergraph [35] is a pair ${G}_{k}=(V,E)$ consisting of a set $V=\{1,2,\ldots ,N\}$ and a nonempty set E of some k-element subsets of V. The elements of V and E are called the vertices and k-hyperedges of G_k, respectively. When $e=({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E$, we say that the vertices ${i}_{1},{i}_{2},\ldots ,{i}_{k}$ are connected by e.

Thus, a graph in the common sense is just a 2-uniform hypergraph.

Given a k-uniform hypergraph ${G}_{k}=(V,E)$, the k-uniform hypergraph state $| {G}_{k}\rangle $ was defined in [35], as follows:(8)$\begin{eqnarray}| {G}_{k}\rangle =\displaystyle \prod _{({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E}{C}^{k}{Z}_{{i}_{1},{i}_{2},\ldots ,{i}_{k}}\mathop{\underbrace{| +\rangle | +\rangle \cdots | +\rangle }}\limits_{N},\end{eqnarray}$where$\begin{eqnarray*}{C}^{k}{Z}_{{i}_{1},{i}_{2},\ldots ,{i}_{k}}={(I-P)}_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}\otimes {I}_{{i}_{k}}+{P}_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}\otimes {Z}_{{i}_{k}},\end{eqnarray*}$$\begin{eqnarray*}{P}_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}=| 11\ldots 1{\rangle }_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}\langle 11\ldots 1| ,\end{eqnarray*}$$\begin{eqnarray*}| +\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle ).\end{eqnarray*}$

In fact,$\begin{eqnarray*}\begin{array}{l}{C}^{k}{Z}_{{i}_{1},{i}_{2},\ldots ,{i}_{k}}| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle \\ \quad =\,\left\{\begin{array}{ll}-| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle , & {j}_{{i}_{1}}={j}_{{i}_{2}}=\ldots ={j}_{{i}_{k}}=1;\\ | {j}_{1}{j}_{2}\ldots {j}_{N}\rangle , & {\rm{otherwise}},\end{array}\right.\end{array}\end{eqnarray*}$for all ${j}_{1},{j}_{2},\ldots ,{j}_{N}=0,1$. Thus,(9)$\begin{eqnarray}{C}^{k}{Z}_{{i}_{1},{i}_{2},\ldots ,{i}_{k}}| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle ={\left(-1\right)}^{{j}_{{i}_{1}}{j}_{{i}_{2}}\cdots {j}_{{i}_{k}}}| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle ,\end{eqnarray}$for all ${j}_{1},{j}_{2},\ldots ,{j}_{N}=0,1$.

Here, we try to construct neural network representations for any k-uniform hypergraph state $| {G}_{k}\rangle $, that is to find an NNQS $| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $ such that $| {G}_{k}\rangle =z| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $ for some normalized constant z.

First, we reduce equation (8) for a k-uniform hypergraph state by the next procedure.

Since$\begin{eqnarray*}| +{\rangle }^{\otimes N}=\mathop{\underbrace{| +\rangle | +\rangle \cdots | +\rangle }}\limits_{N}=\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\sum _{{j}_{1},{j}_{2},\ldots ,{j}_{N}=0,1}| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle ,\end{eqnarray*}$we see from (9) that$\begin{eqnarray*}\begin{array}{rcl}| {G}_{k}\rangle & = & \prod _{({i}_{1},\ldots ,{i}_{k})\in E}{C}^{k}{Z}_{{i}_{1},\ldots ,{i}_{k}}\mathop{\underbrace{| +\rangle | +\rangle \cdots | +\rangle }}\limits_{N}\\ & = & \sum _{{j}_{1},\ldots ,{j}_{N}=0,1}\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}{C}^{k}{Z}_{{i}_{1},\ldots ,{i}_{k}}| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle \\ & = & \sum _{{j}_{1},\ldots ,{j}_{N}=0,1}\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{{j}_{{i}_{1}}\cdots {j}_{{i}_{k}}}| {j}_{1}{j}_{2}\ldots {j}_{N}\rangle .\end{array}\end{eqnarray*}$Given that$\begin{eqnarray*}{\left(-1\right)}^{{j}_{{i}_{1}}{j}_{{i}_{2}}\cdots {j}_{{i}_{k}}}={\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\left(1-{\lambda }_{{j}_{{i}_{2}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)}{{2}^{k}}},\end{eqnarray*}$we obtain(10)$\begin{eqnarray}\begin{array}{rcl}| {G}_{k}\rangle & = & \sum _{{{\rm{\Lambda }}}_{{j}_{1}{j}_{2}\ldots {j}_{N}}\in \{1,-1\}{}^{N}}{{\rm{\Psi }}}_{{G}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ & & \times | {\psi }_{{j}_{1}}\rangle \otimes | {\psi }_{{j}_{2}}\rangle \otimes \ldots \otimes | {\psi }_{{j}_{N}}\rangle ,\end{array}\end{eqnarray}$where(11)$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{G}_{k}}({\lambda }_{{j}_{1}},\ldots ,{\lambda }_{{j}_{N}})\\ =\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)}{{2}^{k}}},\end{array}\end{eqnarray}$and ${\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}},| {\psi }_{{j}_{1}}\rangle ,\ldots ,| {\psi }_{{j}_{N}}\rangle $ are shown in equation (6). We see that the simplified expression equation (10) is simpler and easier to use. Given a k-uniform hypergraph, we can use this expression to obtain a k-uniform hypergraph state associated with it very quickly. For example, for the 3-uniform hypergraph in figure 3, the corresponding 3-uniform hypergraph state is$\begin{eqnarray*}\begin{array}{rcl}| {G}_{3}\rangle & = & \displaystyle \frac{1}{4}\left(| 0000\rangle +| 0001\rangle +| 0010\rangle +| 0011\rangle \right.\\ & & +| 0100\rangle +| 0101\rangle +| 0110\rangle -| 0111\rangle \\ & & +| 1000\rangle +| 1001\rangle +| 1010\rangle +| 1011\rangle \\ & & \left.+| 1100\rangle +| 1101\rangle -| 1110\rangle +| 1111\rangle \right).\end{array}\end{eqnarray*}$

Figure 3.

