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arXiv:2106.10688 · Europhysics Letters (EPL)

Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer

Kh. P. Gnatenko, N. A. Susulovska

Published Nov 1, 2021• 17 citations
arXivHTMLPDF

Abstract

Abstract Multi-qubit graph states generated by the action of controlled phase shift operators on a separable quantum state of a system, in which all the qubits are in arbitrary identical states, are examined. The geometric measure of entanglement of a qubit with other qubits is found for the graph states represented by arbitrary graphs. The entanglement depends on the degree of the vertex representing the qubit, the absolute values of the parameter of the phase shift gate and the parameter of state the gate is acting on. Also the geometric measure of entanglement of the graph states is quantified on the quantum computer . The results obtained on the quantum device are in good agreement with analytical ones.

DOI
10.1209/0295-5075/ac419b

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Abstract Multi-qubit graph states generated by the action of controlled phase shift operators on a separable quantum state of a system, in which all the qubits are in arbitrary identical states, are examined. The geometric measure of entanglement of a qubit with other qubits is found for the graph states represented by arbitrary graphs. The entanglement depends on the degree of the vertex representing the qubit, the absolute values of the parameter of the phase shift gate and the parameter of state the gate is acting on. Also the geometric measure of entanglement of the graph states is quantified on the quantum computer . The results obtained on the quantum device are in good agreement with analytical ones.
Quantum entanglement gives rise to a range of non-classical effects, which are extensively exploited in quantum computing and quantum communication. Therefore, detection and quantification of entanglement as well as preparation of highly entangled quantum states remain the fundamental objectives in these fields. Much attention has been devoted to the studies of graph states, which play a role of a central resource in quantum error correction, quantum cryptography and practical quantum metrology in the presence of noise.We examine multi-qubit graph states generated by the action of controlled phase shift operators on a separable quantum state of a system, in which all the qubits are in arbitrary identical states. Analytical expression is obtained for the geometric measure of entanglement of a qubit with other qubits in graph states represented by arbitrary graphs. We conclude that this quantity depends on the degree of the vertex corresponding to the qubit, the absolute values of the parameter of the phase shift gate and the parameter of the initial state the gate is acting on. Moreover, the geometric measure of entanglement of certain types of graph states is quantified on the IBM’s quantum computer ibmq_athens based on the measurements of the mean spin. Namely, we consider states associated with the native connectivity of ibmq_athens, the claw and the complete graphs. Appropriate protocols are proposed to prepare these states on the quantum computer. The results of quantum computations verify our theoretical findings [1].
We will consider graph states generated by the action of controlled phase shift operators on a separable state of a multiqubit system. The case where all the qubits are initially prepared in arbitrary states is investigated. We will obtain the geometric measure of entanglement of a qubit with the remaining system in graph states represented by arbitrary weighted graphs and will establish its relationship with state parameters. For two-qubit graph states, the geometric measure of entanglement is also quantified on IBM’s simulator Qiskit Aer and quantum processor ibmq lima based on auxiliary mean spin measurements. The results of quantum computations verify our analytic predictions.
The geometric measure of entanglement of variational quantum states is studied on the basis of its relation with the mean value of spin. We examine n-qubit quantum states prepared by a variational circuit with a layer formed by the rotational gates and two-qubit controlled phase gates. The variational circuit is a generalization of that used for preparing quantum Generative Adversarial Network states. The entanglement of a qubit with other qubits in the variational quantum states is determined by the angles of rotational gates that act on the qubit and qubits entangled with it by controlled phase gates and also their parameters. In the case of one layer variational circuit, the states can be associated with graphs with vertices representing qubits and edges corresponding to two-qubit gates. The geometric measure of entanglement of a qubit with other qubits in the quantum graph state depends on the properties of the vertex that represents it in the graph, namely it depends on the vertex degree. The dependence of the geometric measure of entanglement of variational quantum states on their parameters is quantified on IBM's quantum computer.
We study multi-qubit variational quantum states that can be considered as vertex- and edge-weighted graph. These states are constructed as single-layer variational circuits with RXRXRX rotations and RZZRZZRZZ entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are related to the edge-weight structure around the corresponding vertices in the graph (i.e., edge weights incident to the vertices and vertex weights associated with their closed neighborhoods). In the special case of quantum states representing unweighted graphs, these quantities are related to the degrees of the corresponding vertices in the graph. As an example, we analyze the state associated with the star graph K1,4K_{1,4}K1,4​ using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of properties of classical graphs via quantum computing.
We propose a method for constructing multi-qubit entangled quantum states that represent weighted tripartite graphs, and develop approaches for investigating their structural properties using quantum computing. In the general case of multi-qubit states corresponding to arbitrary tripartite graph structures, we derive an expression for the entanglement distance. We establish a connection between entanglement and properties of the corresponding tripartite graphs. Namely, we show that the entanglement of a qubit with the rest of the system in a quantum tripartite graph state depends on the weights of the arcs in the neighborhood of the corresponding vertex, as well as on its degree with respect to the vertex sets of the tripartite graph. As an illustrative example, we consider a tripartite graph forming a triangle and evaluate the entanglement distance with quantum computing. We also compute quantum correlators for the general case of tripartite quantum graph states and relate these quantities to structural features of the underlying graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. Obtained relationships between the quantum properties of multi-qubit states and the structural features of tripartite graphs opens up the possibility of investigating classical systems, such as tripartite graphs, using quantum computing. It is worth emphasizing that tripartite graphs have applications in practical problems, including resource allocation, scheduling, and database and hypergraph modeling.