# Rydberg Quantum Wires for Maximum Independent Set Problems
with Nonplanar and High-Degree Graphs

Minhyuk Kim, Kangheun Kim, Jaeyong Hwang, Eun-Gook Moon, and Jaewook Ahn
<EMAIL_ADDRESS>Department of Physics, KAIST, Daejeon 34141, Republic of
Korea

###### Abstract

One prominent application of near-term quantum computing devices is to solve
combinatorial optimization such as non-deterministic polynomial-time hard (NP-
hard) problems. Here we present experiments with Rydberg atoms to solve one of
the NP-hard problems, the maximum independent set (MIS) of graphs. We
introduce the Rydberg quantum wire scheme with auxiliary atoms to engineer
long-ranged networks of qubit atoms. Three-dimensional (3D) Rydberg-atom
arrays are constructed, overcoming the intrinsic limitations of two-
dimensional arrays. We demonstrate Kuratowski subgraphs and a six-degree
graph, which are the essentials of non-planar and high-degree graphs. Their
MIS solutions are obtained by realizing a programmable quantum simulator with
the quantum-wired 3D arrays. Our construction provides a way to engineer many-
body entanglement, taking a step toward quantum advantages in combinatorial
optimization.

One of the latest efforts in quantum computation research is to use a sizable
quantum many-body system [1, 2, 3, 4] to solve non-deterministic polynomial-
time (NP)-optimization problems [5, 6, 7, 8]. Combinatorial optimization
problems are to find an optimal solution from feasible solutions: for example,
the maximum-independent-set (MIS) problem seeks to find an independent vertex
set of maximal size for a graph [9]. While the MIS problem is classical by
definition, their computational complexity (NP-complete) makes it intractable
to classical Turing machines [10, 11]. Alternatively, if a quantum many-body
system allows an intrinsic mapping of the problem to, for example, the many-
body ground state, its evolution might be engineerable for the benefit of
computational speedup [12, 13].

Rydberg atom systems provide an intrinsic Hamiltonian for the MIS problem [8,
14], of particular relevance in the context of the present paper. Consider $N$
atoms are arranged to a graph $G=G(V,E)$ with vertices ($V$) and edges ($E$),
representing atoms and Rydberg-blockaded atom pairs, respectively. Their many-
body ground state gives the MIS solution of $G$, $\mathbb{M}(G)$ [15, 4]. The
Hamiltonian is approximately given by

$\hat{H}_{G}=U\sum_{(j,k)\in
E}\hat{n}_{j}\hat{n}_{k}-\frac{\hbar\Delta}{2}\sum_{j\in
V}\hat{\sigma}_{j}^{z}$ (1)

where $U$ is the nearest-neighbor interaction, $\Delta$ is the laser detuning,
and $\hat{n}=(\hat{\sigma}^{z}+1)/2$ is the Rydberg excitation. The
configuration $n=1$ ($n=0$) is for the Rydberg (ground) state of each atom,
respectively [16]. It is easy to find that the many-body ground state of
$\hat{H}_{G}$ gives $\mathbb{M}(G)$ with anti-ferromagnetic strong coupling
($U>\hbar\Delta$) and positive detuning field ($\Delta>0$). Thus, quantum
simulations for the ground state of the Schrödinger equation,
$\hat{H}_{G}|\mathbb{M}(G)\rangle=E_{g}(G)|\mathbb{M}(G)\rangle$, find the MIS
solutions.

Figure 1: (a) Graph representation of a Rydberg-atom array: Four ($N=4$) atoms
are arranged in the nearest-neighbor Rydberg-blockade regime for $G(V,E)=$ the
3-pan graph of $V=\\{1,2,3,4\\}$ and
$E=\\{\\{1,2\\},\\{2,3\\},\\{2,4\\},\\{3,4\\}\\}$, of which the MIS solution
is $\mathbb{M}(G)=\\{\\{1,3\\},\\{1,4\\}\\}$. (b) Rydberg quantum-wire
concept: Qubit atoms (blue) are arranged to represent an initial graph
$G_{0}={\bf X_{101}}$. When a chain of auxiliary atoms (red) makes a new edge
between nonadjacent atoms (vertices A and B of $G_{0}$), the combined graph
$G_{0+w}$ (qubit and wire atoms) constructs a target graph $G_{T}$, the Moser
spindle graph, by sharing the same MIS solutions.

In Fig. 1(a), an example of two-dimensional (2D) Rydberg array simulations is
presented with the 4-vertex graph, $G=$ the 3-pan graph in the nomenclature of
ISGCI (Information System on Graph Classes and their Inclusion) [17]. The atom
arrangement, illustrated as the numbering in Fig. 1(a), is suitable to utilize
the Rydberg blockade with the radius, $r_{b}$, and its ground state
$(\left|1010\right\rangle+\left|1001\right\rangle)/\sqrt{2}$ gives the MIS
solution, $\mathbb{M}($3-pan$)=\\{\\{1,3\\},\\{1,4\\}\\}$ by counting the
$n=1$ atoms.

We remark two intrinsic limitations of 2D Rydberg-atom arrays for the MIS
problems. First, non-planar graphs cannot be simulated by 2D Rydberg atoms, as
the mathematical theorem by Kuratowski indicates [18]. Second, graphs with
high-degree vertices cannot be encoded, because the size of Rydberg-atom
interactions is set by the blockade radius. In this work, we demonstrate the
two limitations are overcome by introducing a new scheme of quantum wires to
Rydberg-atom arrays.

Quantum wires are a chain of auxiliary wire atoms, proposed to mediate strong
interactions between distant atoms in such a way that a complex target graph,
$G_{T}$, can be synthesized from a simple initial graph, $G_{0}$. Such quantum
wires are in principle realizable by using local addressing fields [19], but
implementing them in the current Rydberg atom experiments is daunting. We,
instead, propose an alternative quantum-wire scheme, termed as Rydberg quantum
wire, which require no local addressing. Our scheme, as to be shown below,
demonstrates that the MIS problems with non-planar or high-degree graphs are
readily accessible.

The Rydberg quantum wire scheme consists of the three steps, construction of a
wired array, quantum simulation for a MIS problem, and projection of the wire
information. The construction step is illustrated in Fig. 1(b), in which the
quantum wire of an even number ($M=6$) of wire atoms (red spheres) is added to
the initial graph, $G_{0}={\bf X_{101}}$ (blue spheres), to couple the $A$ and
$B$ qubit atoms. If the quantum wire is treated as an edge, the combined graph
$G_{0+w}$ is equivalent to the target graph, $G_{T}=$ the Moser spindle graph.
In the quantum simulation step, the ground state of the Hamiltonian is
obtained by solving the Schrödinger equation,
$\hat{H}_{G_{0+w}}|\mathbb{M}(G_{0+w})\rangle=E_{g}(G_{0+w})|\mathbb{M}(G_{0+w})\rangle$
experimentally, for example, with quantum annealing. Note that the size of the
Hilbert space of $G_{T}$ is different from the one of $G_{0+w}$ by the factor
of $2^{M}$, where $M$ is the number of added atoms for the quantum wire. Thus,
additional operations of quantum states of $G_{0+w}$ is performed in the final
step. The MIS solution of a target graph, $\mathbb{M}(G_{T})$, can be
schematically written as,

$\displaystyle|\mathbb{M}(G_{T})\rangle=\mathbb{P}_{F}\mathbb{P}_{H}\big{[}|\mathbb{M}(G_{0+w})\rangle\big{]}.$

The first operation ($\mathbb{P}_{H}$) is to make a projection of quantum
states in the enlarged Hilbert space of $G_{0+w}$ onto the Hilbert space of
the target graph $G_{T}$, which can be readily done by measuring the qubit
information of $G_{T}$. Hereafter, we introduce the bar notation to specify
the projection, for example, $\mathbb{M}(G)\rightarrow\mathbb{\bar{M}}(G)$.
The second operation ($\mathbb{P}_{F}$) is to remove configurations with
frustration between qubits at the boundaries, when some elements
$\mathbb{F}(G_{0+w})$ of $\mathbb{\bar{M}}(G_{0+w})$ violate the Rydberg
blockade condition. Then, the MIS solution of $G_{T}$ is obtained as

$\displaystyle\mathbb{M}(G_{T})$ $\displaystyle=$
$\displaystyle\mathbb{\bar{M}}(G_{0+w})-\mathbb{\bar{F}}(G_{0+w})$ (2)

(see Appendix for detailed discussions). It is noted that our Rydberg quantum
wire scheme utilizes quantum entanglement of the two quantum many-body
systems, original graph and wire. Namely, qubits of the two systems become
entangled so that the MIS solution of a target graph is accessible.

Figure 2: Experimental tests of Rydberg quantum wires. (a-c) A test graph set,
$\\{G_{0},G_{T},G_{0+w}\\}$, for the algorithm with no frustration,
$\mathbb{M}(G_{T})=\mathbb{\bar{M}}(G_{0+w})$: (a) $G_{0}={\bf P_{4}}$, the
four-vertex path graph; (b) $G_{T}={\bf C_{4}}$, the four-vertex circle graph;
(c) $G_{0+w}={\bf C_{6}}$, the six-vertex circle graph. (d-f) Experimental MIS
solutions (filled bars) in the probabilities of all qubit-atom states in the
bare-atom basis enumerated and sorted by $N_{e}=\sum_{j\in V}n_{j}$ (the total
Rydberg excitation): (d)
$\mathbb{M}(G_{0})=\\{\\{2,4\\},\\{1,4\\},\\{1,3\\}\\}$; (e)
$\mathbb{M}(G_{T})=\\{\\{2,4\\},\\{1,3\\}\\}$; and (f)
$\mathbb{\bar{M}}(G_{0+w})=\\{\\{2,4\\},\\{1,3\\}\\}$. (g-l) A test graph with
the frustration,
$\mathbb{M}(G_{T})=\mathbb{\bar{M}}(G_{0+w})-\mathbb{F}(G_{0+w})$: (g)
$G_{0}={\bf S_{4}}$, the four-atom star graph; (h) $G_{T}=$ the 3-pan graph;
and (i) $G_{0+w}=$ the 5-pan graph. (j-l) Experimental MIS solutions: (j)
$\mathbb{M}(G_{0})=\\{\\{1,2,4\\}\\}$; (k)
$\mathbb{M}(G_{T})=\\{\\{2,4\\},\\{1,4\\}\\}$; and (l)
$\mathbb{\bar{M}}(G_{0+w})=\\{\\{2,4\\},\\{1,4\\},\\{1,2,4\\}\\}$. In (d-f)
and (j-l), state preparation and measurement errors of $P(0|1)=0.03$ and
$P(1|0)=0.18$ are taken into account. In (f,l), numerical simulation for
possible experimental errors are estimated for comparison.

Quantum simulation of MIS problems is performed with quantum annealing of 3D
atom arrays [20]. In experiments, neutral 87Rb atoms are arranged in free
space in such a way that all nearest-neighbor atom pairs, which describe the
edges of the graphs, are kept at a fixed interatomic distance $d$ smaller than
the Rydberg blockade radius, i.e., $d<r_{b}=(C_{6}/\hbar\Omega)^{1/6}=9.8$
$\mu$m, and that all other atom pairs, which are not connected by edges, are
at longer distances [20, 21]. The ground state
$\left|5S_{1/2},F=2,m_{F}=2\right\rangle=\left|n=0\right\rangle$ and the
Rydberg state $\left|71S_{1/2},m_{J}=1/2\right\rangle=\left|n=1\right\rangle$
of each atom are used for the qubit two-state system. An effective Hamiltonian
of the 3D atom arrays is

$\hat{H}(t)=U\sum_{(j,k)\in E}\hat{n}_{j}\hat{n}_{k}-\frac{\hbar}{2}\sum_{j\in
V}\left(\delta(t)\hat{\sigma}^{z}_{j}-\Omega(t)\hat{\sigma}^{x}_{j}\right),$
(3)

where the Pauli matrices with the two states at sites $j,k$ are introduced
with $\hat{n}_{j,k}=(\hat{\sigma}_{j,k}^{z}+1)/2$. Each term on the right hand
side describes the van der Waals interaction at the fixed distance $d$, the
time-dependent detuning, and the time-dependent Rabi frequency, respectively.
Initially, the atoms are prepared in the paramagnetic down spins at $t=0$,
$\left|00\cdots 0\right\rangle$, with $\delta(0)=-\Delta_{i}<0$ and
$\Omega(0)=0$. To find the MIS solutions of $G$, these atoms are quasi-
adiabatically driven to the many-body ground state of $\hat{H}_{G}$, by
turning on and off the Rabi frequency while the detuning is gradually
increased to $\delta(t=t_{f})=\Delta_{f}<U$ [20] (see Appendix for details).

