In less than a century, electronic computers have become essential to everyday life and have transformed scientific research. Their use in solving scientific problems has integrated theoretical and experimental approaches, often blurring the lines between the two. Computers are especially crucial for tackling problems that are too complex to solve analytically or too difficult to explore experimentally.

Among these challenges, the description of the properties of matter stands out as particularly difficult. Although the law governing the behavior of atoms, the Schroedinger equation,has been known for nearly a century, solutions are limited to a few specific cases. This limitation arises especially from the computational resources it demands. As a result, advanced approximate algorithms have been developed over the past decades. In 1982, Feynman argued that addressing this challenge would require not just a change in algorithm, but a shift in paradigm. His proposal was to leverage the properties of quantum mechanics to replicate the behavior of quantum systems. From this insight, the field of quantum computing was born.

In the past forty years, we have seen significant experimental and theoretical advancements, with current devices consisting of hundreds of quantum bits (qubits). While these devices hold great promise, they are still in their early stages, with limited capabilities in terms of system size and the number of operations they can perform. To work within these limitations, recent research has focused on hybrid quantum-classical algorithms. In this approach, quantum devices handle specific subroutines, while a classical computer oversees the overall algorithm.

This thesis contributes to research in hybrid quantum-classical algorithms by developing novel methods for simulating quantum systems. In the first part, which focuses on studying equilibrium properties, we introduce two new methods. The first method belongs to the class of sample-based algorithms and extends them to the study of the low-lying energy spectrum of fermionic systems, achieving greater efficiency and accuracy than previous approaches. Its performance is demonstrated on molecular systems of up to 58 qubits, marking the largest excited-state calculation performed with a quantum device to date. Next, we present a second method that integrates quantum and classical variational models to describe systems with naturally partitioned, weakly interacting clusters. This approach integrates quantum circuits with machine learning models in the study of spin and molecular systems.

The second part of the thesis focuses on the study of quantum dynamics. We introduce a new technique which overcomes several limitations of previous variational approaches, such as the quadratic scaling in parameter count and numerical instability arising from the inversion of ill-conditioned matrices. The method has already been used to simulate the time evolution of quantum systems beyond the capabilities of standard algorithms, which are limited by qubit coherence times. Additionally, we present a method that maps the full dynamics of a physical quantum system onto the ground state of an extended system, approximating it with a variational quantum circuit. This method builds on the Feynman-Kitaev proposal to interpret time as a quantum degree of freedom. Finally, we leverage the history state to measure physical properties and detect dynamical phase transitions.