  Simulating the unitary dynamics of a quantum system is a fundamental problem
of quantum mechanics, in which quantum computers are believed to have
significant advantage over their classical counterparts. One prominent such
instance is the simulation of electronic dynamics, which plays an essential
role in chemical reactions, non-equilibrium dynamics, and material design.
These systems are often time-dependent, which requires that the corresponding
simulation algorithm can be successfully concatenated with itself over
different time intervals to reproduce the overall coherent quantum dynamics of
the system. In this paper, we quantify such simulation algorithms by a property
called fully-coherent: the algorithm succeeds with arbitrarily high success
probability $1-\delta$, while only requiring a single copy of the initial
state. We subsequently develop fully-coherent simulation algorithms based on
quantum signal processing (QSP), including a novel algorithm that circumvents
the use of amplitude amplification while also achieving a query complexity
additive in time $t$, $\ln(1/\delta)$, and $\ln(1/\epsilon)$ for error
tolerance $\epsilon$: $\Theta\big( \|\mathcal{H}\| |t| + \ln(1/\epsilon) +
\ln(1/\delta)\big)$. Furthermore, we numerically analyze these algorithms by
applying them to the simulation of the spin dynamics of the Heisenberg model
and the correlated electronic dynamics of an H$_2$ molecule. Since any
electronic Hamiltonian can be mapped to a spin Hamiltonian, our algorithm can
efficiently simulate time-dependent ab initio electronic dynamics in the
circuit model of quantum computation. Accordingly, it is also our hope that the
present work serves a bridge between QSP-based quantum algorithms and chemical
dynamics, stimulating a cross-fertilization between these exciting fields.
