Extended Lipkin model: Proposal for implementation in a quantum platform and machine learning analysis of its phase diagram

Background: In recent years, the implementation of nuclear physics models in quantum computers has emerged as a promising and novel area of research. Simultaneously, the study of quantum shape phase transitions in nuclear models has gained significant attention. Specifically, the phase diagram of the interacting boson approximation (IBA) has been extensively explored, particularly in connection with large-particle-number-limit considerations. Interestingly, the extended Lipkin model (ELM) serves as a valuable alternative for mimicking the IBA phase diagram and holds the advantage of being more straightforward to implement within a quantum computing platform.Purpose: We explore the ELM and provide: (i) calculations of the ground-state energy using a variational quantum eigensolver; (ii) a comprehensive formulation for implementing the dynamics of the ELM within a quantum computing platform, enabling the experimental exploration of the IBA phase diagram for systems with a small number of particles; and (iii) a determination of the phase diagram of the model using different machine learning (ML) methods. We note that in the ELM, unlike the usual Lipkin model, both first- and second-order quantum shape phase transitions take place depending on the model parameters.Methods: Initially, we employ the adaptive derivative-assembled pseudo-Trotter ansatz variational quantum eigensolver (ADAPT-VQE) to calculate the ground-state energy of the ELM. Next, we introduce the essential formulation and procedures required to implement this model effectively in a quantum computing environment. Finally, we use ML techniques to identify the different phases and critical points of the ELM.Results: We successfully reproduce the ground-state energy of the ELM across the complete phase space of the model using the ADAPT-VQE algorithm. We provide the necessary framework for implementing the ELM in a quantum computing platform, ensuring that the model can be executed with controlled errors. Finally, we obtain meaningful ML predictions that allow us to determine the phase diagram of the model.Conclusions: Our findings offer compelling evidence that the implementation of a nuclear model like the ELM in a quantum computing environment is not only feasible but can also be achieved with manageable error rates. This realization opens the door to detailed experimental investigations of the phase diagram of the ELM (and indirectly of the IBA) in a quantum computer, further advancing our understanding of quantum shape phase transitions and nuclear structure.