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Figure 3.A 3-uniform hypergraph with four vertices.


In addition, using this simplification equation (10), the k-uniform hypergraph state obtained is similar to a spin-z NNQS equation (7), which sets the stage for our follow-up work.

Besides, the wave function of the k-uniform hypergraph state $| {G}_{k}\rangle $ is given by (11). By writing$\begin{eqnarray*}\begin{array}{rcl} & & \left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\left(1-{\lambda }_{{j}_{{i}_{2}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)\\ & = & \sum _{\begin{array}{c}\{{i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},{i}_{2},\ldots ,{i}_{k}\}\\ m=0,1,\ldots ,k\end{array}}{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}}\\ & = & 1+{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})+{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}}),\end{array}\end{eqnarray*}$where$\begin{eqnarray*}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})=-({\lambda }_{{j}_{{i}_{1}}}+\ldots +{\lambda }_{{j}_{{i}_{k}}})+{\left(-1\right)}^{k}{\lambda }_{{j}_{{i}_{1}}}\ldots {\lambda }_{{j}_{{i}_{k}}},\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})\\ =\sum _{\begin{array}{c}\{{i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},\ldots ,{i}_{k}\}\\ m=2,\ldots ,k-1\end{array}}{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}},\end{array}\end{eqnarray*}$we get(12)$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{G}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ \quad =\,\displaystyle \frac{{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | E| }{{2}^{k}}}}{{\left(\sqrt{2}\right)}^{N}}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}{{2}^{k}}}\\ \quad \times \,{\left(-1\right)}^{\tfrac{{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}{{2}^{k}}}.\end{array}\end{eqnarray}$

Next, we try to construct an NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})$, such that$\begin{eqnarray*}{{\rm{\Psi }}}_{{G}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})=z{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}),\end{eqnarray*}$for some constant z, where$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ =\sum _{{h}_{q}=\pm 1}\exp \left(\sum _{q=1}^{N}{a}_{q}{\lambda }_{{j}_{q}}+\sum _{p=1}^{M}{b}_{p}{h}_{p}+\sum _{p=1}^{M}\sum _{q=1}^{N}{W}_{{pq}}{h}_{p}{\lambda }_{{j}_{q}}\right)\\ =\prod _{q=1}^{N}{{\rm{e}}}^{{a}_{q}{\lambda }_{{j}_{q}}}\cdot \prod _{p=1}^{M}2\cosh \left({b}_{p}+\sum _{q=1}^{N}{W}_{{pq}}{\lambda }_{{j}_{q}}\right).\end{array}\end{eqnarray*}$

${\boldsymbol{Case}}\,{\bf{1}}.$ k=1. Let $E=\{({m}_{1}),({m}_{2}),\cdots ,({m}_{s})\}$, $| E| =s$. From the discussion above, we can easily find that the wave function of the 1-uniform hypergraph state $| {G}_{1}\rangle $ is$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{{G}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ \quad =\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{({i}_{1})\in E}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)}{2}}\\ \quad =\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | E| }{2}}{{\rm{e}}}^{-\tfrac{\pi {\rm{i}}\left({\lambda }_{{j}_{{m}_{1}}}+{\lambda }_{{j}_{{m}_{2}}}+\ldots +{\lambda }_{{j}_{{m}_{s}}}\right)}{2}}.\end{array}\end{eqnarray*}$

Let ${{\rm{\Omega }}}_{1}=(a,b,W)$, where$\begin{eqnarray*}a=[{a}_{v}]\in {{\mathbb{C}}}^{N},{a}_{v}=\left\{\begin{array}{ll}-\displaystyle \frac{\pi {\rm{i}}}{2}, & v={m}_{1},\cdots ,{m}_{s},\\ 0, & \mathrm{otherwise},\end{array}\right.\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}b & = & [{b}_{v}]\in {{\mathbb{C}}}^{M},{b}_{v}=\displaystyle \frac{\pi {\rm{i}}}{3},W\,=\,0\in {{\mathbb{C}}}^{M\times N},\\ z & = & \displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | E| }{2}}.\end{array}\end{eqnarray*}$The NNQWF with these parameters then reads(13)$\begin{eqnarray}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})={{\rm{e}}}^{-\tfrac{\pi {\rm{i}}\left({\lambda }_{{j}_{{m}_{1}}}+{\lambda }_{{j}_{{m}_{2}}}+\ldots +{\lambda }_{{j}_{{m}_{s}}}\right)}{2}},\end{eqnarray}$and so$\begin{eqnarray*}{{\rm{\Psi }}}_{{G}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})=z{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}).\end{eqnarray*}$This implies that any 1-uniform hypergraph state $| {G}_{1}\rangle $ can be represented by a spin-z NNQS.

${\boldsymbol{Case}}\,{\bf{2}}.$ $k\geqslant 2$. Given ${z}_{0}=\tfrac{1}{{\left(\sqrt{2}\right)}^{N}}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | E| }{{2}^{k}}}$, we only need to construct the NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}})$ to satisfy the following equation:(14)$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ \quad =\,\displaystyle \frac{{z}_{0}}{z}\prod _{({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}\\ \qquad \times {\left(-1\right)}^{\tfrac{1}{{2}^{k}}{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}.\end{array}\end{eqnarray}$

To construct the NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{k}_{1}},{\lambda }_{{k}_{2}},\ldots ,{\lambda }_{{k}_{N}}),$ we first represent the function$\begin{eqnarray*}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})},\end{eqnarray*}$as some small NNQWFs for each $({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E$, and then proceed to construct the NNQWF that we needed.

