Calculation of entangling capability of quantum circuits with quantum programming | QSD2023

Quantum Science Days is an annual, international, and virtual scientific conference organized by QWorld (Association) to provide opportunities to the quantum community to present and discuss their research results at all levels (from short projects to thesis work to research publications), and to get to know each other. The third edition (QSD2023) included 7 invited speakers, 10 thematic talks on “Building an Open Quantum Ecosystem”, 31 contributed talks, an industrial demo session by Classiq, and a career talk on quantum. QSD2023 was sponsored by Unitary Fund & Classiq and supported by Latvian Quantum Initiative.

Entangling capability of multi-qubit parameterized quantum circuits and its calculation with quantum programming
Speaker: Kh. P. Gnatenko
Abstract: We study the geometric measure of entanglement of the multi-qubit quantum states generated by parameterized circuits on the basis of analytical calculations and with programming on quantum devices of IBM, and Riggetti [1,2]. Quantum states generated by the rotational gates (RY gate) and two-qubit controlled phase gates (CP gate) are examined [3]. The states can be associated with graphs with the vertices corresponding to qubits and edges representing the action of two-qubit gates. Also, we study the entanglement of quantum graph states prepared with parameterized quantum circuits with rotational gates (RY, RZ gates) and two-qubit RXX gates. We find analytically an expression for the geometric measure of entanglement of the states corresponding to the arbitrary graph structure [3,4]. We show that the measure of entanglement of a qubit in a graph state is related to the degree of vertex representing it in a graph and analyze the dependencies of the entanglement on the circuit's parameters. Also, the average geometric measure of entanglement of the graph states is calculated and analyzed.
In addition, we examine the geometric measure of entanglement of Generative Adversarial Network states [5] and their generalization (states prepared with a circuit with a layer formed by the rotational gates and two-qubit controlled phase gates) [6].
Quantum protocols for studies of the entanglement of the states on quantum devices are constructed. To detect the entanglement we use its relation with the mean spin obtained in [7]. Protocols are realized on IBM's and Riggetti's quantum computers. The results of quantum calculations are in agreement with the theoretical ones.
[3] Gnatenko, Kh. P. Susulovska, N. A.: Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer. EPL (Europhys. Lett.) 136, 40003 (2021).
[4] Gnatenko, Kh. P. Tkachuk, V. M.: Entanglement of graph states of spin system with Ising interaction and its quantifying on IBM’s quantum computer. Phys. Lett. A. 396, 127248 (2021).
[5] Zoufal, Ch. Lucchi, A. Woerner, S.: Quantum Generative Adversarial Networks for learning and loading random distribution. npj Quantum Information 5, 103 (2019).
[6] Gnatenko, Kh. P.: Evaluation of variational quantum states entanglement on a quantum computer by the mean value of spin, arXiv:2301.03885 (2023).
[7] Frydryszak, A. M. Samar, M. I. Tkachuk, V. M.: Quantifying geometric measure of entanglement by mean value of spin and spin correlations with application to physical systems. Eur. Phys. J. D 71, 233 (2017).

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