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arXiv:2605.00259

Entanglement Distance of Two- and Multi-Qubit Variational States and Its Quantification with Quantum Computing

Kh. P. Gnatenko, R. O. Hredil, V. Y. Pinchuk, M. Z. Seniak, Y. T. Shevchuk

quant-phSubmitted Apr 30, 2026• Announced May 4, 2026• 0 citations• Revised May 28, 2026 (v3)
v1 · Apr 30, 2026v2 · May 25, 2026v3 · May 28, 2026
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Abstract

We study the entanglement distance of variational quantum states for two-qubit and multi-qubit systems. These states are constructed using variational quantum circuits with $R_Y$ rotations and entangling $CZ$ gates. For the two-qubit case, we analytically derive recurrence relations for expectation values of Pauli observables. This approach allows us to calculate quantum correlators and evaluate the entanglement distance as a function of the circuit parameters and depth. The analysis was extended to a closed one-dimensional chain of $N$ qubits. An explicit analytical expression for the entanglement is derived for the case of two layers. We conclude that the entanglement of a qubit with the rest of the system depends on the parameters of the gates acting on first- and second-nearest neighbors in the chain topology of the entangling layers. We also quantify the entanglement of the variational quantum states using quantum computing on the AerSimulator. The corresponding quantum protocols are constructed, and the dependence of the entanglement on the parameters of the variational quantum states is studied. The results of the quantum programming are in good agreement with the theoretical predictions.

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We study the entanglement distance of variational quantum states for two-qubit and multi-qubit systems. These states are constructed using variational quantum circuits with RYR_YRY​ rotations and entangling CZCZCZ gates. For the two-qubit case, we analytically derive recurrence relations for expectation values of Pauli observables. This approach allows us to calculate quantum correlators and evaluate the entanglement distance as a function of the circuit parameters and depth. The analysis was extended to a closed one-dimensional chain of NNN qubits. An explicit analytical expression for the entanglement is derived for the case of two layers. We conclude that the entanglement of a qubit with the rest of the system depends on the parameters of the gates acting on first- and second-nearest neighbors in the chain topology of the entangling layers. We also quantify the entanglement of the variational quantum states using quantum computing on the AerSimulator. The corresponding quantum protocols are constructed, and the dependence of the entanglement on the parameters of the variational quantum states is studied. The results of the quantum programming are in good agreement with the theoretical predictions.
We consider multi-qubit entangled quantum states known in the literature as one-layer variational quantum states. The states can also be considered as quantum graph states representing directed and weighted graphs. The corresponding variational quantum protocols includeRY,RZR Y, R ZRY,RZ, and entangling blocks withRXXR X XRXXgates. We examine the general case in which the entangling blocks have an arbitrary structure, namely,RXXR X XRXXgates are applied to arbitrary pairs of qubits. This case corresponds to an arbitrary structure of the associated weighted and directed graphs. Variational quantum states, as well as quantum graph states, have broad applications in various fields of quantum computing, including quantum machine learning, quantum error correction, and optimization algorithms, and others. One of the key resources in quantum computing is entanglement. We analytically calculate the entanglement distance for variational quantum states with entangling blocks of arbitrary structure. The dependence of entanglement on the parameters of the variational quantum state and on the characteristics of the corresponding weighted and directed graphs is investigated. We find that the entanglement of a qubit with other qubits in the state depends on the outdegree and indegree of the corresponding vertex in the graph, as well as on the weights of the incoming and outgoing arcs associated with that vertex. Quantum protocols for quantifying the entanglement distance of variational quantum states have been constructed. For particular cases of variational quantum states, the dependence of entanglement distance on the state parameters has been examined using quantum computation on the Aer simulator.
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.
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 entanglement distance of evolutionary quantum states of a two-spin system with the XXZ model has been studied. The analysis has been conducted both analytically and using quantum computing. An analytical dependence of the entanglement distance on the values of the model coupling constants and the parameters of the initial states has been obtained. The speed of evolution of a two-spin system has been investigated. The analysis has been performed analytically and using quantum computing. An explicit dependence of the speed of evolution on the coupling constants and on the parameters of the initial state has been obtained. The results of quantum computations are in good agreement with the theoretical predictions.
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.