Quantum parameter estimation for early Fault-tolerant quantum simulation

Recent years have seen quantum computers move from theory to practice, with the first prototypes of \emph{fault-tolerant} quantum computers capable of correcting their own errors. This marks the beginning of an exciting era of \emph{early fault tolerance}, with the first practical applications on the horizon. Among the most promising is quantum simulation, where a quantum computer is used to study another quantum system, such as molecules or materials. However, a central challenge is how to reliably extract accurate information from these early imperfect devices under realistic constraints, including noise, limited qubit numbers, circuit depth, and control. This thesis addresses this challenge by developing and analyzing algorithms for quantum phase estimation in these regimes, a key primitive for determining the energy spectra. It also considers the problem of learning the underlying dynamics of a quantum system, demonstrating the advantage of active control over passive observation. Together, these results contribute to ongoing efforts to overcome the limitations of early quantum hardware through improved algorithms and classical post-processing.