We first consider in Fig. 2 experimental tests of Rydberg quantum wires for
the cases with and without the frustration,
${\mathbb{\bar{F}}(G_{0+w})}=\emptyset$ and $\neq\emptyset$ in Eq. (2),
respectively. For graphs without the frustration, we consider the initial
graph, $G_{0}={\bf P}_{4}$ in Fig. 2(a), and the target graph, $G_{T}={\bf
C}_{4}$ in Fig. 2(b). The construction of the wired graph, $G_{0+w}={\bf
C}_{6}$, is done by adding an $M=2$ Rydberg quantum wire (red spheres) as
shown in Fig. 2(c). Quantum simulations observe high-population states as in
Figs. 2(d,e,f), of which the MIS solutions are summarized as
$\mathbb{M}(G_{0})=\\{\\{2,4\\},\\{1,4\\},\\{1,3\\}\\}$,
$\mathbb{\bar{M}}(G_{0+w})=\\{\\{2,4\\},\\{1,3\\}\\}$, and
$\mathbb{M}(G_{T})=\\{\\{2,4\\},\\{1,3\\}\\}$. It is easy to verify that
$\mathbb{F}(G_{0+w})=\emptyset$ and
$\mathbb{M}(G_{T})=\mathbb{\bar{M}}(G_{0+w})$, satisfying the MIS solution in
Eq. (2). Note that the population difference among the MIS solutions of
$G_{0}$, in Fig. 2(d), is due to the fact that the quantum annealing results
in a coherent superposition of the MIS solutions,
$\left|\mathbb{M}(G_{0})\rangle\right\rangle=(\left|0101\right\rangle+\left|1010\right\rangle)/\sqrt{2}+\left|0110\right\rangle/2$.

For graphs with the frustration, i.e., $\mathbb{F}(G_{0+w})\neq\emptyset$, we
consider the initial and target graphs, $G_{0}={\bf S_{4}}$ and $G_{T}=$ 3-pan
as in Fig. 2(g,h). The wired graph is $G_{0+w}=$ 5-pan, constructed with the
Rydberg quantum wire as shown in Fig. 2(i). Quantum simulations of $G_{0}$,
$G_{T}$, and $G_{0+w}$ are respectively shown in Figs. 2(j,k,l), of which the
high-population states are given by $\mathbb{M}(G_{0})=\\{\\{1,2,4\\}\\}$,
$\mathbb{M}(G_{T})=\\{\\{1,4\\},\\{2,4\\}\\}$, and
$\mathbb{\bar{M}}(G_{0+w})=\\{\\{1,4\\},\\{2,4\\},\\{1,2,4\\}\\}$. The results
also confirm the MIS solution,
$\mathbb{M}(G_{T})=\mathbb{\bar{M}}(G_{0+w})-\mathbb{\bar{F}}(G_{0+w})$, where
$\mathbb{\bar{F}}(G_{0+w})=\\{{\\{1,2,4\\}}\\}$ is the frustrated
configuration.

Figure 3: Nonplanar graphs. Experimental construction of Kuratowski subgraphs:
(a) ${\bf K_{5}}$. (b) ${\bf K_{3,3}}$. (c) Rydberg quantum-wire
implementation of ${\bf K_{5}^{exp}}$ with ${\bf K_{5}}\mbox{-}e$ (blue) and a
${\bf P_{6}}$ quantum wire (red). (d) ${\bf K_{3,3}^{exp}}$ with the A graph
(blue) and three quantum wires (yellow, orange, and red). (e) Experimental MIS
solutions $\mathbb{\bar{M}}({\bf
K_{5}^{exp}})=\\{\\{1\\},\\{2\\},\cdots,\\{5\\},\\{2,5\\}\\}$, in which {2,5}
has a frustrated quantum wire. (f) $\mathbb{\bar{M}}({\bf
K_{3,3}^{exp}})=\\{\\{1,2,3\\},\\{4,5,6\\}\\}$.

Next, we consider nonplanar graphs, focusing on Kuratowski subgraphs, ${\bf
K_{5}}$ and ${\bf K_{3,3}}$. The seminal work by Kuratowski shows that a graph
$G$ is nonplanar if and only if $G$ contains any of these two Kuratowski
subgraphs [18]. ${\bf K_{5}}$ in Fig. 3(a) is a complete graph with each
vertex edged to all other vertices and ${\bf K_{3,3}}$ in Fig. 3(b) is a
bipartite graph with three vertices on one side completely connected to the
vertices on the other side. It has been shown that both of these graphs
require quantum wiring as well as 3D atom arrangements [22, 23].

We construct wired graphs, $G_{0+w}={\bf K_{5}^{exp}}$ and ${\bf
K_{3,3}^{exp}}$, as shown in Figs. 3(c,d), to simulate the target graphs
$G_{T}={\bf K_{5}}$ and ${\bf K_{3,3}}$, respectively. In Fig. 3(c), the
initial graph is $G_{0}={\bf K_{5}}\mbox{-}e$ of five qubit atoms (blue
spheres) in the tetrahedral configuration, of which the two qubit atoms (2 and
5) are coupled by the quantum wire of six wire atoms (red spheres). With the
constructed ${\bf K_{5}^{exp}}$, quantum annealing is performed and the result
is shown in Fig. 3(e). Six peaks are observed corresponding to the five
singly-excited solutions (black bars), $\\{1\\}$, $\\{2\\}$, $\cdots$,
$\\{5\\}$, and a doubly-excited $\\{2,5\\}$ (gray bar). The last solution,
$\\{2,5\\}$, is frustrated, and the MIS solution of ${\bf K_{5}}$ is obtained
by Eq. (2), $\mathbb{M}({\bf K_{5}})=\\{\\{1\\},\\{2\\},\cdots\\{5\\}\\}$. We
note that the uneven probabilities of the singly-excited states are attributed
to the difference of their higher-order interactions, which are ignored in Eq.
(1), to other qubit atoms and the quantum wire, which are mainly due to the
asymmetries of physical implementations (for example, atom 3 is kept closer to
atom 5 than 2 to avoid the interference of holographic optical potentials
[22]). In Fig. 3(d) for ${\bf K_{3,3}^{exp}}$, the initial graph is the A
graph (blue) and three quantum wires (yellow, orange, and red) are used to
connect $(1,6)$, $(1,4)$, and $(3,4)$ pairs. Quantum annealing of the
constructed ${\bf K_{3,3}^{exp}}$ results in two peaks as shown in Fig. 3(f),
corresponding to the two MIS solutions (black bars), $\\{1,2,3\\}$ and
$\\{4,5,6\\}$ of ${\bf K_{3,3}^{exp}}$. Both the solutions are not frustrated
and the Rydberg quantum-wire algorithm in Eq. (2) experimentally finds the MIS
solution of ${\bf K_{3,3}}$ as $\mathbb{M}({\bf
K_{3,3}})=\\{\\{1,2,3\\},\\{4,5,6\\}\\}$.

Figure 4: Vertex-splitting demonstration: (a) A star graph ${\bf S_{6}}$ of
the six-degree center vertex. (b) 2D implementation of ${\bf S_{6}}$ results
in a wheel graph ${\bf W_{7}}$. (c) Three quantum wires are used to split the
center 6-degree vertex to three extended 3-degree vertices. (d) Experimental
MIS solution $\mathbb{\bar{M}}({\bf
S_{6}^{exp}})=\\{\\{2,3,\cdots,7\\}\\}=\mathbb{M}({\bf S_{6}})$.

As an additional application of Rydberg quantum wires, we consider a graph of
high-degree vertex in Fig. 4. Implementation of a high-degree vertex is
crucial in quantum simulations with Rydberg-atom arrays. For example, the star
graph ${\bf S_{6}}$ in Fig. 4(a), which has a six-degree vertex at the center,
cannot be simulated in any 2D arrays without a quantum wire scheme, because a
naive 2D implementation of atoms results in a different graph, the wheel graph
${\bf W_{7}}$, as shown in Fig. 4(b). Our strategy is to reduce the degree of
the high-degree vertex, by using the Rydberg quantum wires, also known as
vertex-splitting in graph theory [24]. As shown in Fig. 4(c), the 13-vertex
extended tree-like graph (red and blue spheres), ${\bf S_{6}^{exp}}$, is
constructed from the initial graph, $7{\bf K_{1}}$ of seven isolated vertices.
Three quantum wires are used to split the 6-degree center vertex by adding the
three 3-degree vertices. Quantum annealing results of the as-constructed ${\bf
S_{6}^{exp}}$ are shown in Fig. 4(d). A single peak is observed corresponding
to the MIS solution of ${\bf S_{6}^{exp}}$, which is not frustrated
($\mathbb{\bar{F}}(G_{0+w})=\emptyset$), i.e., $\mathbb{\bar{M}}({\bf
S_{6}^{exp}})=\\{\\{2,3,\cdots,7\\}\\}$, and the Rydberg quantum wire scheme
successfully constructs the high-degree graph.

In summary, we have proposed and experimentally demonstrated the Rydberg
quantum wire scheme, which utilizes Rydberg many-body interactions along a
chain of neutral atoms to program the complex connections of nonplanar and
high-degree graphs necessary for general MIS problems. We have used 3D arrays
of qubit and quantum-wire atoms, to construct the Kuratowski subgraphs, ${\bf
K_{5}}$ and ${\bf K_{3,3}}$, and the six-degree graph, ${\bf S_{6}}$, and
probed their many-body ground states using the near-adiabatic quantum
annealing procedure. The observed ground-states of the quantum-wired systems
have exhibited excellent agreements with, or algorithmically retrieved, the
MIS solutions of the target graphs. Our demonstration suggests that a general
graph of $N$-by-$N$ couplings is in principle implementable by Rydberg quantum
wires, while there remain unresolved issues such as limited physical resources
[25], efficient many-body ground-state probing [26], and technical issues
involved with tangled 3D wires. It is hoped that our quantum-wire scheme
demonstrated for MIS problems shall be useful to further developments for
other optimization problems.

##  Appendix A Quantum Mechanics of the wired MIS problems

Our goal is to find the MIS solution of a target graph, $G_{T}$, which can be
obtained by finding the ground state of the Schrödinger equation of the target
Hamiltonian,

$\displaystyle\hat{H}_{T}|\mathbb{M}(G_{T})\rangle=E_{g}(G_{T})|\mathbb{M}(G_{T})\rangle.$

However, the construction of $G_{T}$ is fundamentally limited by non-planar
graphs and high-degree vertices.