${\boldsymbol{Step}}\,{\bf{1}}.$ Noting that$\begin{eqnarray*}\begin{array}{l}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}\\ =\mathop{{\rm{\Pi }}}\limits_{\displaystyle \genfrac{}{}{0em}{}{\{{i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},{i}_{2},\ldots ,{i}_{k}\}}{m=2,\ldots ,k-1}}{\left(-1\right)}^{{2}^{-k}{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}}},\end{array}\end{eqnarray*}$we first write each factor$\begin{eqnarray*}{\left(-1\right)}^{{2}^{-k}{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}}},\end{eqnarray*}$as an NNQWF. To do this, for $m=2,3,\ldots ,k-1$ and $\{{i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},{i}_{2},\ldots ,{i}_{k}\}$, we write$\begin{eqnarray*}{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=({a}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},{b}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},{W}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}),\end{eqnarray*}$$\begin{eqnarray*}{a}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=0\in {{\mathbb{C}}}^{m},\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{b}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=\left(\begin{array}{c}{b}_{1}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}\\ {b}_{2}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}\end{array}\right),{b}_{1}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}\\ \quad ={b}_{2}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=b{\rm{i}},\end{array}\end{eqnarray*}$$\begin{eqnarray*}b=\arctan \left({{\rm{e}}}^{\tfrac{{\left(-1\right)}^{m+1}\pi {\rm{i}}}{{2}^{k}}}\right)-\displaystyle \frac{m\pi }{4},\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{W}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}\\ \quad =\,\left(\begin{array}{cccc}{W}_{1{i}_{1}^{{\prime} }}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })} & {W}_{1{i}_{2}^{{\prime} }}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })} & \ldots & {W}_{1{i}_{m}^{{\prime} }}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}\\ {W}_{2{i}_{1}^{{\prime} }}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })} & {W}_{2{i}_{2}^{{\prime} }}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })} & \ldots & {W}_{2{i}_{m}^{{\prime} }}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}\end{array}\right)\\ \quad =\,\left(\begin{array}{cccc}\displaystyle \frac{{\rm{i}}\pi }{4} & \displaystyle \frac{{\rm{i}}\pi }{4} & \ldots & \displaystyle \frac{{\rm{i}}\pi }{4}\\ \displaystyle \frac{{\rm{i}}\pi }{4} & \displaystyle \frac{{\rm{i}}\pi }{4} & \ldots & \displaystyle \frac{{\rm{i}}\pi }{4}\end{array}\right)\in {{\mathbb{C}}}^{2\times m},\end{array}\end{eqnarray*}$then the NNQWF ${{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},{\lambda }_{{j}_{{i}_{2}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}})$ generated by these parameters is$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},{\lambda }_{{j}_{{i}_{2}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}})\\ \quad =\,\sum _{{h}_{1}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},{h}_{2}^{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=\pm 1}\exp \left(b{\rm{i}}({h}_{1}^{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}+{h}_{2}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })})\right.\\ \quad \left.+\,\displaystyle \frac{\pi {\rm{i}}}{4}({h}_{1}^{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}+{h}_{2}^{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })})\cdot ({\lambda }_{{j}_{{i}_{1}^{{\prime} }}}+{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}+\ldots +{\lambda }_{{j}_{{i}_{m}^{{\prime} }}}\right)\\ \quad =\,4{\cos }^{2}\left(b+\displaystyle \frac{\pi }{4}\sum _{s=1}^{m}{\lambda }_{{j}_{{i}_{s}^{{\prime} }}}\right)\\ \quad =\,{D}_{m}{\left(-1\right)}^{\tfrac{{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}}}{{2}^{k}}},\end{array}\end{eqnarray*}$where$\begin{eqnarray*}{D}_{m}=2\sin \left(2\arctan \left({{\rm{e}}}^{\tfrac{{\left(-1\right)}^{m+1}\pi {\rm{i}}}{{2}^{k}}}\right)\right).\end{eqnarray*}$This shows that(15)$\begin{eqnarray}\begin{array}{l}{D}_{m}{\left(-1\right)}^{\tfrac{{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}}}{{2}^{k}}}\\ \quad ={{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},{\lambda }_{{j}_{{i}_{2}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}}),\end{array}\end{eqnarray}$for any $m=2,3,\ldots ,k-1$. This implies that the function$\begin{eqnarray*}{D}_{m}{\left(-1\right)}^{\tfrac{{\left(-1\right)}^{m}{\lambda }_{{j}_{{i}_{1}^{{\prime} }}}{\lambda }_{{j}_{{i}_{2}^{{\prime} }}}\cdots {\lambda }_{{j}_{{i}_{m}^{{\prime} }}}}{{2}^{k}}},\end{eqnarray*}$can be implemented by an NNQWF ${{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}})$ for any $\{{i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset $ $ \{{i}_{1},{i}_{2},\ldots ,{i}_{k}\}(m=2,3,\ldots ,k-1)$, which is the neural network with two hidden neurons ${h}_{1}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}$ and ${h}_{2}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}$.

