  The quantum dense output problem is the process of evaluating
time-accumulated observables from time-dependent quantum dynamics using quantum
computers. This problem arises frequently in applications such as quantum
control and spectroscopic computation. We present a range of algorithms
designed to operate on both early and fully fault-tolerant quantum platforms.
These methodologies draw upon techniques like amplitude estimation, Hamiltonian
simulation, quantum linear Ordinary Differential Equation (ODE) solvers, and
quantum Carleman linearization. We provide a comprehensive complexity analysis
with respect to the evolution time $T$ and error tolerance $\epsilon$. Our
results demonstrate that the linearization approach can nearly achieve optimal
complexity $\mathcal{O}(T/\epsilon)$ for a certain type of low-rank dense
outputs. Moreover, we provide a linearization of the dense output problem that
yields an exact and finite-dimensional closure which encompasses the original
states. This formulation is related to the Koopman Invariant Subspace theory
and may be of independent interest in nonlinear control and scientific machine
learning.