Our strategy to overcome the limitations is to investigate a wired graph,
$G_{0+w}$, and obtain the MIS solution of $G_{T}$. The basic fact is the
relation, $\mathbb{M}(G_{T})\subset\mathbb{\bar{M}}(G_{0+w})$, which can be
easily shown by replacing an edge of $G_{T}$ by a quantum wire. The MIS
solution of $G_{0+w}$ is obtained by solving the Schrödinger equation,

$\displaystyle\hat{H}_{0+w}|\mathbb{M}(G_{0+w})\rangle=E_{g}(G_{0+w})|\mathbb{M}(G_{0+w})\rangle.$

The ground state can be written as

$\displaystyle|\mathbb{M}(G_{0+w})\rangle=\sum_{j}C_{j}|\Psi_{j}\rangle=\sum_{j}C_{j}|\psi_{j}\rangle\otimes|\phi_{j}\rangle.$

Note that the qubit state $|\psi_{j}\rangle$ (the combined state
$|\Psi_{j}\rangle$) is located in the Hilbert space whose size is $2^{N}$
($2^{N+M}$), where $N$ and $M$ are the numbers of qubits of the target graph
and wires, respectively. The MIS solution of the wired graph is

$\displaystyle\mathbb{M}(G_{0+w})$ $\displaystyle=$
$\displaystyle\\{(\psi_{1},\phi_{1}),(\psi_{2},\psi_{2}),\cdots\\}.$

We introduce the projection operator, $\mathbb{P}_{H}$, defined as

$\displaystyle\mathbb{P}_{H}|\Psi_{j}\rangle=|\psi_{j}\rangle.$

Mathematically, a Fock space with different qubit numbers is necessary, and
yet it is safe to use the above notation for the MIS problems. The projection
state is

$\displaystyle\mathbb{P}_{H}|\mathbb{M}(G_{0+w})\rangle=\sum_{j}C_{j}|\psi_{j}\rangle$

and the corresponding solution set is

$\displaystyle\mathbb{\bar{M}}(G_{0+w})$ $\displaystyle=$
$\displaystyle\\{\psi_{1},\psi_{2},\cdots,\tilde{\psi}_{1},\tilde{\psi}_{2},\cdots\\}.$

The bar notation is introduced to specify the projection. Some of the elements
of $\mathbb{\bar{M}}(G_{0+w})$ are not a solution of the MIS problem of
$G_{T}$, which are specified by the tilde notation. One can introduce the
frustration function defined as $f(n_{A},n_{B})=n_{A}n_{B}$, where $n_{A}$ and
$n_{B}$ are the number operator eigenvalues of the boundaries where a wire is
connected. For an element of $\mathbb{\bar{M}}(G_{0+w})$, the Rydberg blockade
condition, $f(n_{A},n_{B})=1$, can be tested, and a set with the condition is
obtained,
$\mathbb{\bar{F}}(G_{0+w})=\\{\tilde{\psi}_{1},\tilde{\psi}_{2},\cdots\\}$.
Then, one can introduce the second projection operator,
$\mathbb{P}_{F}=\mathbb{I}-\sum_{a}|\tilde{\psi}_{a}\rangle\langle\tilde{\psi}_{a}|$,
and the final state is

$\displaystyle|\mathbb{M}(G_{T})\rangle$ $\displaystyle=$
$\displaystyle\mathbb{P}_{F}\mathbb{P}_{H}\big{[}|\mathbb{M}(G_{0+w})\rangle\big{]}.$

Then, the solution set can be written as

$\displaystyle\mathbb{M}(G_{T})$ $\displaystyle=$
$\displaystyle\mathbb{\bar{M}}(G_{0+w})-\mathbb{\bar{F}}(G_{0+w})=\\{\psi_{1}\psi_{2},\cdots\\}.$
(4)

##  Appendix B Rydberg-atom quantum simulator

Experiments of Rydberg quantum wires are performed with a 3D Rydberg-atom
quantum simulator, which consists of a magneto-optical trap (MOT) of 87Rb
atoms, an optical system for holographical optical tweezers, a laser system
for Rydberg-atom excitations, and a single-atom detection system.

The atoms are cooled in the MOT down to 30 $\mu$K by Doppler and polarization
gradient cooling, and optically pumped to the ground hyperfine state
$\left|0\right\rangle=\left|5S_{1/2},F=2,m_{F}=2\right\rangle$. After the MOT
is turned off, optical tweezers (far-off resonant optical dipole traps) are
turned on to capture single atoms. Up to 250 optical tweezers are created at
predetermined 3D locations of 5-10 $\mu$m spacing, by using an 820-nm laser
(Ti:Sapphire CW laser of Avesta), a spatial light modulator (SLM, ODPDM512 of
Meadowlark optics), and a microscope objective lens (Mitutoyo G Plan Apo
50$\times$). Each optical tweezer has the trap depth of 1 mK, the diameter of
2 $\mu$m, and the lifetime of 40(10) s.

To make a defect-free arrangement of $N$ atoms, we use $N$ optical tweezers on
target atom positions and another $N$ optical tweezers as a reservoir around
the targets. The atom occupations of the optical tweezers are determined by
fluorescence imaging of the
$\left|5S_{1/2},F=2\right\rangle-\left|5P_{3/2},F=3\right\rangle$ transition
with an electron-multiplied-CCD camera and an electrically-tunable-lens (ETL,
EL-16-40-TC of Optotune). The lateral and axial resolutions are 0.3 $\mu$m and
0.5 $\mu$m, respectively. After the occupations are checked, the captured
atoms are rearranged by optical tweezers steered along a set of paths obtained
by Hungarian algorithm [27]. 3D Gerchberg-Saxton (GS) algorithm is used to
program the dynamic holograms, of which the real-time calculation is performed
with a GPU (NVIDIA, Titan-X pascal). The 3D GS algorithm uses for fast
convergence the weights of target site intensity feedback, and the initial
phase pattern of SLM for every iteration utilizes the calculated phase pattern
from the former iteration [28]. As example, a 35 times iteration takes about
700 ms and it is sufficient to move traps about 20 $\mu$m with an over 90 %
occupation probability of each site. The 3D atom positions of all experimental
graphs are listed in Table 1.

Table 1: Atom positions of the graphs demonstrated in the main text.  Figure | Graph | Positions ($x,y,z$) ($\mu$m) |  |  |
---|---|---|---|---|---
2a | ${\bf P_{4}}$ | 1 : (-7.0, 3.0, 0.0) | 2 : (-3.5, -3.0, 0.0) | 3 : (3.5, -3.0, 0.0) | 4 : (7.0, 3.0, 0.0)
2b | ${\bf C_{4}}$ | 1 : (-3.5, 3.5, 0.0) | 2 : (-3.5, -3.5, 0.0) | 3 : (3.5, -3.5, 0.0) | 4 : (3.5, 3.5, 0.0)
2c | ${\bf C_{6}}$ | 1 : (-7.0, 0.0, 0.0) | 2 : (-3.5, -6.1, 0.0) | 3 : (3.5, -6.1, 0.0) | 4 : (7.0, 0.0, 0.0)
|  | 5 : (-3.5, 6.1, 0.0) | 6 : (3.5, 6.1, 0.0) |  |
2g | ${\bf S_{4}}$ | 1 : (-3.8, 6.5, 0.0) | 2 : (-3.8, -6.5, 0.0) | 3 : (0.0, 0.0, 0.0) | 4 : (7.5, 0.0, 0.0)
2h | 3-pan | 1 : (-6.5, 3.8, 0.0) | 2 : (-6.5, -3.8, 0.0) | 3 : (0.0, 0.0, 0.0) | 4 : (7.5, 0.0, 0.0)
2i | 5-pan | 1 : (-3.8, 6.5, 0.0) | 2 : (-3.8, -6.5, 0.0) | 3 : (0.0, 0.0, 0.0) | 4 : (7.5, 0.0, 0.0)
|  | 5 : (-10.7, 3.8, 0.0) | 6 : (-10.7, -3.8, 0.0) |  |
3a | ${\bf K_{5}}$ | 1 : (-2.0, 0.0, -2.1) | 2 : (-5.0, 0.0, 2.5) | 3 : (5.0, 0.0, -2.5) | 4 : (1.0, 2.9, 1.5)
|  | 5 : (1.0, -3.5, 1.5) | 6 : (-9.6, 5.0, 4.0) | 7 : (-8.8, 12.1, 3.6) | 8 : (-5.1, 12.1, -0.8)
|  | 9 : (1.0, 12.1, -3.8) | 10 : (8.1, 12.1, -4.7) | 11 : (8.8, 5.0, -5.1) |
3b | ${\bf K_{3,3}}$ | 1 : (-6.9, 8.0, 0.0) | 2 : (0.0, -4.0, 0.0) | 3 : (8.0, 4.0, 0.0) | 4 : (-6.9, -8.0, 0.0)
|  | 5 : (0.0, 4.0, 0.0) | 6 : (8.0, -4.0, 0.0) | 7 : (-13.9, 4.0, 0.0) | 8 : (-13.9, -4.0, 0.0)
|  | 9 : (-6.9, -13.7, -5.7) | 10 : (0.0, -16.0, -7.0) | 11 : (7.0, -13.0, -7.0) | 12 : (14.5, -10.5, -7.0)
|  | 13 : (17.0, -3.0, -7.0) | 14 : (13.7, 4.0, -5.7) | 15 : (-6.9, 13.7, 5.7) | 16 : (0.0, 16.0, 7.0)
|  | 17 : (7.0, 13.0, 7.0) | 18 : (14.5, 10.5, 7.0) | 19 : (17.0, 3.0, 7.0) | 20 : (13.7, -4.0, 5.7)
4c | ${\bf S_{6}}$ | 1 : (0.0, 0.0, 0.0) | 2 : (6.1, 17.5, 0.0) | 3 : (-6.1, 17.5, 0.0) | 4 : (-18.2, -3.5, 0.0)
|  | 5 : (-12.1, -14.0, 0.0) | 6 : (12.1, -14.0, 0.0) | 7 : (18.2, -3.5, 0.0) | 8 : (0.0, 14.0, 0.0)
|  | 9 : (-12.1, -7.0, 0.0) | 10 : (12.1, -7.0, 0.0) | 11 : (0.0, 7.0, 0.0) | 12 : (-6.1, -3.5, 0.0)
|  | 13 : (6.1,-3.5,0.0) |  |  |

The atoms are two-photon excited from the ground state
$\left|0\right\rangle=\left|5S_{1/2},F=2,m_{F}=2\right\rangle$ to the Rydberg
state $\left|1\right\rangle=\left|71S_{1/2},m_{j}=1/2\right\rangle$ via the
intermediate state
$\left|m\right\rangle=\left|5P_{3/2},F=3,m_{F}=3\right\rangle$. We use 780 nm
(home-made extra-cavity diode laser) and 480 nm (TA-SHG Pro from Toptica)
lasers for the $\left|0\right\rangle\rightarrow\left|m\right\rangle$ and
$\left|m\right\rangle\rightarrow\left|1\right\rangle$ transitions,
respectively. Both the lasers are frequency-stabilized to an ultra-low-
expansion cavity (Stable laser systems, finesse 15000) by Pound-Drever-Hall
method. The effective Rabi frequency of the two-photon transition is
$\Omega=\Omega_{\rm 780}\Omega_{\rm 480}/2\Delta_{m}=2\pi\times 0.88(1)$ MHz,
where $\Omega_{\rm 780}=2\pi\times 50$ MHz and $\Omega_{\rm 480}=2\pi\times
20$ MHz are the Rabi frequencies are the 780 nm and 480 nm lasers,
respectively, and $\Delta_{m}=2\pi\times 570$ MHz is the intermediate
detuning. The laser power of 780(480) nm beam is 50 $\mu$W (550 mW) and the
$1/e^{2}$ beam diameter is 180 $\mu$m (100 $\mu$m), which is sufficient to
cover the atomic array we used with near-uniform Rabi frequencies less than 10
$\%$ deviation. The typical decay time of the single-atom Rabi oscillation is
7(2) $\mu$s [21].

For quantum annealing, the control parameters $\delta(t)$ and $\Omega(t)$ of
the Hamiltonian $\hat{H}(t)$ in Eq. (3) are adjusted from the paramagnetic
phase condition, $\delta(t=0)=\Delta_{i}<0$ and $\Omega(t=0)=0$, to the MIS
phase condition, $\delta(t=t_{f})=\Delta_{f}<U$ and $\Omega(t=t_{f})=0$.
Within the time of $t_{f}=4~{}\mu$s, we use three regions of operations. At
the first region during $t:0\rightarrow t_{f}/10$, $\delta(t)$ is fixed to
$\Delta_{i}$, and $\Omega(t)$ is linearly increased from zero to $\Omega_{0}$,
for the initial state to evolve to the ground state of quantum Ising
Hamiltonian. Then at the second region $\delta(t)$ is linearly increased from
$\Delta_{i}<0$ to $\Delta_{f}>0$, while $\Omega(t)=\Omega_{0}$ is constant
during $t:t_{f}/10\rightarrow 9t_{f}/10$. Finally at the third region, during
$t:9t_{f}10\rightarrow t_{f}$, $\Omega(t)$ gradually decreases to zero while
$\delta(t)=\Delta_{f}$ is kept constant [20]. Table 2 shows the interatomic
distance, $d$, initial detuning, $\Delta_{i}$, final detuning, $\Delta_{f}$,
and measurement repetitions of each experiment.

