Quantum Simulation and Variational Quantum Methods to solve the Alternated Exchange Heisenberg Model.

In this Master's Thesis, we conduct an in-depth study of the Alternating-Exchange Heisenberg Model (AEHM), a one-dimensional system of spins $S = \frac{1}{2}$ that exhibits a interesting phenomenology associated with dimerization, the opening of a spectral gap, and the emergence of topological edge modes. The behavior of the model is analyzed from a dual perspective: through classical numerical simulations and hybrid quantum algorithms, with the aim of evaluating its spectral structure, correlation properties, and its potential to generate spin currents analogous to the bulk photovoltaic effect. First, a classical study is carried out using the QuSpin library, where the properties of the AEHM are explored for different chain lengths, boundary conditions, and coupling ratios. We characterize the transition between the uniform gapless regime and the strongly dimerized regime, as well as the appearance of edge states under certain conditions. Local correlations and magnetization profiles are also computed, allowing us to distinguish between trivial and nontrivial topological phases through the bulk-boundary correspondence. Second, a variational scheme based on the Variational Quantum Eigensolver (VQE) algorithm is implemented to approximate the ground state of the AEHM using ansatz inspired by valence-bond states. We analyze the convergence of the method, its dependence on circuit depth, and its ability to faithfully reproduce the results obtained through exact diagonalization. This study makes it possible to assess the feasibility of hybrid algorithms for investigating strongly correlated systems in the NISQ era. Finally, we discuss the relevance of the AEHM as a platform for exploring spin-transport phenomena, particularly the generation of optically induced spin currents analogous to the bulk photovoltaic effect. The results show that the combination of classical simulation and quantum algorithms provides an effective framework for investigating many-body systems and opens the door to future applications in quantum computing, spintronics, and topological materials.