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arXiv:2401.14997 · Ukrainian Journal of Physics

Geometric Measure of Entanglement of Quantum Graph States Prepared with Controlled Phase Shift Operators

N. A. Susulovska

Published Mar 19, 2025• 0 citations
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Abstract

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.

DOI
10.15407/ujpe70.3.161

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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.
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].
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.
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.
The geometric measure of entanglement of mixed states is investigated based on its relation to spin correlations. We derive an analytical expression for the entanglement of rank-2 mixed states composed of arbitrary multiqubit pure states within a two-dimensional Hilbert subspace. The dependence of the entanglement of rank-2 mixed multiqubit states on the parameters characterizing arbitrary pure states in the mixture is studied both analytically and using quantum computing. For a particular class of mixed states, namely, equal-probability mixtures of an arbitrary multiqubit state and the Schrödinger cat state, we construct quantum protocols for detecting the geometric measure of entanglement based on its relation to spin correlations. We quantify the entanglement using both the AerSimulator and the ibm_marrakesh quantum computer. The effects of quantum-computing errors are also analyzed. The simulation results reproduce the analytical dependencies, while measurements on the quantum device demonstrate good agreement with theoretical predictions within estimated error bounds.
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.