Table 2: Experimental parameters Figure | Graph |  $d$ ($\mu$m) |  $\Delta_{i}$ (MHz) |  $\Delta_{f}$ (MHz) | repetition
---|---|---|---|---|---
2a | ${\bf P_{4}}$ | 7.0 | -3.0 | +3.0 | 678
2b | ${\bf C_{4}}$ | 7.0 | -3.0 | +3.0 | 672
2c | ${\bf C_{6}}$ | 7.0 | -3.0 | +3.0 | 734
2g | ${\bf S_{4}}$ | 7.5 | -3.0 | +3.0 | 425
2h | 3-pan | 7.5 | -3.0 | +3.0 | 525
2i | 5-pan | 7.5 | -3.0 | +3.0 | 570
3a | ${\bf K_{5}}$ | 7.0 | -3.0 | +1.5 | 1961
3b | ${\bf K_{3,3}}$ | 8.0 | -3.0 | +3.0 | 1279
4c | ${\bf S_{6}}$ | 7.0 | -3.0 | +3.0 | 1640
Table 3: Experimental time budget Process | Time budget
---|---
Single atom loading (PGC) | 100 ms
Initial occupancy checking |  $40\times 2\times N_{p}$ ms
| (for 3D array with $N_{p}$ planes)
Arom rearrangements | 800 ms
Final occupancy checking |  $40\times 2\times N_{p}$ ms
Bias field ON, optical pumping | 20 ms
Quantum annealing |  $5~{}\mu$s
| (including trap off)
Bias field OFF | 20 ms
Final state detection |  $40\times 2\times N_{p}$ ms
Table 4: Experimental errors  Error sources | Treatment
---|---
Spontaneous decay from intermediate level $\left|m\right\rangle$ | Lindblad master equation [29, 30]
|
$\frac{d\rho}{dt}=-\frac{i}{\hbar}\left[H,\rho\right]+\sum_{j=1}^{N}{\left(L_{j}\rho
L_{j}^{\dagger}-\frac{1}{2}\left\\{L_{j}^{\dagger}L_{j},\rho\right\\}\right)}$
| with $L_{j}=\sqrt{\gamma_{m}/2}\sigma_{z}^{(j)}$ and $\gamma_{m}=2\pi\times
10$ kHz
Laser phase noise | Monte-Carlo simulation with spectral noises [30]
|  $10^{4}$ (Hz2$/$Hz) (780 nm) and $10^{3}$ (Hz2$/$Hz) (480 nm)
Laser intensity fluctuation | Monte-Carlo simulation with measured value (2$\%$)
Finite temperature | Monte-Carlo simulation with positional fluctuation [31]
| standard deviation $\sigma_{r}=0.1~{}\mu$m (radial) and
$\sigma_{z}=0.6~{}\mu$m (axial)
| where the temperature 30 $\mu$K, trap depth 1 mK
State-preparation-and-measurement (SPAM) error | Correction of measured microstate probabilities [20]
|  $P(\left|0\right\rangle|\left|1\right\rangle)=0.18$ and
$P(\left|1\right\rangle|\left|0\right\rangle)=0.03$

The experimental time budget is summarized in Table 3. After the procedure,
resulting atom states are detected with the fluorescence of ground-state atoms
during 40 ms cyclic transitions to $\left|5P_{3/2},F=3\right\rangle$, and the
procedure is repeated until the probability distribution is obtained. In our
experiment, state-preparation-and-detection (SPAM) errors are
$P(\left|0\right\rangle|\left|1\right\rangle)=0.18$ and
$P(1|\left|0\right\rangle)=0.03$ [20]. With these parameters the measured
microstates $S$ can be reconstructed to the error-corrected microstates
$S^{\prime}$ by $S^{\prime}=(M^{-1})^{\otimes N}S$ for $N$ atoms, where $M$ is
the SPAM error matrices, given by

$M=\begin{bmatrix}1-P(\left|0\right\rangle|\left|1\right\rangle)&P(\left|1\right\rangle|\left|0\right\rangle)\\\
P(\left|0\right\rangle|\left|1\right\rangle)&1-P(\left|1\right\rangle|\left|0\right\rangle)\end{bmatrix}.$
(5)

Experimental errors are listed in Table 4. Dominant errors are from the
spontaneous decay rate of the intermediate state and laser phase noises [29,
30]. While the the spontaneous decay from the Rydberg state (of the lifetime
$10^{4}$ times longer than $\left|m\right\rangle$) is ignorable, the
scattering rate to $\left|m\right\rangle$ is $\gamma_{m}=2\pi\times 10$ kHz.
The phase noise of Rydberg excitation lasers are measured by the frequency
noise spectrum of the error signal while both lasers are stabilized. The
overall spectral densities of frequency noise are measured to $10^{4}$
(Hz2$/$Hz) for 780 nm and $10^{3}$ (Hz2$/$Hz) for 480 nm around the Fourier
frequency near the Rabi frequency $\Omega_{0}$. Numerical simulations with
Lindblad master equation with these effects taken into account are shown (with
red dots and purple dots) in Figs. 2(f,l). Additional experimental errors are
from the finite temperature of atoms and the intensity fluctuations of Rydberg
excitation lasers. The finite temperature of atoms in the tweezer induces the
uncertainty of exact positions of atomic sites. For the temperature 30 $\mu$K
and the trap depth 1 mK, the standard deviation of the position is calculated
to $\sigma_{r}=0.1~{}\mu$m for radial direction and $\sigma_{z}=0.6~{}\mu$m
for axial direction (tweezer propagation direction). The intensity
fluctuations of lasers are measured about 2$\%$. Monte Carlo numerical
simulation are performed with these effect taken into to account for the
probabilities (green dots) in Figs. 2(f,l).

###### Acknowledgements.

This research was supported by Samsung Science and Technology Foundation
(SSTF-BA1301-52) and National Research Foundation of Korea (NRF)
(2017R1E1A1A01074307, 2019M3E4A1080411, 2020R1A4A3079707, and
2021R1A2C4001847).

## References

  * [1] S. Ebadi, T. T. Wang, H. Levine, H. et al., “Quantum phases of matter on a 256-atom programmable quantum simulator,” Nature 595, 227 (2021).
  * [2] F. Arute, K. Arya, R. Babbush, et al., “Quantum supremacy using a programmable superconducting processor,” Nature 574, 505 (2019).
  * [3] C. Monroe, W. C. Campbell, L.-M. Duan, Z. -X. Gong, A. V. Gorshkov, P. Hess, R. Islam, K. Kim, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, “Programmable Quantum Simulations of Spin Systems with Trapped Ions,” Rev. Mod. Phys. 93, 025001 (2021).
  * [4] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, “Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms,” Nature 595, 233 (2021).
  * [5] H. Chen, X. Kong, B. Chong, G. Qin, X. Zhou, X. Peng, and J. Du, “Experimental demonstration of a quantum annealing algorithm for the traveling salesman problem in a nuclear-magnetic-resonance quantum simulator,” Phys. Rev. A 83, 032314 (2011).
  * [6] S. Yarkoni, A. Plaat, and T. Back, “First results solving arbitrarily structured maximum independent set problems using quantum annealing,” IEEE Congress on Evolutionary Computation (CEC), 1 (2018).
  * [7] F. Centrone, N. Kumar, E. Diamanti, et al., “Experimental demonstration of quantum advantage for NP verification with limited information,” Nat. Commun. 12, 850 (2021).
  * [8] A. Lucas, “Ising formulations of many NP problems,” Frontiers in Phys. 2, 5 (2014).
  * [9] B. Korte and J. Vygen, _Combinatorial Optimization_ (Springer, New York, 2017).
  * [10] F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Mathematical and General 15, 3241 (1982).
  * [11] S. Arora and B. Barak, _Computational Complexity: A Modern Approach_ (Cambridge University Press, New York, 2009).
  * [12] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science 292, 472 (2001).
  * [13] N. G. Dickson and M. H. S. Amin, “Does adiabatic quantum optimization fail for NP-complete problems?” Phys. Rev. Lett. 106, 050502 (2011).
  * [14] H. Pichler, S.-T. Wang, L. Zhou, S. Choi, and M. D. Lukin, “Quantum optimization for maximum independent set using Rydberg atom arrays,” arXiv:1808.10816 (2018).
  * [15] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579 (2017).
  * [16] H. Labuhn, D. Barredo, S. Ravets, S. Léséleuc, T. Macrí, T. Lahaye, and A. Browaeys, ”Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature 534, 667 (2016).
  * [17] https://www.graphclasses.org
  * [18] K. Kuratowski, “Sur le probléme des courbes gauches en topologie,” Fund. Math. (in French) 15, 271 (1930).
  * [19] X. Qiu, P. Zoller, and X. Li, “Programmable quantum annealing architectures with Ising quantum wires,” PRX Quantum 1, 020311 (2020).
  * [20] Y. Song, M. Kim, H. Hwang, W. Lee, and J. Ahn, “Quantum simulation of Cayley-tree Ising Hamiltonians with three-dimensional Rydberg atoms,” Phys. Rev. Research 3, 013286 (2021).
  * [21] M. Kim, Y. Song, J. Kim, and J. Ahn, “Quantum-Ising Hamiltonian programming in trio, quartet, and sextet qubit systems,” PRX Quantum 1, 020323 (2020).
  * [22] D. Barredo, V. Lienhard, S. Léséleuc, T. Lahaye, and A. Browaeys, ”Synthetic three-dimensional atomic structures assembled atom by atom,” Nature 561, 79 (2018).
  * [23] W. Lee, H. Kim, and J. Ahn, ”Three-dimensional rearrangement of single atoms using actively controlled optical microtraps,” Optics Express 24, 9816 (2016).
  * [24] M. Garey and D. Johnson, “The rectilinear Steiner tree problem is NP-complete,” SIAM J. Appl. Math. 32, 826 (1977).
  * [25] J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2, 79 (2018).
  * [26] V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Babbush, V. Smelyanskiy, J. Martinis, and H. Neven, “What is the computational value of finite-range tunneling?” Phys. Rev. X 6, 031015 (2016).
  * [27] W. Lee, H. Kim, and J. Ahn, “Defect-free atomic array formation using the Hungarian matching algorithm,” Phys. Rev. A 95, 053424 (2017).
  * [28] H. Kim, M. Kim, W. Lee, and J. Ahn, “Gerchberg-Saxton algorithm for tweezer-trap atom arrangements,” Opt. Express 27(3), 2184 (2019).
  * [29] S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, “Analysis of imperfections in the coherent optical excitation of single atoms to Rydberg states,” Phys. Rev. A 97, 053803 (2018).
  * [30] W. Lee, M. Kim, H. Jo, Y. Song, and J. Ahn, “Coherent and dissipative dynamics of entangled few-body systems of Rydberg atoms,” Phys. Rev. A 99, 043404 (2019).
  * [31] H. Jo, Y. Song, M. Kim, and J. Ahn, “Rydberg Atom Entanglements in the Weak Coupling Regime,” Phys. Rev. Lett. 124, 033603 (2020).