${\boldsymbol{Step}}\,{\bf{2}}.$ To represent$\begin{eqnarray*}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})},\end{eqnarray*}$as an NNQWF for each $({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E$, we write$\begin{eqnarray*}{{\rm{\Omega }}}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}=({a}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})},{b}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})},{W}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}),\end{eqnarray*}$$\begin{eqnarray*}{a}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}=\left(\begin{array}{c}{a}_{1}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ {a}_{2}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ \vdots \\ {a}_{k}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ \end{array}\right)\in {{\mathbb{C}}}^{k},{a}_{i}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}=-\displaystyle \frac{\pi {\rm{i}}}{{2}^{k}},\end{eqnarray*}$$\begin{eqnarray*}{b}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}=\left(\begin{array}{c}{b}_{1}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ {b}_{2}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ \end{array}\right),\ {b}_{1}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}={b}_{2}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}=b{\rm{i}},\end{eqnarray*}$$\begin{eqnarray*}b=\arctan \left({{\rm{e}}}^{\tfrac{{\left(-1\right)}^{k+1}\pi {\rm{i}}}{{2}^{k}}}\right)-\displaystyle \frac{k\pi }{4},\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{W}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ =\ \left(\begin{array}{cccc}{W}_{1{i}_{1}}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})} & {W}_{1{i}_{2}}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})} & \ldots & {W}_{1{i}_{k}}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\\ {W}_{2{i}_{1}}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})} & {W}_{2{i}_{2}}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})} & \ldots & {W}_{2{i}_{k}}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}\end{array}\right)\\ =\ \left(\begin{array}{cccc}\displaystyle \frac{{\rm{i}}\pi }{4} & \displaystyle \frac{{\rm{i}}\pi }{4} & \ldots & \displaystyle \frac{{\rm{i}}\pi }{4}\\ \displaystyle \frac{{\rm{i}}\pi }{4} & \displaystyle \frac{{\rm{i}}\pi }{4} & \ldots & \displaystyle \frac{{\rm{i}}\pi }{4}\end{array}\right)\in {{\mathbb{C}}}^{2\times k},\end{array}\end{eqnarray*}$then the NNQWF ${{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}}({\lambda }_{{j}_{{i}_{1}}},{\lambda }_{{j}_{{i}_{2}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})$ generated by these parameters is(16)$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}}({\lambda }_{{j}_{{i}_{1}}},{\lambda }_{{j}_{{i}_{2}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})\\ =\sum _{{h}_{x}^{({i}_{1},\ldots ,{i}_{k})}=\pm 1}\exp (\sum _{t=1}^{k}{a}_{t}^{({i}_{1},\ldots ,{i}_{k})}{\lambda }_{{j}_{{i}_{t}}})\exp \left(b{\rm{i}}({h}_{1}^{({i}_{1},\ldots ,{i}_{k})}\right.\\ +{h}_{2}^{({i}_{1},\ldots ,{i}_{k})})+\displaystyle \frac{\pi {\rm{i}}}{4}({h}_{1}^{({i}_{1},\ldots ,{i}_{k})}+{h}_{2}^{({i}_{1},\ldots ,{i}_{k})})\left.\cdot ({\lambda }_{{j}_{{i}_{1}}}+\ldots +{\lambda }_{{j}_{{i}_{k}}},),\right)\\ ={D}_{k}{\left(-1\right)}^{-\tfrac{{\lambda }_{{j}_{{i}_{1}}}+{\lambda }_{{j}_{{i}_{2}}}+\ldots +{\lambda }_{{j}_{{i}_{k}}}}{{2}^{k}}+\displaystyle \frac{{\left(-1\right)}^{k}{\lambda }_{{j}_{{i}_{1}}}{\lambda }_{{j}_{{i}_{2}}}\cdots {\lambda }_{{j}_{{i}_{k}}}}{{2}^{k}}}\\ ={D}_{k}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})},\end{array}\end{eqnarray}$where$\begin{eqnarray*}{D}_{k}=2\sin \left(2\arctan \left({{\rm{e}}}^{\tfrac{{\left(-1\right)}^{k+1}\pi {\rm{i}}}{{2}^{k}}}\right)\right).\end{eqnarray*}$

This implies that$\begin{eqnarray*}{D}_{k}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})},\end{eqnarray*}$can be implemented by the NNQWF ${{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}}({\lambda }_{{j}_{{i}_{1}}},{\lambda }_{{j}_{{i}_{2}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})$ for any $({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E$, which is the neural network with two hidden neurons ${h}_{1}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}$ and ${h}_{2}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}$.

${\boldsymbol{Step}}\,{\bf{3}}.$ Furthermore, using equations (15) and (16), the right-hand side of equation (14) becomes(17)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{z}_{0}}{z}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{1}{{2}^{k}}{f}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})+\displaystyle \frac{1}{{2}^{k}}{g}_{{i}_{1}\ldots {i}_{k}}({\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})}\\ =\displaystyle \frac{{z}_{0}}{z}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}\left[\displaystyle \frac{1}{{D}_{k}}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}}({\lambda }_{{j}_{{i}_{1}}},{\lambda }_{{j}_{{i}_{2}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}})\right.\\ \left.\times \prod _{\begin{array}{c}\{{i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},\ldots ,{i}_{k}\}\\ m=2,\ldots ,k-1\end{array}}\displaystyle \frac{1}{{D}_{m}}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}})\right]\\ =\displaystyle \frac{{{Az}}_{0}}{z}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}\prod _{\begin{array}{c}\{{i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},\ldots ,{i}_{k}\}\\ m=2,\ldots ,k\end{array}}\\ {{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}}),\end{array}\end{eqnarray}$where $A={\left({\prod }_{m=2}^{k}\tfrac{1}{{\left({D}_{m}\right)}^{{C}_{k}^{m}}}\right)}^{| E| }.$

