One of the central goals of condensed matter physics is to understand the emergence of collective phenomena in strongly correlated quantum systems. However, the exponential complexity of many-body quantum states poses fundamental challenges to both analytical and classical numerical methods. This thesis explores how classical computation can support and extend the capabilities of quantum simulation, with a focus on both analog and digital quantum simulators, guided by and aligned with recent experimental advancements. The first part of this thesis focuses on analog quantum simulation in solid-state moiré materials. In particular, we investigate WSe$_2$/WTe$_2$ heterobilayers as promising platforms for simulating extended Hubbard models on a triangular lattice. Using large-scale density matrix renormalization group (DMRG) simulations, we map out the quantum phase diagram as a function of kinetic energy and Coulomb interactions. Our results identify several exotic phases, including chiral spin liquids and generalized Wigner crystals, and provide theoretical guidance for experimental exploration of interaction-driven quantum phase transitions in these systems. The second part addresses digital quantum simulators and the challenge of quantum error correction. We propose a machine learning-based decoder for fault-tolerant quantum computation in the presence of logical circuits, motivated by the architecture of neutral atom-based quantum processors. By designing a modular neural network architecture and training it on realistic error models, we achieve competitive decoding accuracy and efficiency. Our work highlights the potential of classical machine learning to enhance the performance of quantum error correction, bridging algorithmic design with practical hardware considerations. Together, these studies demonstrate the essential role of classical computation in supporting quantum simulation, from characterizing novel phases in analog platforms to enabling fault tolerance in digital quantum devices. The interplay between theoretical modeling, numerical simulation, and experimental collaboration provides a path forward for probing and harnessing quantum many-body phenomena.