# Overview of STEM Science as
process, method, material, and data Named Entities

Jennifer D’Souza
TIB Leibniz Information Centre for Science and Technology,
Hannover, Germany
<EMAIL_ADDRESS>

###### Abstract

We are faced with an unprecedented production in scholarly publications
worldwide. Stakeholders in the digital libraries posit that the document-based
publishing paradigm has reached the limits of adequacy. Instead, structured,
machine-interpretable, fine-grained scholarly knowledge publishing as
Knowledge Graphs (KG) is strongly advocated. In this work, we develop and
analyze a large-scale structured dataset of STEM articles across 10 different
disciplines, viz. Agriculture, Astronomy, Biology, Chemistry, Computer
Science, Earth Science, Engineering, Material Science, Mathematics, and
Medicine. Our analysis is defined over a large-scale corpus comprising 60K
abstracts structured as four scientific entities process, method, material,
and data. Thus our study presents, for the first-time, an analysis of a large-
scale multidisciplinary corpus under the construct of four named entity labels
that are specifically defined and selected to be domain-independent as opposed
to domain-specific. The work is then inadvertently a feasibility test of
characterizing multidisciplinary science with domain-independent concepts.
Further, to summarize the distinct facets of scientific knowledge per concept
per discipline, a set of word cloud visualizations are offered. The STEM-
NER-60k corpus, created in this work, comprises over 1M extracted entities
from 60k STEM articles obtained from a major publishing platform and is
publicly released https://github.com/jd-coderepos/stem-ner-60k.

Overview of STEM Science as
process, method, material, and data Named Entities

Jennifer D’Souza TIB Leibniz Information Centre for Science and Technology,
Hannover, Germany<EMAIL_ADDRESS>

##  1 Introduction

In the quest for knowledge Schubert (2006), significant progress has been made
toward the automated understanding of the meaning of text in the commonsense
domain. Some state-of-the-art systems that power commonsense machine
interpretability or readability are Babelfy Moro et al. (2014), DBpedia
Spotlight Mendes et al. (2011), NELL Mitchell et al. (2018), and FRED Gangemi
et al. (2017), to name a few. In contrast, scholarly literature remains
relatively understudied for its intelligible machine interpretability.
Consequently, fine-grained scholarly knowledge remains largely inaccessible
for machine reading. In terms of data standards, particularly the FAIR guiding
principles for scientific data creation Wilkinson et al. (2016), this implies
a wider unexplored scope for obtaining scientific resources that are findable,
accessible, interpretable, and reusable.

There are a multitude of recent emerging large-scale initiatives to build
specialised Scholarly Knowledge Graphs (SKGs) capable of serving specific user
needs. Consider Google Scholar, Web of Science Birkle et al. (2020), Microsoft
Academic Graph Wang et al. (2020), Open Research Knowledge Graph Auer (2018a);
Auer et al. (2020), Semantic Scholar Fricke (2018), etc. These initiatives
already create a practical systems need for automated graph encoding methods
(e.g., to associate IRIs as URIs or URLs with graph nodes for their
RDFization—a W3C labeled graph standard).

We position this paper within the broader aim of supporting the machine
reading of scientific terms multidisciplinarily as dereferencable Web
resources. Specifically, in this work, we release a large-scale,
multidisciplinary, structured dataset of scientific named entities called
STEM-NER-60k comprising over a 1M extracted entities from 60k articles across
the 10 most prolific STEM disciplines on Elsevier, viz. Agriculture (agr),
Astronomy (ast), Biology (bio), Chemistry (chem), Computer Science (cs), Earth
Science (es), Engineering (eng), Materials Science (ms), and Mathematics
(mat). The NER extraction objective was based on the STEM-ECR corpus in our
prior work Brack et al. (2020); D’Souza et al. (2020) which consisted of 110
abstracts from open access publications with each of aforementioned 10 domains
equally represented with 11 abstracts per domain. Each abstract was structured
in terms of the scientific entities, the entities were classified within a
four-class formalism (viz. process, method, material, and data) and were
resolved to Wikipedia and Wiktionary respectively. The 4-class formalism had
the following objective: to be restricted within the single broad domain of
Science, while still bridging terms multidisciplinarily, thus, in a sense,
validating the terms themselves and automatic systems developed for their
semantic adaptability. To this end, while our prior gold-standard STEM-ECR
corpus D’Souza et al. (2020) initiated the development of semantically-
adaptable systems, the silver-standard STEM-NER-60k corpus of this work
(https://github.com/jd-coderepos/stem-ner-60k) fosters the development of not
just semantically-adaptable, but also scalable solutions.

##  2 Background

Scholarly domain NER is not entirely new: the flagship Semantic Evaluation
(SemEval) series has so far seen five related tasks organized Kim et al.
(2010); Moro and Navigli (2015); Augenstein et al. (2017a); Gábor et al.
(2018a); D’Souza et al. (2021)—however, none of this work has been done so far
in the broad multidisciplinary setting of Science. Extending our prior work on
this theme Brack et al. (2020); D’Souza et al. (2020), we explore scholarly
NER on a large-scale, silver-standard corpus of structured STEM articles which
has a wide-ranging application scope in the emerging field of the creation and
discovery of SKGs that strive toward representing knowledge from scholarly
articles in machine-interpretable form. The characteristics of our corpus are
unique and noteworthy. 1) The linguistic phenomenon of interdisciplinary word-
sense switching is pervasive. E.g., consider the term “the Cloud” which in cs
takes on the meaning of a technological solution for hosting software, versus
in ast where it takes the common interpretation of the mass of water vapor we
see in the sky. 2) There is a seemingly evident shift of the sense
interpretation of terms to take on our corpus domain-specific scientific word
senses as opposed to their common sense interpretations which may be more
widely known. E.g., “power” in mat refers to exponentiation, which, otherwise
in a common sense, takes on a human social interpretation. Thus, our work on
multidisciplinary NER with semantically bridging concepts w.r.t both our prior
gold-standard STEM-ECR corpus (https://data.uni-hannover.de/dataset/stem-
ecr-v1-0) and the STEM-NER-60k corpus, discussed in this paper, facilitates
designing novel solutions attempting multidisciplinary NER within a generic
four-class formalism capable of bridging semantic concepts
multidisciplinarily.

The paper is structured in two main parts. First, related work in terms of
existing class formalisms for annotating scientific entities is discussed.
Second, insights to the STEM-NER-60k corpus released in this work
(https://github.com/jd-coderepos/stem-ner-60k) are given.

##  3 Related Work: Scientific Named Entity Recognition (NER) Formalisms

The structuring of unstructured articles as an NER task has been taken up at a
wide-scale in three scientific disciplines: Computer Science (CS), Biomedicine
(Bio), and Chemistry (Chem).

In this section, we discuss the different NER conceptual formalisms defined in
the three domains.

###  3.1 Computer Science NER (CS NER)

CS NER corpora can be compared along five dimensions: (1) domain, (2)
annotation coverage, (3) semantic concepts, (4) size, and (5) annotation
method. Most of the corpora consist of relatively short documents. The
shortest is the CL-Titles corpus (2021) with only paper titles. The longer
ones have sentences from full-text articles, viz. ScienceIE (2017b), NLP-TDMS
(2019), SciREX (2020), and ORKG-TDM (2021). We see that the corpora have had
from one D’Souza et al. (2021) to atmost seven NER concepts QasemiZadeh and
Schumann (2016). Each corpora’ concepts purposefully informs an overarching
knowledge extraction objective. E.g., the concepts focus, technique, and
domain in the FTD corpus (2011) helped examine the influence between research
communities; ACL-RD-TEC (2016) made possible a broader trends analysis with
seven concepts. Eventually, corpora began to shed light on a novel scientific
community research direction toward representing the entities as knowledge
graphs Auer (2018b) with hierarchical relation annotations such as synonymy
(2017b) or semantic relations such ‘Method Used-for a Task’ (2018); otherwise,
concepts were combined within full-fledged semantic constructs as Leaderboards
with between three to four concepts Hou et al. (2019); Jain et al. (2020);
Mondal et al. (2021); Kabongo et al. (2021), viz. rp, dataset, meth, metric,
and score; or were in extraction objectives with solely contributions-focused
entities of a paper Färber et al. (2021); D’Souza and Auer (2021).

###  3.2 Biomedical NER (BioNER)

BioNER predates CS NER. It was one of the earliest domains taken up for text
mining of fine-grained entities from scholarly publications to enhance search
engine performance in health applications. It aims to recognize concepts in
bioscience and medicine. E.g., protein, gene, disease, drug, tissue, body part
and location of activity such as cell or organism. The most frequently used
corpora are GENETAG (full-text articles annotated with protein/gene entities)
Tanabe et al. (2005), JNLPBA (~2400 abstracts annotated with DNA, RNA,
protein, cell type and cell line concepts) Collier and Kim (2004), GENIA (~200
Medline abstracts annotated with 36 different concepts from the Genia ontology
and several levels of linguistic/semantic features) Kim et al. (2003), NCBI
disease corpus (793 abstracts annotated with diseases in the MeSH taxonomy)
Doğan et al. (2014), CRAFT (the second largest corpus with 97 full text papers
annotated with over 4000 corpus) Bada et al. (2012) linking to the NCBI
Taxonomy, the Protein, Gene, Cell, Sequence ontologies etc. Finally, the
MedMentions corpus Mohan and Li (2018) as the largest dataset with ~4000
abstracts with ~34,724 concepts from the UMLS ontology. By leveraging
ontologies such as the Gene Ontology Ashburner et al. (2000), UMLS Bodenreider
(2004), MESH, or the NCBI Taxonomy Schoch et al. (2020), for the semantic
concepts, these corpora build on years of careful knowledge representation
work and are semantically consistent with a wide variety of other efforts that
exploit these community resources. This differs from CS NER which is evolving
toward standardized concepts.

Structured knowledge as knowledge bases (KB) were early seen as necessary in
organizing biomedical scientific findings. Biomedical NER was applied to build
such KBs. E.g., protein-protein (PPI) interaction databases as MINT Chatr-
Aryamontri et al. (2007) and IntAct Kerrien et al. (2007) or the more detailed
KBs as pathway Bader et al. (2006) or Gene Ontology Annotation Camon et al.
(2004). Community challenges help curate these KBs via text mining at a large-
scale. E.g., BioCreative for PPI Krallinger et al. (2008, 2011), protein-
mutation associations Krallinger et al. (2009), and gene-disease relations
Krallinger et al. (2010); or BioNLP Kim et al. (2011) for complex n-ary bio
events. CS NER is also been addressed in equivalent series such as SemEval
(2017b; 2018b; 2021) which is promising to foster rapid task progress.

###  3.3 Chemistry NER (ChemNER)

BioNER in part fosters Chemistry NER. Text mining for drug and chemical
compound entities Herrero-Zazo et al. (2013); Krallinger et al. (2015) are
indispensable to mining chemical disease relations Li et al. (2016), and drug
and chemical-protein interactions Krallinger et al. (2017, 2021). Obtaining
this structured knowledge has implications in precision medicine, drug
discovery as well as basic biomedical research. Corpora for ChemNER are
Corbett et al.’s dataset (42 full-text papers with ~7000 chemical entities),
ChemDNER (10,000 PubMed abstracts with 84,355 chemical entities) (2015), and
NLM-Chem (150 full-text papers with 38,342 chemical entities normalized to
2,064 MeSH identifiers) Islamaj et al. (2021).

The high-level identification of entities in text is a staple of most modern
NLP pipelines over commonsense knowledge. This, in the context of the
scientific entities formalisms presented above is pertinent to scholarly
knowledge as well. While being structurally domain-wise related, our work has
a unique objective: to extract entities across STEM Science which follow a
generic four-entity conceptual formalism. Thus apart from having an impact in
the emerging field of the discovery of science graphs, the STEM-NER-60k corpus
can have specific applications in higher-level NLP tasks including information
extraction Shah et al. (2003), factuality ascertainment of statements in
knowledge base population Adel (2018), and question answering over linked data
Unger et al. (2014).

##  4 Our STEM-NER-60k Corpus

We now describe our corpus: 1) definitions of the four scientific concepts
(2020), viz. process, method, material, and data, are given; 2) the process by
which the silver-standard STEM-NER-60k corpus is created is explained; and 3)
corpus insights are offered specifically in terms of the multidisciplinary
entities annotated under the formalism of four concepts bridging the 10
domains.

