  Time evolution of quantum systems is of interest in physics, in chemistry,
and, more recently, in computer science. Quantum computers are suggested as one
route to propagating quantum systems far more efficiently than ordinary
numerical methods. In the past few years, researchers have actively been
improving quantum simulation algorithms, especially those in second
quantization. This work continues to advance the state-of-the-art by unifying
several diverging approaches under a common framework. In particular, it
highlights the similarities and differences of the first and second quantized
algorithms which are usually presented in a distinct fashion. By combining
aspects of the two approaches, this work moves towards an online second
quantized algorithm operating within a single-Fock space. This paper also
unifies a host of approaches to algorithmic quantum measurement by removing
superficial differences. The aim of the effort is not only to give a high-level
understanding of quantum simulation, but to move towards experimentally
realizable algorithms with practical applications in chemistry and beyond.