To label the elements of E, we write$\begin{eqnarray*}E=\{{e}_{1},{e}_{2},\ldots ,{e}_{| E| }\}.\end{eqnarray*}$When ${e}_{t}=({i}_{1},{i}_{2},\ldots ,{i}_{k})$, we let$\begin{eqnarray*}{a}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}={a}_{t},\end{eqnarray*}$and label the set$\begin{eqnarray*}{F}_{t}:= \underset{m=2}{\overset{k}{\cup }}\{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }):\{{i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},{i}_{2},\ldots ,{i}_{k}\}\},\end{eqnarray*}$as$\begin{eqnarray*}{F}_{t}=\{{f}_{t1},{f}_{t2},\ldots ,{f}_{t({2}^{k}-k-1)}\}.\end{eqnarray*}$When $({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })={f}_{{ts}}$, we let$\begin{eqnarray*}{\widetilde{W}}_{{pq}}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=\left\{\begin{array}{ll}\displaystyle \frac{\pi {\rm{i}}}{4}, & p=1,2;q={i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} };\\ 0, & \mathrm{otherwise},\end{array}\right.\end{eqnarray*}$$\begin{eqnarray*}{\widetilde{W}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=[{\widetilde{W}}_{{pq}}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}]\in {{\mathbb{C}}}^{2\times k},\end{eqnarray*}$where $p=1,2;q=1,2,\ldots ,k$ and define$\begin{eqnarray*}{b}_{{ts}}={b}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},{W}_{{ts}}={\widetilde{W}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},\end{eqnarray*}$and let$\begin{eqnarray*}{b}_{t}=\left(\begin{array}{c}{b}_{t1}\\ {b}_{t2}\\ \vdots \\ {b}_{t({2}^{k}-k-1)}\\ \end{array}\right),\ {W}_{t}=\left(\begin{array}{c}{W}_{t1}\\ {W}_{t2}\\ \vdots \\ {W}_{t({2}^{k}-k-1)}\\ \end{array}\right),\end{eqnarray*}$$\begin{eqnarray*}{{\rm{\Omega }}}_{t}=({a}_{t},{b}_{t},{W}_{t}).\end{eqnarray*}$Using proposition 2, we have(18)$\begin{eqnarray}\begin{array}{l}\prod _{\displaystyle \genfrac{}{}{0em}{}{\{{i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},\ldots ,{i}_{k}\}}{m=2,\ldots ,k}}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}})\\ \quad =\,{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{t}}({\lambda }_{{j}_{{i}_{1}}},{\lambda }_{{j}_{{i}_{2}}},\ldots ,{\lambda }_{{j}_{{i}_{k}}}),\end{array}\end{eqnarray}$for every ${e}_{t}=({i}_{1},{i}_{2},\ldots ,{i}_{k}).$

${\boldsymbol{Step}}\,{\bf{4}}.$ Furthermore, we let$\begin{eqnarray*}{\hat{a}}_{t}={\hat{a}}_{({i}_{1},{i}_{2},\ldots ,{i}_{k})}=[{\hat{a}}_{{tq}}]\in {{\mathbb{C}}}^{N},\end{eqnarray*}$$\begin{eqnarray*}{\hat{a}}_{{tq}}=\left\{\begin{array}{ll}{a}_{v}^{({i}_{1},{i}_{2},\ldots ,{i}_{k})}, & q={i}_{v},v=1,2,\ldots ,k,\\ 0, & \mathrm{otherwise},\end{array}\right.\end{eqnarray*}$$\begin{eqnarray*}{\hat{W}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=[{\hat{W}}_{{pq}}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}]\in {{\mathbb{C}}}^{2\times N},\end{eqnarray*}$$\begin{eqnarray*}{\hat{W}}_{{pq}}^{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}=\left\{\begin{array}{ll}\displaystyle \frac{\pi {\rm{i}}}{4}, & p=1,2;q={i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} };\\ 0, & {\rm{otherwise,}}\end{array}\right.\end{eqnarray*}$for $p=1,2;q=1,2,\ldots ,N$, and$\begin{eqnarray*}{\hat{W}}_{{ts}}={\hat{W}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},\ {\hat{W}}_{t}=[{\hat{W}}_{{ts}}],\end{eqnarray*}$$\begin{eqnarray*}{b}_{{ts}}={b}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })},{b}_{t}=\left(\begin{array}{c}{b}_{t1}\\ {b}_{t2}\\ \vdots \\ {b}_{t({2}^{k}-k-1)}\\ \end{array}\right),\end{eqnarray*}$for $t=1,2,\ldots ,| E| ,s=1,2,\ldots ,{2}^{k}-k-1$. We write$\begin{eqnarray*}b=\left(\begin{array}{c}{b}_{11}\\ \vdots \\ {b}_{1({2}^{k}-k-1)}\\ {b}_{21}\\ \vdots \\ {b}_{2({2}^{k}-k-1)}\\ \vdots \\ {b}_{| E| 1}\\ \vdots \\ {b}_{| E| ({2}^{k}-k-1)}\\ \end{array}\right),\ W=\left(\begin{array}{c}{\hat{W}}_{11}\\ \vdots \\ {\hat{W}}_{1({2}^{k}-k-1)}\\ {\hat{W}}_{21}\\ \vdots \\ {\hat{W}}_{2({2}^{k}-k-1)}\\ \vdots \\ {\hat{W}}_{| E| 1}\\ \vdots \\ {\hat{W}}_{| E| ({2}^{k}-k-1)}\\ \end{array}\right),\end{eqnarray*}$and$\begin{eqnarray*}a={\hat{a}}_{1}+{\hat{a}}_{2}+\ldots +{\hat{a}}_{| E| }\in {{\mathbb{C}}}^{N},{\rm{\Omega }}=(a,b,W).\end{eqnarray*}$Using proposition 2, we obtain:(19)$\begin{eqnarray}\begin{array}{l}\prod _{({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E}\prod _{\begin{array}{c}\{{i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{{i}_{1},{i}_{2},\ldots ,{i}_{k}\}\\ m=2,\ldots ,k\end{array}}\\ {{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },{i}_{2}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},{\lambda }_{{j}_{{i}_{2}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}})\\ \quad =\,{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}).\end{array}\end{eqnarray}$

Let$\begin{eqnarray*}z={z}_{0}{\left(\prod _{m=2}^{k}{\left({D}_{m}\right)}^{{C}_{k}^{m}}\right)}^{| E| }.\end{eqnarray*}$Using equations (17) and (19) then yields that$\begin{eqnarray*}{{\rm{\Psi }}}_{{G}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})=z{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}),\end{eqnarray*}$for all $({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\in \{-1,1\}{}^{N}.$

We have constructed now an NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})$ that satisfies equation (14). This leads to the following conclusion.