###  4.1 Concept Definitions

Following an iterative process of concept refinement D’Souza et al. (2020) via
a process that involved expert adjudication of which scientific concepts were
multidisciplinarily semantically meaningful, the following four concepts were
agreed upon to be relevant for entities across STEM, specifically across the
following 10 domains, viz. agri, ast, bio, chem, cs, es, eng, ms, and math.

  * •

process. Natural phenomenon, or independent/dependent activities. E.g.,
growing (bio), cured (ms), flooding (es).

  * •

method. A commonly used procedure that acts on entities. E.g., powder X-ray
(chem), the PRAM analysis (cs), magnetoencephalography (med).

  * •

material. A physical or digital entity used for scientific experiments. E.g.,
soil (agri), the moon (ast), the set (math).

  * •

data. The data themselves, or quantitative or qualitative characteristics of
entities. E.g., rotational energy (eng), tensile strength (ms), vascular risk
(med).

###  4.2 Corpus Creation

The silver-standard STEM-NER-60k corpus was created as follows. Roughly 60,000
articles in text format and restricted only to the articles with the CC-BY
redistributable license on Elsevier were first downloaded
https://tinyurl.com/60k-raw-dataset. Next, our aim was to obtain the four-
concept entity annotations for the Abstracts in this corpus of publications.
We leveraged our prior-developed state-of-the-art NER system for this purpose
Brack et al. (2020). The Brack et al. system was based on Beltagy et al.’s
SciBERT which in turn for the specific NER configuration makes use of the
original BERT NER Devlin et al. (2018); Ma and Hovy (2016) prediction
architecture. The Brack et al. system, however, was pretrained on a smaller
gold-standard STEM corpus (2020) expert-annotated with the four generic
scientific entities described in subsection 4.1. Applying this system on the
Abstracts on the newly downloaded 60k articles produced the silver-standard
STEM-NER-60k corpus of this work. The resulting silver-standard corpus
statistics are shown in Table 1.

|  articles |  process |  method |  material |  data
---|---|---|---|---|---
agr |  4,944 |  20,532 |  3,252 |  62,043 |  33,608
ast |  15,003 |  31,104 |  10,423 |  55,753 |  97,011
bio |  9,038 |  54,029 |  6,777 |  100,454 |  43,418
chem |  5,232 |  18,185 |  6,044 |  48,779 |  30,596
cs |  5,389 |  17,014 |  13,650 |  35,554 |  33,575
es |  4,363 |  28,432 |  5,129 |  56,571 |  50,211
eng |  2,441 |  12,705 |  3,293 |  24,633 |  24,238
ms |  2,144 |  10,102 |  2,437 |  23,054 |  16,981
math |  1,765 |  8,002 |  1,941 |  11,381 |  15,631
med |  15,054 |  89,637 |  19,580 |  148,059 |  134,249
Table 1: STEM-NER-60k corpus statistics comprising 59,984 articles structured
in terms of process, method, material, and data concepts

Our corpus is publicly released (https://github.com/jd-coderepos/stem-ner-60k)
to support related R&D endeavors and for other researchers interested in
further investigating the task of scholarly NER.

###  4.3 Corpus Insights

Here, details of the STEM-NER-60k corpus are offered.

But first, given our large-scale multidisciplinary scientific corpus, we take
the opportunity to briefly examine the difference between scientific writing
and non-scientific writing, if any, with the help of entropy formulations for
each of our data domains. The entropies obtained for our corpus per domain is
as follows: med (4.58) > chem (4.58) > bio (4.56) > ast (4.53) > agri (4.5) >
ms (4.5) > es (4.48) > eng (4.42) > math (4.4). Intriguingly, these numbers
are close to non-scientific English (4.11 bits Cover (1999)). Thus, given our
corpus scientific English, one can rule out any atypical English usage syntax
other than domain-specific jargon vocabulary.

Next, briefly, we address which general scientific entity annotation patterns
can one anticipate in our silver-standard corpus? 1) Entity annotations can be
expected as definite noun phrases whenever found. 2) Coreferring lexical units
for scientific entities in the context of a single Abstract can be expected to
be annotated with the same concept type. 3) Quantifiable lexical units such as
numbers (e.g., years 1999, measurements 4km) or even phrases (e.g., vascular
risk) should be data. 4) Where found, the most precise text reference (i.e.
including qualifiers) regarding materials used in the experiment should be
annotated. E.g., the term “carbon atoms in graphene” was annotated as a single
material entity and not separately as “carbon atoms”, “graphene”. 5) The
precedence of annotation of the scientific concepts in the original corpus in
case of any confusion in classifying the four classes of scientific entities
was resolved as follows: method > process > data > material, where the concept
appearing earlier in the list was selected as the preferred class.

####  4.3.1 STEM scientific terms as process

Verbs (e.g., measured), verb phrases (e.g., integrating results), or noun
phrases (e.g., an assessment, future changes, this transport process, the
transfer) can be expected to be scientific entity candidates for process.
Generally, it can be one of two things: an occurrence natural to the
state/properties of the entity or an action performed by the investigators. In
the latter case, however, it is better aptly expected as a method entity when
the action is a named instance.

Some examples are offered for scientific entities as process candidates. 1) In
the sentence, “The transfer of a laboratory process into a manufacturing
facility is one of the most critical steps required for the large scale
production of cell-based therapy products”, the terms “The transfer”, “a
laboratory process”, and “the large scale production” all are of type process.
2) In “The transterminator ion flow in the Venusian ionosphere is observed at
solar minimum for the first time.”, the terms “The transterminator ion flow”
and “solar minimum” process entities. The verb “observed”, however, is not
considered a process since it doesn’t act upon another object. 3) On the other
hand, in “It is suggested that this ion flow contributes to maintaining the
nightside ionosphere.”, the terms “this ion flow” and “maintaining” are both
considered valid process candidates. Finally, 4) in the sentence “Cellular
morphology, pluripotency gene expression and differentiation into the three
germ layers have been used compare the outcomes of manual and automated
processes” the terms “pluripotency gene expression”, “differentiation”,
“compare”, and “manual and automated processes” are each annotated as process.

####  4.3.2 STEM scientific terms as method

Phrases which can be expected to be annotated as a method entity are those
which contain the following trigger words: simulation, method, algorithm,
scheme, technique, system, function, derivative, proportion, strategy, solver,
experiment, test, computation, program. As an example, consider the sentence
“Here finite-element modelling has demonstrated that once one silica
nanoparticle debonds then its nearest neighbours are shielded from the applied
stress field, and hence may not debond.” In this sentence, the term “finite-
element modelling” is annotated as a method.

####  4.3.3 STEM scientific terms as material

This concept is expounded merely with the following three examples. 1) In
“Based on the results of the LUCAS topsoil survey we performed an assessment
of plant available P status of European croplands.” the term “European
croplands” should be material. 2) In “The transfer of a laboratory process
into a manufacturing facility is one of the most critical steps required for
the large scale production of cell-based therapy products.” there are two
material terms, viz. “a manufacturing facility” and “cell-based therapy
products”. Finally, 3) in the sentence “Cellular morphology, pluripotency gene
expression and differentiation into the three germ layers have been used to
compare the outcomes of manual and automated processes.” the phrase “the three
germ layers” is a material.

####  4.3.4 STEM scientific terms as data

Phrases satisfying the patterns in the following examples can be expected to
be data. 1) In “Based on the results of the LUCAS topsoil survey we performed
an assessment of plant available P status of European croplands.”, the phrases
“the results” and “plant available P status” are considered as data in the
original annotation scheme. 2) In “Our analysis shows a status of a baseline
period of the years 2009 and 2012, while a repeated LUCAS topsoil survey can
be a useful tool to monitor future changes of nutrient levels, including P in
soils of the EU.” the phrases: “a status of a baseline period”, “nutrient
levels”, and “P” are data items. 3) Further, in “Observations near the
terminator of the energies of ions of ionospheric origin showed asymmetry
between the noon and midnight sectors, which indicated an antisunward ion flow
with a velocity of (2.5±1.5)kms-1.” the terms “asymmetry between the noon and
midnight sectors”, “a velocity”, and “(2.5±1.5)kms-1” are data. Finally, 4) in
“We established a P fertilizer need map based on integrating results from the
two systems.” the phrase “a P fertilizer need map” is data which should
override the selection of “a P fertilizer” as material by the concept
precedence stated earlier.

Next, in Figures 1 to 10, word clouds of the top 100 entities for the 10 STEM
disciplines is shown.

(a) process

(b) method

(c) material

(d) data

Figure 1: Agriculture domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 2: Astronomy domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 3: Biology domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 4: Chemistry domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 5: Computer Science domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 6: Earth Science domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 7: Engineering domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 8: Material Science domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 9: Mathematics domain word clouds

(a) process

(b) method

(c) material

(d) data

Figure 10: Medicine domain word clouds

##  5 Applications

In this section, we offer some directions for practical applications developed
based on STEM-ECR NER. With this we hope to offer inspiration for similar or
related usages of the STEM-NER-60k corpus developed in this work.

###  5.1 STEM Entities Recommendation Service in the Open Research Knowledge
Graph

Figure 11: STEM Entities-based Abstract Annotator Recommendation Service for
process, method, material, and data entities in the Add-paper wizard of the
next-generation Open Research Knowledge Graph (ORKG) digital library front-end
service.

The STEM-ECR annotation project (https://data.uni-hannover.de/dataset/stem-
ecr-v1-0) was initiated to support the population of structured scientific
concepts in the Open Research Knowledge Graph (ORKG). Figure 11 demonstrates
the integration of our prior developed machine learning model of STEM entities
Brack et al. (2020) in the ORKG frontend. The model takes as input the
Abstract of a new incoming publication and structures the Abstract per the
four concepts. The user can then flexibly select and deselect annotations
based on their prediction confidences or the user preference to automatically
structure the contribution data of the work.

###  5.2 Knowledge Graph Construction for Fine-Grained Structured Search

Knowledge Graphs (KG) play a crucial role in many modern applications
(https://developers.google.com/knowledge-graph) as solutions to the
information access and search problem. There have been several initiatives in
the NLP Ammar et al. (2018); Luan et al. (2018); D’Souza and Auer (2021);
D’Souza et al. (2021), and the Semantic Web Auer et al. (2018); Jaradeh et al.
(2019), communities suggesting an increasing trend toward adoption of KGs for
scientific articles. The automatic construction of KGs from text is a
challenging problem, more so owing to the multidisciplinary nature of Science
at large. While machines can better handle the volume of scientific
literature, they need supervisory signals to determine which elements of the
text have value. The STEM-NER-60k corpus can be leveraged to construct
knowledge graphs underlying fine-grained search over publications. Figure 12
demonstrates an example KG that was manually annotated with relations between
the entities. In the figure, the entity nodes are color-coded by their concept
type: orange corresponds to process, purple for method, green corresponds to
material, and blue for data.

Figure 12: Structured Knowledge Graph (KG) representation of a Material
Science domain publication Abstract Chen et al. (2013) as process, method,
material, and data typed entities. For KGs in the remaining 9 STEM domains we
consider, see Appendix A.

##  6 Conclusion and Future Work Directions

In this paper, we have systematically introduced a large-scale
multidisciplinary STEM corpus spanning 10 disciplines with structured
abstracts in terms of generic process, method, material, and data entities.
The corpus called STEM-NER-60k is publicly released for facilitating future
research https://github.com/jd-coderepos/stem-ner-60k. Based on the
applications presented in section 5, we envision two main future research
directions: 1) extending the set of generic concepts beyond the four proposed
in this work. E.g., our prior work Brack et al. (2020) proposed concepts such
as task, objective, and result. And, 2) for automatic KG construction
multidisciplinarily, annotating semantic relations between the entities.

