| Sumario: | The Bohr-Mottelson model (BMM) and the Interacting Boson Model (IBM) are two fundamental approaches in nuclear physics that offer complementary perspectives on the structure and dynamics of nuclei. The Bohr Hamiltonian, formulated in terms of collective variables in five dimensions, is key to describing low-energy quadrupole states and nuclear shape transitions, including spherical, ?-unstable, and axially symmetric deformed configurations. This framework has gained renewed interest due to the increasing availability of experimental data and the discovery of symmetries at critical points that characterize phase transitions in nuclear systems. This thesis develops new solutions for the Bohr Hamiltonian applied to different classes of nuclei, using quasi-exact analytical methods and quantum formalisms. By solving the Schrödinger equation, energy spectra and wave functions are derived, enabling the precise reproduction of experimental observables such as excitation energies and reduced electric quadrupole transition probabilities B(E2). The IBM describes the structure and dynamics of medium and heavy nuclei in the lowest part of their spectrum, truncating the shell model space and treating pairs of nucleons as bosons. Although the IBM is formulated from the outset in terms of second quantization, there is a formalism that provides a geometric view of the model, allowing for comparisons with the shapes obtained in the Bohr model. IBM usually operates with a single configuration (a fixed number of bosons), but the formalism has been extended to include configuration mixing, which enhances its ability to describe nuclear shape coexistence and phase transitions. In this work, the IBM is applied to the Ru isotope chain: i) with a single configuration, and ii) with configuration mixing. The goal is to show the differences between these descriptions in a nuclear region of interest for analyzing the evolution and coexistence of nuclear shapes. Finally, quantum computing techniques are applied to study quantum phase transitions in finite systems. In this work, the Extended Lipkin Model (ELM) is used as a simplified alternative to the IBM. This model has a phase diagram equivalent to that of the IBM and allows exploration of shape evolution and the corresponding phase transitions. For the model’s implementation on quantum platforms, adaptive methods of pseudo-Trotter derived variational quantum optimizer assembly (ADAPT-VQE) were used to address large model spaces. Additionally, a proposal for implementation on a quantum computer is presented, and it is shown how machine learning techniques can be used to classify nuclear shapes, integrating advanced computational tools into nuclear structure studies. The results of this work significantly advance the theoretical understanding of nuclear shape phase transitions, establishing a bridge between theory and experiment. This work has resulted in several peer-reviewed publications and numerous presentations, contributing to the fields of low-energy nuclear physics and quantum technologies.
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