Theorem 1. Any k-uniform hypergraph state $| {G}_{k}\rangle $ can be represented by a spin-z NNQS (7) given a neural network with a $\{1,-1\}$ input.

Example 1. Consider a hypergraph G with three vertices and a 3-hyperedge ${e}_{1}=(1,2,3)$ which is represented by the left-hand side of figure 4. In this case, the wave function of the 3-uniform hypergraph state $| G\rangle $ reads$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{{G}_{3}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},{\lambda }_{{j}_{3}})\\ \quad =\,\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{3}}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{1}}\right)\left(1-{\lambda }_{{j}_{2}}\right)\left(1-{\lambda }_{{j}_{3}}\right)}{{2}^{3}}}\\ \quad =\,\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{3}}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi }{{2}^{3}}}\prod _{m=2}^{3}\displaystyle \frac{1}{{\left({D}_{m}\right)}^{{C}_{3}^{m}}}\prod _{\begin{array}{c}\{{i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{1,2,3\}\\ m=2,3\end{array}}\\ {{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}}),\end{array}\end{eqnarray*}$which is a constant multiple of the NNQWF$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},{\lambda }_{{j}_{3}})\\ \quad =\,\prod _{\begin{array}{c}\{{i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} }\}\subset \{1,2,3\}\\ m=2,3\end{array}}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{({i}_{1}^{{\prime} },\ldots ,{i}_{m}^{{\prime} })}}({\lambda }_{{j}_{{i}_{1}^{{\prime} }}},\ldots ,{\lambda }_{{j}_{{i}_{m}^{{\prime} }}}),\end{array}\end{eqnarray*}$with the parameter ${{\rm{\Omega }}}_{1}=({a}_{1},{b}_{1},{W}_{1})$ where$\begin{eqnarray*}{a}_{1}=\left(\begin{array}{c}-\displaystyle \frac{\pi {\rm{i}}}{8}\\ -\displaystyle \frac{\pi {\rm{i}}}{8}\\ -\displaystyle \frac{\pi {\rm{i}}}{8}\\ \end{array}\right),\end{eqnarray*}$$\begin{eqnarray*}{b}_{1}={\rm{i}}\left(\begin{array}{c}\arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{\pi }{2}\\ \arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{\pi }{2}\\ \arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{\pi }{2}\\ \arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{\pi }{2}\\ \arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{\pi }{2}\\ \arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{\pi }{2}\\ \arctan \left({{\rm{e}}}^{\tfrac{\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{3\pi }{4}\\ \arctan \left({{\rm{e}}}^{\tfrac{\pi {\rm{i}}}{8}}\right)-\displaystyle \frac{3\pi }{4}\end{array}\right),{W}_{1}=\left(\begin{array}{ccc}\displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4} & 0\\ \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4} & 0\\ \displaystyle \frac{\pi {\rm{i}}}{4} & 0 & \displaystyle \frac{\pi {\rm{i}}}{4}\\ \displaystyle \frac{\pi {\rm{i}}}{4} & 0 & \displaystyle \frac{\pi {\rm{i}}}{4}\\ 0 & \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4}\\ 0 & \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4}\\ \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4}\\ \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4} & \displaystyle \frac{\pi {\rm{i}}}{4}\end{array}\right).\end{eqnarray*}$That is,$\begin{eqnarray*}{{\rm{\Psi }}}_{{G}_{3}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},{\lambda }_{{j}_{3}})=z{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},{\lambda }_{{j}_{3}}),\end{eqnarray*}$where$\begin{eqnarray*}z=\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{3}}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi }{{2}^{3}}}\prod _{m=2}^{3}\displaystyle \frac{1}{{\left({D}_{m}\right)}^{{C}_{3}^{m}}}.\end{eqnarray*}$

Figure 4.

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Figure 4.Neural network representation of the hypergraph state which corresponds to a hypergraph consisting of three vertices $1,2,3$, ${S}_{i}={\sigma }_{i}^{z},i=1,2,3$.


The neural network that generates ${{\rm{\Psi }}}_{{G}_{3}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},{\lambda }_{{j}_{3}})$ is given on the right-hand side of figure 4.

