Studies of properties of bipartite graphs with quantum programming

TL;DR

This work studies multi-qubit quantum states represented by bipartite graphs $G(U,V,E)$ generated via $CNOT$ gates on a separable state and analyzes how entanglement and quantum correlators reflect graph structure. It derives analytic expressions for the entanglement distance $E^{ED}_u(\ket{\psi})$ for $u\in U$ and $E^{ED}_v(\ket{\psi})$ for $v\in V$, revealing explicit dependence on vertex degrees and initial-state parameters, with a symmetric reduction $E^{ED}_u(\ket{\psi})=\sin^2 \theta^{(U)}_u\left(1-(\cos \phi^{(V)} \sin \theta^{(V)})^{2n_u}\right)$. The paper also links quantum correlators, such as $\langle\prod_{u\in U}\prod_{v\in V}\sigma^x_u\sigma^x_v\rangle$ and $\langle\prod_{u\in U}\sigma^x_u\rangle$, to counts of odd and even vertex degrees in $U$ and $V$, and provides quantum protocols to estimate these graph properties on near-term devices. Validation on the AerSimulator for a star graph demonstrates agreement with theory, illustrating a pathway for quantum-assisted graph-property estimation with potential advantages as quantum hardware scales.

Abstract

Multi-qubit quantum states corresponding to bipartite graphs $G(U,V,E)$ are examined. These states are constructed by applying $CNOT$ gates to an arbitrary separable multi-qubit quantum state. The entanglement distance of the resulting states is derived analytically for an arbitrary bipartite graph structure. A relationship between entanglement and the vertex degree is established. Additionally, we identify how quantum correlators relate to the number of vertices with odd and even degrees in the sets $U$ and $V$. Based on these results, quantum protocols are proposed for quantifying the number of vertices with odd and even degrees in the sets $U$ and $V$. For a specific case where the bipartite graph is a star graph, we analytically calculate the dependence of entanglement distance on the state parameters. These results are also verified through quantum simulations on the AerSimulator, including noise models. Furthermore, we use quantum calculations to quantify the number of vertices with odd degrees in $U$ and $V$. The results agree with the theoretical predictions.

Figures (5)

• Figure 1: Quantum protocol for quantifying the entanglement distance of $q[l]$ with other qubits in quantum graph state (\ref{['state']}) calculating mean values $\bra{\psi} \sigma^x_l \ket{\psi}$, $\bra{\psi} \sigma^y_l \ket{\psi}$, $\bra{\psi} \sigma^z_l \ket{\psi}$
• Figure 2: Bipartite graph $G(U,V,E)$ with $U={0}$, $V={1,2,3}$ and $E={(0,1), (0,2), (0,3)}$ corresponding to quantum state (\ref{['state_star4']}).
• Figure 3: Entanglement distance of qubit $q[0]$ with other qubits in state (\ref{['state_star4']}) (a) for $\phi^{(V)}=0$ and different values of $\theta^{(U)}$, $\theta^{(V)}$; (b) for $\theta^{(V)}=\pi/2$ and different values of $\theta^{(U)}$, $\phi^{(V)}$; (c) for $\theta^{(U)}=\pi/2$ and different values of $\theta^{(V)}$, $\phi^{(V)}$. The results obtained using the AerSimulator which includes a readout error of the order $10^{-2}$, a Pauli-X error of $10^{-4}$, and a CNOT error of $10^{-2}$ are indicated by cross markers, while the continuous surface represents the corresponding analytical calculations.
• Figure 4: Entanglement distance of qubit $q[1]$ with other qubits in state (\ref{['state_star4']}) (a) for $\phi^{(V)}=0$ and different values of $\theta^{(U)}$, $\theta^{(V)}$; (b) for $\theta^{(V)}=\pi/2$ and different values of $\theta^{(U)}$, $\phi^{(V)}$; (c) for $\theta^{(U)}=\pi/2$ and different values of $\theta^{(V)}$, $\phi^{(V)}$. The results obtained using the AerSimulator which includes a readout error of the order $10^{-2}$, a Pauli-X error of $10^{-4}$, and a CNOT error of $10^{-2}$ are indicated by cross markers, while the continuous surface represents the corresponding analytical calculations.
• Figure 5: Quantum protocol for quantifying cardinality of $V_{odd}$, $U_{odd}$, $V_{even}$, $U_{even}$ in bipartite graph on the basis of studies of $\bra{\psi} \prod_{u\in U} \prod_{v\in V} \sigma^{\alpha}_u \sigma^{\alpha}_v \ket{\psi}$ in quantum graph state \ref{['state']}. For $\alpha=x$ operator $R_{\alpha}$ reads $R_{\alpha}=RY(-\pi/2)$, $R_{\alpha}=RX(\pi/2)$ for $\alpha=y$ and for $\alpha=z$ we have $R_{\alpha}=I$.