One of the most promising applications of quantum computers is to simulate quantum mechanical systems and deliver an advantage to classical computation by leveraging their inherent quantum behaviour. In this thesis I make simulations of recently developed quantum algorithms for ground state energy estimation under different quantum noise levels. I present a new approach to achieve a noise tolerant Hamiltonian simulation algorithm for ground state energy estimation which also surmounts stochastic limitations most of its counterparts face. This algorithm is based on an adaptive set of fuzzy bisection searches to estimate the ground state energy digit by digit that can get to any arbitrary target precision. It builds upon the Quantum Eigenvalue Transformation of Unitary Matrices (QETU) algorithm and it delivers good approximations in simulations with quantum depolarizing probability up to 1e-3, particularly for the Transverse-Field Ising Model (TFIM). Simulations were run with different system Hamiltonians, system sizes and time evolution encoding methods on IBM qiskit.