Example 2. Neural network representation of k-uniform hypergraph state corresponding to a given k-uniform hypergraph. The representation process is shown in figure 5 below. In this case, the parameters are$\begin{eqnarray*}{\rm{\Omega }}=(a,b,W),a={\left(-\displaystyle \frac{\pi {\rm{i}}}{8},-\displaystyle \frac{\pi {\rm{i}}}{4},-\displaystyle \frac{\pi {\rm{i}}}{4},-\displaystyle \frac{\pi {\rm{i}}}{8}\right)}^{{\rm{T}}},\end{eqnarray*}$$\begin{eqnarray*}b={\rm{i}}{\left(x,x,x,x,x,x,y,y,x,x,x,x,x,x,y,y\right)}^{{\rm{T}}},\end{eqnarray*}$where $x=\arctan \left({{\rm{e}}}^{\tfrac{-\pi {\rm{i}}}{8}}\right)-\tfrac{\pi }{2},y=\arctan \left({{\rm{e}}}^{\tfrac{\pi {\rm{i}}}{8}}\right)-\tfrac{3\pi }{4},$$\begin{eqnarray*}W={\left({r}_{1},{r}_{1},{r}_{2},{r}_{2},{r}_{3},{r}_{3},{r}_{4},{r}_{4},{r}_{5},{r}_{5},{r}_{6},{r}_{6},{r}_{7},{r}_{7},{r}_{8},{r}_{8}\right)}^{{\rm{T}}},\end{eqnarray*}$with$\begin{eqnarray*}{r}_{1}={\left(\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4},\mathrm{0,0}\right)}^{{\rm{T}}},{r}_{2}={\left(\displaystyle \frac{\pi {\rm{i}}}{4},0,\displaystyle \frac{\pi {\rm{i}}}{4},0\right)}^{{\rm{T}}},\end{eqnarray*}$$\begin{eqnarray*}{r}_{3}={\left(0,\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4},0\right)}^{{\rm{T}}},{r}_{4}={\left(\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4},0\right)}^{{\rm{T}}},\end{eqnarray*}$$\begin{eqnarray*}{r}_{5}={\left(0,\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4},0\right)}^{{\rm{T}}},{r}_{6}={\left(0,\displaystyle \frac{\pi {\rm{i}}}{4},0,\displaystyle \frac{\pi {\rm{i}}}{4}\right)}^{{\rm{T}}}\end{eqnarray*}$$\begin{eqnarray*}{r}_{7}={\left(\mathrm{0,0},\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4}\right)}^{{\rm{T}}},{r}_{8}={\left(0,\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4},\displaystyle \frac{\pi {\rm{i}}}{4}\right)}^{{\rm{T}}}.\end{eqnarray*}$

Figure 5.

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Figure 5.Neural network representation of k-uniform hypergraph states. The first figure is a hypergraph representation of a 3-uniform hypergraph state. The second one is an idea of the process; it shows a neural network representation of the 3-uniform hypergraph state, where $E=\{(1,2,3),(2,3,4)\}$, ${S}_{i}={\sigma }_{i}^{z},i=1,\ldots ,8$.


Remark 1. We see from Example 1 and Example 2 that the number of visible-layer neurons is equal to the number of vertices of the k-uniform hypergraph, and the number of hidden-layer neurons is $| E| ({2}^{k+1}-2k-2)$. This is a general rule for the neural network representation of any k-uniform hypergraph state.

4. Neural network representations of hypergraph states

A hypergraph is a generalization of the concept of a k-uniform hypergraph state, defined as follows.

A hypergraph [35] is a pair $G=(V,E)$ consisting of a set $V=\{1,2,\ldots ,N\}$ and a nonempty set E of subsets of V. The elements of V and E are called the vertices and hyperedges of G, respectively. When $e=({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E$, we say that the vertices ${i}_{1},{i}_{2},\ldots ,{i}_{k}$ are connected by e. Hence, E is a set of any k-hyperedges, where k is no longer fixed but may range from 1 to N.

Given a mathematical hypergraph $G=(V,E)$ [35], one can construct the corresponding hypergraph state as follows [35]:(20)$\begin{eqnarray}| G\rangle =\prod _{k=1}^{N}\prod _{({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E}{C}^{k}{Z}_{{i}_{1},{i}_{2},\ldots ,{i}_{k}}\mathop{\underbrace{| +\rangle | +\rangle \cdots | +\rangle }}\limits_{N},\end{eqnarray}$where$\begin{eqnarray*}{C}^{k}{Z}_{{i}_{1},{i}_{2},\ldots ,{i}_{k}}={(I-P)}_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}\otimes {I}_{{i}_{k}}+{P}_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}\otimes {Z}_{{i}_{k}},\end{eqnarray*}$$\begin{eqnarray*}{P}_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}=| 11\ldots 1{\rangle }_{{i}_{1}{i}_{2}\ldots {i}_{k-1}}\langle 11\ldots 1| ,\end{eqnarray*}$$\begin{eqnarray*}| +\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle ).\end{eqnarray*}$

Here, we try to construct neural network representations for any hypergraph state $| G\rangle $, that is, we try to find an NNQS $| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $ such that $| G\rangle =z| {{\rm{\Psi }}}_{S,{\rm{\Omega }}}\rangle $ for some normalized constant z.

At first, we reduce equation (20) for the hypergraph state using equation (11) and obtain that(21)$\begin{eqnarray}\begin{array}{l}| G\rangle =\sum _{{{\rm{\Lambda }}}_{{j}_{1}\ldots {j}_{N}}\in \{1,-1\}{}^{N}}\left(\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{k=1}^{N}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}\right.\\ \quad \left(-1\right)\left.{}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)}{{2}^{k}}}\right)| {\psi }_{{j}_{1}}\rangle \otimes \ldots \otimes | {\psi }_{{j}_{N}}\rangle ,\end{array}\end{eqnarray}$where ${\lambda }_{{j}_{{i}_{1}}},\ldots ,{\lambda }_{{j}_{{i}_{1}}},| {\psi }_{{j}_{1}}\rangle ,\ldots ,| {\psi }_{{j}_{N}}\rangle ,$ are shown in equation (6). We see that the simplified equation (21) is simpler and easier to use. Given a hypergraph, we can use this expression to obtain a hypergraph state associated with it very quickly. For example, for the hypergraph in figure 6, the corresponding hypergraph state is$\begin{eqnarray*}\begin{array}{rcl}| G\rangle & = & \displaystyle \frac{1}{4}\left(| 0000\rangle +| 0001\rangle +| 0010\rangle +| 0011\rangle \right.\\ & & +| 0100\rangle -| 0101\rangle +| 0110\rangle -| 0111\rangle \\ & & +| 1000\rangle +| 1001\rangle +| 1010\rangle +| 1011\rangle \\ & & \left.+| 1100\rangle -| 1101\rangle -| 1110\rangle -| 1111\rangle \right).\end{array}\end{eqnarray*}$

Figure 6.

