Variational M-Partite Geometric Entanglement Algorithm

Variational quantum algorithms have emerged as a powerful tool for harnessing the potential of near-term quantum devices to address complex challenges across quantum science and technology. Yet, the robust and scalable quantification of entanglement in many-body quantum systems remains a significant challenge, crucial for both advancing theoretical understanding and enabling practical applications. In this work, we propose a variational quantum algorithm to evaluate the $M$-partite geometric entanglement across arbitrary partitions of an $N$-qubit system into $M$ parties. By constructing tailored variational ansatz circuits for both single- and multi-qubit parties, we optimize the overlap between a target quantum state and an $M$-partite variational separable state. This method provides a flexible and scalable approach for characterizing arbitrary $M$-partite entanglement in complex quantum systems of a given dimension. The accuracy of the proposed method is assessed by reproducing known analytical results. We further demonstrate its capability to evaluate entanglement among $M$ parties for any given conventional or unconventional partitions of one- and two-dimensional spin systems, both near and at a quantum critical point. Our results establish the versatility of the variational approach in capturing different types of entanglement in various quantum systems, surpassing the capabilities of existing methods. Our approach offers a powerful methodology for advancing research in quantum information science, condensed matter physics, and quantum field theory. Additionally, we discuss its advantages, highlighting its adaptability to diverse system architectures in the context of near-term quantum devices.