## References

  * Adel (2018) Heike Adel. 2018.  _Deep learning methods for knowledge base population_.  Ph.D. thesis, lmu.
  * Ammar et al. (2018) Waleed Ammar, Dirk Groeneveld, Chandra Bhagavatula, Iz Beltagy, Miles Crawford, Doug Downey, Jason Dunkelberger, Ahmed Elgohary, Sergey Feldman, Vu Ha, et al. 2018.  Construction of the literature graph in semantic scholar.  In _NAACL-HLT (3)_.
  * Ashburner et al. (2000) Michael Ashburner, Catherine A Ball, Judith A Blake, David Botstein, Heather Butler, J Michael Cherry, Allan P Davis, Kara Dolinski, Selina S Dwight, Janan T Eppig, et al. 2000.  Gene ontology: tool for the unification of biology.  _Nature genetics_ , 25(1):25–29.
  * Auer et al. (2018) Sören Auer, Viktor Kovtun, Manuel Prinz, Anna Kasprzik, Markus Stocker, and Maria Esther Vidal. 2018.  Towards a knowledge graph for science.  In _Proceedings of the 8th International Conference on Web Intelligence, Mining and Semantics_ , pages 1–6.
  * Auer et al. (2020) Sören Auer, Allard Oelen, Muhammad Haris, Markus Stocker, Jennifer D’Souza, Kheir Eddine Farfar, Lars Vogt, Manuel Prinz, Vitalis Wiens, and Mohamad Yaser Jaradeh. 2020.  Improving access to scientific literature with knowledge graphs.  _Bibliothek Forschung und Praxis_ , 44(3):516–529.
  * Auer (2018a) Sören Auer. 2018a.  Towards an open research knowledge graph.
  * Auer (2018b) Sören Auer. 2018b.  Towards an open research knowledge graph.
  * Augenstein et al. (2017a) Isabelle Augenstein, Mrinal Das, Sebastian Riedel, Lakshmi Vikraman, and Andrew McCallum. 2017a.  Semeval 2017 task 10: Scienceie - extracting keyphrases and relations from scientific publications.  In _SemEval@ACL_.
  * Augenstein et al. (2017b) Isabelle Augenstein, Mrinal Das, Sebastian Riedel, Lakshmi Vikraman, and Andrew McCallum. 2017b.  SemEval 2017 task 10: ScienceIE - extracting keyphrases and relations from scientific publications.  In _Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017)_ , pages 546–555, Vancouver, Canada. Association for Computational Linguistics.
  * Bada et al. (2012) Michael Bada, Miriam Eckert, Donald Evans, Kristin Garcia, Krista Shipley, Dmitry Sitnikov, William A Baumgartner, K Bretonnel Cohen, Karin Verspoor, Judith A Blake, et al. 2012.  Concept annotation in the craft corpus.  _BMC bioinformatics_ , 13(1):1–20.
  * Bader et al. (2006) Gary D Bader, Michael P Cary, and Chris Sander. 2006.  Pathguide: a pathway resource list.  _Nucleic acids research_ , 34(suppl_1):D504–D506.
  * Beltagy et al. (2019) Iz Beltagy, Kyle Lo, and Arman Cohan. 2019.  Scibert: A pretrained language model for scientific text.  In _EMNLP-IJCNLP_ , pages 3606–3611.
  * Birkle et al. (2020) Caroline Birkle, David A Pendlebury, Joshua Schnell, and Jonathan Adams. 2020.  Web of science as a data source for research on scientific and scholarly activity.  _Quantitative Science Studies_ , 1(1):363–376.
  * Bodenreider (2004) Olivier Bodenreider. 2004.  The unified medical language system (umls): integrating biomedical terminology.  _Nucleic Acids Research_ , 32(Database issue):D267.
  * Brack et al. (2020) Arthur Brack, Jennifer D’Souza, Anett Hoppe, Sören Auer, and Ralph Ewerth. 2020.  Domain-independent extraction of scientific concepts from research articles.  In _European Conference on Information Retrieval_ , pages 251–266. Springer.
  * Camon et al. (2004) Evelyn Camon, Michele Magrane, Daniel Barrell, Vivian Lee, Emily Dimmer, John Maslen, David Binns, Nicola Harte, Rodrigo Lopez, and Rolf Apweiler. 2004.  The gene ontology annotation (goa) database: sharing knowledge in uniprot with gene ontology.  _Nucleic Acids Research_ , 32(Database issue):D262.
  * Chatr-Aryamontri et al. (2007) Andrew Chatr-Aryamontri, Arnaud Ceol, Luisa Montecchi Palazzi, Giuliano Nardelli, Maria Victoria Schneider, Luisa Castagnoli, and Gianni Cesareni. 2007\.  Mint: the molecular interaction database.  _Nucleic acids research_ , 35(suppl_1):D572–D574.
  * Chen et al. (2013) J Chen, AJ Kinloch, S Sprenger, and AC Taylor. 2013.  The mechanical properties and toughening mechanisms of an epoxy polymer modified with polysiloxane-based core-shell particles.  _Polymer_ , 54(16):4276–4289.
  * Collier and Kim (2004) Nigel Collier and Jin-Dong Kim. 2004.  Introduction to the bio-entity recognition task at JNLPBA.  In _Proceedings of the International Joint Workshop on Natural Language Processing in Biomedicine and its Applications (NLPBA/BioNLP)_ , pages 73–78, Geneva, Switzerland. COLING.
  * Corbett et al. (2007) Peter Corbett, Colin Batchelor, and Simone Teufel. 2007.  Annotation of chemical named entities.  In _Biological, translational, and clinical language processing_ , pages 57–64.
  * Cotton et al. (2014) Alissa Cotton, Kumar Patchigolla, and John E Oakey. 2014.  Minor and trace element emissions from post-combustion co2 capture from coal: Experimental and equilibrium calculations.  _Fuel_ , 117:391–407.
  * Cover (1999) Thomas M Cover. 1999.  _Elements of information theory_.  John Wiley & Sons.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _CoRR_ , abs/1810.04805.
  * Doğan et al. (2014) Rezarta Islamaj Doğan, Robert Leaman, and Zhiyong Lu. 2014.  Ncbi disease corpus: a resource for disease name recognition and concept normalization.  _Journal of biomedical informatics_ , 47:1–10.
  * Dolev et al. (2014) Danny Dolev, Matthias Függer, Markus Posch, Ulrich Schmid, Andreas Steininger, and Christoph Lenzen. 2014.  Rigorously modeling self-stabilizing fault-tolerant circuits: An ultra-robust clocking scheme for systems-on-chip.  _Journal of Computer and System Sciences_ , 80(4):860–900.
  * D’Souza and Auer (2021) Jennifer D’Souza and Soeren Auer. 2021.  Pattern-based acquisition of scientific entities from scholarly article titles.  _arXiv preprint arXiv:2109.00199_.
  * D’Souza et al. (2020) Jennifer D’Souza, Anett Hoppe, Arthur Brack, Mohmad Yaser Jaradeh, Sören Auer, and Ralph Ewerth. 2020.  The STEM-ECR dataset: Grounding scientific entity references in STEM scholarly content to authoritative encyclopedic and lexicographic sources.  In _Proceedings of the 12th Language Resources and Evaluation Conference_ , pages 2192–2203, Marseille, France. European Language Resources Association.
  * D’Souza and Auer (2021) Jennifer D’Souza and Sören Auer. 2021.  Sentence, phrase, and triple annotations to build a knowledge graph of natural language processing contributions—a trial dataset.  _Journal of Data and Information Science_ , 6(3):6–34.
  * D’Souza et al. (2021) Jennifer D’Souza, Sören Auer, and Ted Pedersen. 2021.  Semeval-2021 task 11: Nlpcontributiongraph-structuring scholarly nlp contributions for a research knowledge graph.  In _Proceedings of the 15th International Workshop on Semantic Evaluation (SemEval-2021)_ , pages 364–376.
  * D’Souza et al. (2020) Jennifer D’Souza, Anett Hoppe, Arthur Brack, Mohmad Yaser Jaradeh, Sören Auer, and Ralph Ewerth. 2020.  The stem-ecr dataset: Grounding scientific entity references in stem scholarly content to authoritative encyclopedic and lexicographic sources.  In _Proceedings of The 12th Language Resources and Evaluation Conference_ , pages 2192–2203.
  * Färber et al. (2021) Michael Färber, Alexander Albers, and Felix Schüber. 2021.  Identifying used methods and datasets in scientific publications.  In _Proceedings of the Workshop on Scientific Document Understanding: co-located with 35th AAAI Conference on Artificial Inteligence (AAAI 2021) ; Remote, February 9, 2021. Ed.: A. P. B. Veyseh_ , volume 2831 of _CEUR Workshop Proceedings_. CEUR Workshop Proceedings (CEUR-WS).
  * Fricke (2018) Suzanne Fricke. 2018.  Semantic scholar.  _Journal of the Medical Library Association: JMLA_ , 106(1):145.
  * Gábor et al. (2018a) Kata Gábor, Davide Buscaldi, Anne-Kathrin Schumann, Behrang QasemiZadeh, Haifa Zargayouna, and Thierry Charnois. 2018a.  Semeval-2018 task 7: Semantic relation extraction and classification in scientific papers.  In _SemEval_ , pages 679–688.
  * Gábor et al. (2018b) Kata Gábor, Davide Buscaldi, Anne-Kathrin Schumann, Behrang QasemiZadeh, Haïfa Zargayouna, and Thierry Charnois. 2018b.  SemEval-2018 task 7: Semantic relation extraction and classification in scientific papers.  In _Proceedings of The 12th International Workshop on Semantic Evaluation_ , pages 679–688, New Orleans, Louisiana. Association for Computational Linguistics.
  * Gangemi et al. (2017) Aldo Gangemi, Valentina Presutti, Diego Reforgiato Recupero, Andrea Giovanni Nuzzolese, Francesco Draicchio, and Misael MongiovÃ¬. 2017.  Semantic Web Machine Reading with FRED.  _Semantic Web_ , 8(6):873–893.
  * Gupta and Manning (2011) Sonal Gupta and Christopher Manning. 2011.  Analyzing the dynamics of research by extracting key aspects of scientific papers.  In _Proceedings of 5th International Joint Conference on Natural Language Processing_ , pages 1–9, Chiang Mai, Thailand. Asian Federation of Natural Language Processing.
  * Haution (2012) Olivier Haution. 2012.  Integrality of the chern character in small codimension.  _Advances in Mathematics_ , 231(2):855–878.
  * Herrero-Zazo et al. (2013) María Herrero-Zazo, Isabel Segura-Bedmar, Paloma Martínez, and Thierry Declerck. 2013.  The ddi corpus: An annotated corpus with pharmacological substances and drug–drug interactions.  _Journal of biomedical informatics_ , 46(5):914–920.
  * Hou et al. (2019) Yufang Hou, Charles Jochim, Martin Gleize, Francesca Bonin, and Debasis Ganguly. 2019.  Identification of tasks, datasets, evaluation metrics, and numeric scores for scientific leaderboards construction.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 5203–5213, Florence, Italy. Association for Computational Linguistics.
  * Islamaj et al. (2021) Rezarta Islamaj, Robert Leaman, Sun Kim, Dongseop Kwon, Chih-Hsuan Wei, Donald C Comeau, Yifan Peng, David Cissel, Cathleen Coss, Carol Fisher, et al. 2021.  Nlm-chem, a new resource for chemical entity recognition in pubmed full text literature.  _Scientific Data_ , 8(1):1–12.
  * Jain et al. (2020) Sarthak Jain, Madeleine van Zuylen, Hannaneh Hajishirzi, and Iz Beltagy. 2020.  SciREX: A challenge dataset for document-level information extraction.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 7506–7516, Online. Association for Computational Linguistics.
  * Jaradeh et al. (2019) Mohamad Yaser Jaradeh, Allard Oelen, Kheir Eddine Farfar, Manuel Prinz, Jennifer D’Souza, Gábor Kismihók, Markus Stocker, and Sören Auer. 2019\.  Open research knowledge graph: Next generation infrastructure for semantic scholarly knowledge.  In _Proceedings of the 10th International Conference on Knowledge Capture_ , K-CAP ’19, page 243–246, New York, NY, USA. Association for Computing Machinery.
  * Kabongo et al. (2021) Salomon Kabongo, Jennifer D’Souza, and Sören Auer. 2021.  Automated mining of leaderboards for empirical ai research.  _arXiv preprint arXiv:2109.13089_.
  * Kakimpa et al. (2012) B Kakimpa, DM Hargreaves, and JS Owen. 2012.  An investigation of plate-type windborne debris flight using coupled cfd–rbd models. part ii: Free and constrained flight.  _Journal of wind engineering and industrial aerodynamics_ , 111:104–116.
  * Kerrien et al. (2007) Samuel Kerrien, Yasmin Alam-Faruque, Bruno Aranda, Iain Bancarz, Alan Bridge, Cathy Derow, Emily Dimmer, Marc Feuermann, Anja Friedrichsen, Rachael Huntley, et al. 2007.  Intact—open source resource for molecular interaction data.  _Nucleic acids research_ , 35(suppl_1):D561–D565.
  * Kim et al. (2003) J-D Kim, Tomoko Ohta, Yuka Tateisi, and Jun’ichi Tsujii. 2003.  Genia corpus—a semantically annotated corpus for bio-textmining.  _Bioinformatics_ , 19(suppl_1):i180–i182.
  * Kim et al. (2011) Jin-Dong Kim, Tomoko Ohta, Sampo Pyysalo, Yoshinobu Kano, and Jun’ichi Tsujii. 2011.  Extracting bio-molecular events from literature—the bionlp’09 shared task.  _Computational Intelligence_ , 27(4):513–540.
  * Kim et al. (2010) Su Nam Kim, Olena Medelyan, Min-Yen Kan, and Timothy Baldwin. 2010.  Semeval-2010 task 5: Automatic keyphrase extraction from scientific articles.  In _Proceedings of the 5th International Workshop on Semantic Evaluation_ , pages 21–26.
  * Kivimäki et al. (2012) Mika Kivimäki, Martin J Shipley, Charlotte L Allan, Claire E Sexton, Markus Jokela, Marianna Virtanen, Henning Tiemeier, Klaus P Ebmeier, and Archana Singh-Manoux. 2012.  Vascular risk status as a predictor of later-life depressive symptoms: a cohort study.  _Biological psychiatry_ , 72(4):324–330.
  * Krallinger et al. (2009) Martin Krallinger, Jose MG Izarzugaza, Carlos Rodriguez-Penagos, and Alfonso Valencia. 2009.  Extraction of human kinase mutations from literature, databases and genotyping studies.  _BMC bioinformatics_ , 10(8):1–20.
  * Krallinger et al. (2008) Martin Krallinger, Florian Leitner, Carlos Rodriguez-Penagos, and Alfonso Valencia. 2008.  Overview of the protein-protein interaction annotation extraction task of biocreative ii.  _Genome biology_ , 9(2):1–19.
  * Krallinger et al. (2010) Martin Krallinger, Florian Leitner, and Alfonso Valencia. 2010.  Analysis of biological processes and diseases using text mining approaches.  _Bioinformatics Methods in Clinical Research_ , pages 341–382.
  * Krallinger et al. (2021) Martin Krallinger, Antonio Miranda, Farrokh Mehryary, Jouni Luoma, Sampo Pyysalo, and Alfonso Valencia. 2021.  Drugprot shared task (biocreative vii track 1-2021) text mining drug-protein/gene interactions (drugprot) shared task.
  * Krallinger et al. (2017) Martin Krallinger, Obdulia Rabal, Saber Ahmad Akhondi, Martín Pérez Pérez, Jesus Santamaría, Gael Pérez Rodríguez, Georgios Tsatsaronis, Ander Intxaurrondo, José Antonio Baso López, Umesh K. Nandal, Erin M. van Buel, Anjana Chandrasekhar, Marleen Rodenburg, Astrid Lægreid, Marius A. Doornenbal, Julen Oyarzábal, Anália Lourenço, and Alfonso Valencia. 2017.  Overview of the biocreative vi chemical-protein interaction track.
  * Krallinger et al. (2015) Martin Krallinger, Obdulia Rabal, Florian Leitner, Miguel Vazquez, David Salgado, Zhiyong Lu, Robert Leaman, Yanan Lu, Donghong Ji, Daniel M Lowe, et al. 2015.  The chemdner corpus of chemicals and drugs and its annotation principles.  _Journal of cheminformatics_ , 7(1):1–17.
  * Krallinger et al. (2011) Martin Krallinger, Miguel Vazquez, Florian Leitner, David Salgado, Andrew Chatr-Aryamontri, Andrew Winter, Livia Perfetto, Leonardo Briganti, Luana Licata, Marta Iannuccelli, et al. 2011.  The protein-protein interaction tasks of biocreative iii: classification/ranking of articles and linking bio-ontology concepts to full text.  _BMC bioinformatics_ , 12(8):1–31.
  * Krupp et al. (2012) N Krupp, E Roussos, P Kollmann, C Paranicas, DG Mitchell, SM Krimigis, A Rymer, GH Jones, CS Arridge, TP Armstrong, et al. 2012.  The cassini enceladus encounters 2005–2010 in the view of energetic electron measurements.  _Icarus_ , 218(1):433–447.
  * Li et al. (2016) Jiao Li, Yueping Sun, Robin J Johnson, Daniela Sciaky, Chih-Hsuan Wei, Robert Leaman, Allan Peter Davis, Carolyn J Mattingly, Thomas C Wiegers, and Zhiyong Lu. 2016.  Biocreative v cdr task corpus: a resource for chemical disease relation extraction.  _Database_ , 2016.
  * Luan et al. (2018) Yi Luan, Luheng He, Mari Ostendorf, and Hannaneh Hajishirzi. 2018.  Multi-task identification of entities, relations, and coreference for scientific knowledge graph construction.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 3219–3232.
  * Ma and Hovy (2016) Xuezhe Ma and Eduard H. Hovy. 2016.  End-to-end sequence labeling via bi-directional lstm-cnns-crf.  _CoRR_ , abs/1603.01354.
  * Martin et al. (2012) Sarah L Martin, Sacha J Mooney, Matthew J Dickinson, and Helen M West. 2012.  Soil structural responses to alterations in soil microbiota induced by the dilution method and mycorrhizal fungal inoculation.  _Pedobiologia_ , 55(5):271–281.
  * Mendes et al. (2011) Pablo N Mendes, Max Jakob, Andrés García-Silva, and Christian Bizer. 2011\.  Dbpedia spotlight: shedding light on the web of documents.  In _Proceedings of the 7th international conference on semantic systems_ , pages 1–8. ACM.
  * Mitchell et al. (2018) Tom Mitchell, William Cohen, Estevam Hruschka, Partha Talukdar, Bishan Yang, Justin Betteridge, Andrew Carlson, Bhanava Dalvi, Matt Gardner, Bryan Kisiel, et al. 2018.  Never-ending learning.  _Communications of the ACM_ , 61(5):103–115.
  * Mohan and Li (2018) Sunil Mohan and Donghui Li. 2018.  Medmentions: A large biomedical corpus annotated with umls concepts.  In _Automated Knowledge Base Construction (AKBC)_.
  * Mondal et al. (2021) Ishani Mondal, Yufang Hou, and Charles Jochim. 2021.  End-to-end construction of NLP knowledge graph.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 1885–1895, Online. Association for Computational Linguistics.
  * Moro et al. (2014) Andrea Moro, Francesco Cecconi, and Roberto Navigli. 2014.  Multilingual Word Sense Disambiguation and Entity Linking for Everybody.  In _ISWC_ , pages 25–28, Riva del Garda, Italy.
  * Moro and Navigli (2015) Andrea Moro and Roberto Navigli. 2015.  Semeval-2015 task 13: Multilingual all-words sense disambiguation and entity linking.  In _Proceedings of the 9th international workshop on semantic evaluation (SemEval 2015)_ , pages 288–297.
  * QasemiZadeh and Schumann (2016) Behrang QasemiZadeh and Anne-Kathrin Schumann. 2016.  The acl rd-tec 2.0: A language resource for evaluating term extraction and entity recognition methods.  In _Proceedings of the Tenth International Conference on Language Resources and Evaluation (LREC’16)_ , pages 1862–1868.
  * Schoch et al. (2020) Conrad L Schoch, Stacy Ciufo, Mikhail Domrachev, Carol L Hotton, Sivakumar Kannan, Rogneda Khovanskaya, Detlef Leipe, Richard Mcveigh, Kathleen O’Neill, Barbara Robbertse, Shobha Sharma, Vladimir Soussov, John P Sullivan, Lu Sun, Seán Turner, and Ilene Karsch-Mizrachi. 2020.  NCBI Taxonomy: a comprehensive update on curation, resources and tools.  _Database_ , 2020.  Baaa062.
  * Schubert (2006) Lenhart Schubert. 2006.  Turing’s dream and the knowledge challenge.  In _Proceedings of the national conference on artificial intelligence_ , volume 21, page 1534. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999.
  * Shah et al. (2003) Parantu K Shah, Carolina Perez-Iratxeta, Peer Bork, and Miguel A Andrade. 2003.  Information extraction from full text scientific articles: where are the keywords?  _BMC bioinformatics_ , 4(1):20.
  * Soares et al. (2014) Filipa AC Soares, Amit Chandra, Robert J Thomas, Roger A Pedersen, Ludovic Vallier, and David J Williams. 2014.  Investigating the feasibility of scale up and automation of human induced pluripotent stem cells cultured in aggregates in feeder free conditions.  _Journal of biotechnology_ , 173:53–58.
  * Tanabe et al. (2005) Lorraine Tanabe, Natalie Xie, Lynne H Thom, Wayne Matten, and W John Wilbur. 2005\.  Genetag: a tagged corpus for gene/protein named entity recognition.  _BMC bioinformatics_ , 6(1):1–7.
  * Unger et al. (2014) Christina Unger, Corina Forascu, Vanessa Lopez, Axel-Cyrille Ngonga Ngomo, Elena Cabrio, Philipp Cimiano, and Sebastian Walter. 2014.  Question answering over linked data (qald-4).
  * Wang et al. (2020) Kuansan Wang, Zhihong Shen, Chiyuan Huang, Chieh-Han Wu, Yuxiao Dong, and Anshul Kanakia. 2020.  Microsoft academic graph: When experts are not enough.  _Quantitative Science Studies_ , 1(1):396–413.
  * Wilkinson et al. (2016) Mark D Wilkinson, Michel Dumontier, IJsbrand Jan Aalbersberg, Gabrielle Appleton, Myles Axton, Arie Baak, Niklas Blomberg, Jan-Willem Boiten, Luiz Bonino da Silva Santos, Philip E Bourne, et al. 2016.  The fair guiding principles for scientific data management and stewardship.  _Scientific data_ , 3(1):1–9.