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Figure 6.A hypergraph.


In addition, through this simplification equation (21), the hypergraph state obtained is similar to the spin-z NNQS equation, (7), which sets the stage for our follow-up work.

Besides, we can obtain that the wave function of hypergraph state $| G\rangle $ is$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{G}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ \quad =\,\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{k=1}^{N}\prod _{({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\left(1-{\lambda }_{{j}_{{i}_{2}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)}{{2}^{k}}}.\end{array}\end{eqnarray*}$

Next, we try to construct an NNQWF ${{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})$, such that$\begin{eqnarray*}{{\rm{\Psi }}}_{G}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})=z{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}),\end{eqnarray*}$for some constant z.

When k=1, we let$\begin{eqnarray*}{E}_{1}=\{({i}_{1})| ({i}_{1})\in E\}=\{({m}_{1}),({m}_{2}),\cdots ,({m}_{s})\}.\end{eqnarray*}$We can then see from section 3 that$\begin{eqnarray*}\prod _{({i}_{1})\in E}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)}{2}}={{\rm{e}}}^{\tfrac{{\rm{i}}\pi | {E}_{1}| }{2}}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}),\end{eqnarray*}$where ${{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})$ is given by equation (13).

When k=2,…,N, we let$\begin{eqnarray*}{E}_{k}=\{({i}_{1},{i}_{2},\ldots ,{i}_{k})| ({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E\},\end{eqnarray*}$i.e. E_k is the set of superedges of k vertices. Using equations (12), (14), (17) and (19), we then find that a parameter ${{\rm{\Omega }}}_{k}=({a}_{k},{b}_{k},{W}_{k})$ exists, such that(22)$\begin{eqnarray}\begin{array}{l}\prod _{({i}_{1},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)}{{2}^{k}}}\\ \quad =\,{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | {E}_{k}| }{{2}^{k}}}{\left(\prod _{m=2}^{k}\displaystyle \frac{1}{{\left({D}_{m}\right)}^{{C}_{k}^{m}}}\right)}^{| {E}_{k}| }{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{k}}({\lambda }_{{j}_{1}},\ldots ,{\lambda }_{{j}_{N}}),\end{array}\end{eqnarray}$where $| {E}_{k}| $ is the cardinality of the set E_k.

Furthermore, we have$\begin{eqnarray*}\begin{array}{l}{{\rm{\Psi }}}_{G}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\\ =\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}\prod _{k=1}^{N}\prod _{({i}_{1},{i}_{2},\ldots ,{i}_{k})\in E}{\left(-1\right)}^{\tfrac{\left(1-{\lambda }_{{j}_{{i}_{1}}}\right)\left(1-{\lambda }_{{j}_{{i}_{2}}}\right)\ldots \left(1-{\lambda }_{{j}_{{i}_{k}}}\right)}{{2}^{k}}}\\ =\displaystyle \frac{1}{{\left(\sqrt{2}\right)}^{N}}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | {E}_{1}| }{2}}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{1}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})\prod _{k=2}^{N}\left({{\rm{e}}}^{\tfrac{{\rm{i}}\pi | {E}_{k}| }{{2}^{k}}}\right.\\ \times \left.{\left(\prod _{m=2}^{k}\displaystyle \frac{1}{{\left({D}_{m}\right)}^{{C}_{k}^{m}}}\right)}^{| {E}_{k}| }{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}\right)\\ =\displaystyle \frac{{\prod }_{k=1}^{N}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | {E}_{k}| }{{2}^{k}}}}{{\left(\sqrt{2}\right)}^{N}}\left(\prod _{k=2}^{N}{\left(\prod _{m=2}^{k}\displaystyle \frac{1}{{\left({D}_{m}\right)}^{{C}_{k}^{m}}}\right)}^{| {E}_{k}| }\right)\\ \times \prod _{k=1}^{N}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}).\end{array}\end{eqnarray*}$

Using proposition 2, we can obtain that there exists a set of parameters Ω, such that$\begin{eqnarray*}\prod _{k=1}^{N}{{\rm{\Psi }}}_{S,{{\rm{\Omega }}}_{k}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})={{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}).\end{eqnarray*}$Put$\begin{eqnarray*}{z}_{0}=\displaystyle \frac{{\prod }_{k=1}^{N}{{\rm{e}}}^{\tfrac{{\rm{i}}\pi | {E}_{k}| }{{2}^{k}}}}{{\left(\sqrt{2}\right)}^{N}}\prod _{k=2}^{N}{\left(\prod _{m=2}^{k}\displaystyle \frac{1}{{\left({D}_{m}\right)}^{{C}_{k}^{m}}}\right)}^{| {E}_{k}| },\end{eqnarray*}$thus$\begin{eqnarray*}{{\rm{\Psi }}}_{G}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}})={z}_{0}{{\rm{\Psi }}}_{S,{\rm{\Omega }}}({\lambda }_{{j}_{1}},{\lambda }_{{j}_{2}},\ldots ,{\lambda }_{{j}_{N}}).\end{eqnarray*}$

This leads to the following conclusion.

Theorem 2. Any hypergraph state can be represented as a spin-z NNQS (7) given a neural network with a $\{1,-1\}$ input.

5. Conclusions

In this paper, we have constructed a neural network representation for any hypergraph state. Our method constructively shows that all hypergraph states can be precisely represented by appropriate neural networks as proposed in [Science 355 (2017) 602] and formulated in [Sci. China-Phys. Mech. Astron. 63 (2020) 210312]. The results obtained will provide a theoretical foundation for seeking approximate representations of hypergraph states and solving the quantum many-body problem using machine-learning methods.

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