According to Wikipedia¹, a graph state is a special type of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. In particular, they are a convenient way of representing certain types of entangled states.
Graph states are useful in quantum error-correcting codes, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models.
Graph states can be defined in two equivalent ways: through the notion of quantum circuits and stabilizer formalism. The stabilizer formalism definition is based on defining an operator for each vertex of the graph that commutes with all other operators and has the graph state as its eigenstate.
Graph states are necessarily entangled states, unless the graph is trivial (i.e., has no edges). This can be seen from the fact that any local measurement on a qubit will affect the state of its neighbors, and thus create correlations between them. Graph states can also be used to generate other types of entangled states, such as cluster states or GHZ states, by applying local unitary transformations or measurements.
Source: Conversation with Bing, 2023/5/2 (1) Graph state - Wikipedia. https://en.wikipedia.org/wiki/Graph_state. (2) GraphStateVis: Interactive Visual Analysis of Qubit Graph States and .... https://arxiv.org/abs/2105.12752. (3) Graph states as ground states of two-body frustration-free Hamiltonians. https://iopscience.iop.org/article/10.1088/1367-2630/16/7/073013/pdf. (4) Measurement-based quantum computation using two-component BECs - arXiv. https://arxiv.org/pdf/2203.08993.
A three-qubit W-state is an entangled quantum state of the form:
$$|W\rangle = \frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)$$
One way to write this state in terms of graph states is to use the quantum circuit definition. We can start with a three-qubit graph state corresponding to a triangle graph, which is equivalent to applying a controlled-Z gate between every pair of qubits:
$$|G\rangle = CZ_{12}CZ_{13}CZ_{23}(|+\rangle^{\otimes 3})$$
Then, we can apply local unitary transformations to each qubit to obtain the W-state. For example, we can apply a Hadamard gate to qubit 1, a phase gate with angle $\pi/3$ to qubit 2, and a phase gate with angle $-\pi/3$ to qubit 3:
$$|W\rangle = H_1P_2(\pi/3)P_3(-\pi/3)|G\rangle$$
Alternatively, we can use the stabilizer formalism definition. We can start with a three-qubit graph state corresponding to a star graph, which has the following stabilizer operators:
$$S_1 = X_1Z_2Z_3$$ $$S_2 = Z_1X_2$$ $$S_3 = Z_1X_3$$
Then, we can apply local Clifford operations to each qubit to obtain the W-state. For example, we can apply a Hadamard gate followed by a phase gate with angle $\pi/4$ to qubit 1, and a phase gate with angle $\pi/4$ to qubits 2 and 3:
$$|W\rangle = HP(\pi/4)_1P(\pi/4)_2P(\pi/4)_3|G\rangle$$
These are not the only ways to write the W-state in terms of graph states, but they illustrate the general idea of using local operations to transform one type of entanglement into another.
Source: Conversation with Bing, 2023/4/28 (1) Multipartite Entanglement and Hypergraph states of three qubits. https://www.researchgate.net/publication/234131122_Multipartite_Entanglement_and_Hypergraph_states_of_three_qubits. (2) W state - Wikipedia. https://en.wikipedia.org/wiki/W_state. (3) W state example - IBM Quantum. https://quantum-computing.ibm.com/composer/docs/iqx/example-circuits/w-state. (4) W-state | Quantiki. https://quantiki.org/wiki/W-state.
\\\\section{W-state in graph}
A three-qubit W-state is an entangled quantum state of the form:
\\\\begin{equation}
|W\\\\rangle = \\\\frac{1}{\\\\sqrt{3}}(|100\\\\rangle + |010\\\\rangle + |001\\\\rangle)
\\\\end{equation}
One way to write this state in terms of graph states is to use the quantum circuit definition. We can start with a three-qubit graph state corresponding to a triangle graph, which is equivalent to applying a controlled-Z gate between every pair of qubits:
\\\\begin{equation}
|G\\\\rangle = CZ_{12}CZ_{13}CZ_{23}(|+\\\\rangle^{\\\\otimes 3})
\\\\end{equation}
Then, we can apply local unitary transformations to each qubit to obtain the W-state. For example, we can apply a Hadamard gate to qubit 1, a phase gate with angle $\\\\pi/3$ to qubit 2, and a phase gate with angle $-\\\\pi/3$ to qubit 3:
\\\\begin{equation}
|W\\\\rangle = H_1P_2(\\\\pi/3)P_3(-\\\\pi/3)|G\\\\rangle
\\\\end{equation}
Alternatively, we can use the stabilizer formalism definition. We can start with a three-qubit graph state corresponding to a star graph, which has the following stabilizer operators:
\\\\begin{equation}
S_1 = X_1Z_2Z_3
\\\\end{equation}
\\\\begin{equation}
S_2 = Z_1X_2
\\\\end{equation}
\\\\begin{equation}
S_3 = Z_1X_3
\\\\end{equation}
Then, we can apply local Clifford operations to each qubit to obtain the W-state. For example, we can apply a Hadamard gate followed by a phase gate with angle $\\\\pi/4$ to qubit 1, and a phase gate with angle $\\\\pi/4$ to qubits 2 and 3:
\\\\begin{equation}
|W\\\\rangle = HP(\\\\pi/4)_1P(\\\\pi/4)_2P(\\\\pi/4)_3|G\\\\rangle
\\\\end{equation}
These are not the only ways to write the W-state in terms of graph states, but they illustrate the general idea of using local operations to transform one type of entanglement into another.
Source: Conversation with Bing, 2023/4/28
(1) Multipartite Entanglement and Hypergraph states of three qubits. \\\\cite{hypergraph}
(2) W state - Wikipedia. \\\\cite{wiki_wstate}
(3) W state example - IBM Quantum. \\\\cite{ibm_wstate}
(4) W-state | Quantiki. \\\\cite{quantiki_wstate}