##  Appendix A Appendix

Supplementary to the Knowledge Graph in the Material Science domain presented
in subsection 5.2, in Figures 13 to 20, we demonstrate nine examples of
knowledge graphs for the respective nine remaining STEM domains we consider.
In all graphs, nodes are color-coded by their concept type: orange corresponds
to process, purple for method, green corresponds to material, and blue for
data.

Figure 13: Structured KG representation of a Biology domain publication
Abstract Soares et al. (2014) as process, method, material, and data entities.
Figure 14: Structured KG representation of a Mathematics domain publication
Abstract Haution (2012) as process, method, material, and data entities.
Figure 15: Structured KG representation of an Agriculture domain publication
Abstract Martin et al. (2012) as process, method, material, and data typed
entities. Figure 16: Structured KG representation of an Astronomy domain
publication Abstract Krupp et al. (2012) as process, method, material, and
data typed entities. Figure 17: Structured KG representation of a Chemistry
domain publication Abstract Cotton et al. (2014) as process, method, material,
and data typed entities. Figure 18: Structured KG representation of a Computer
Science domain publication Abstract Dolev et al. (2014) as process, method,
material, and data typed entities. Figure 19: Structured KG representation of
an Engineering domain publication Abstract Kakimpa et al. (2012) as process,
method, material, and data typed entities. Figure 20: Structured KG
representation of a Medical domain publication Abstract Kivimäki et al. (2012)
as process, method, material, and data typed entities.