  We present a systematic study of quantum system compression for the evolution
of generic many-body problems. The necessary numerical simulations of such
systems are seriously hindered by the exponential growth of the Hilbert space
dimension with the number of particles. For a \emph{constant} Hamiltonian
system of Hilbert space dimension $n$ whose frequencies range from $f_{\min}$
to $f_{\max}$, we show via a proper orthogonal decomposition, that for a
run-time $T$, the dominant dynamics are compressed in the neighborhood of a
subspace whose dimension is the smallest integer larger than the time-bandwidth
product $\delf=(f_{\max}-f_{\min})T$. We also show how the distribution of
initial states can further compress the system dimension. Under the stated
conditions, the time-bandwidth estimate reveals the \emph{existence} of an
effective compressed model whose dimension is derived solely from system
properties and not dependent on the particular implementation of a variational
simulator, such as a machine learning system, or quantum device. However,
finding an efficient solution procedure \emph{is} dependent on the simulator
implementation{\color{black}, which is not discussed in this paper}. In
addition, we show that the compression rendered by the proper orthogonal
decomposition encoding method can be further strengthened via a multi-layer
autoencoder. Finally, we present numerical illustrations to affirm the
compression behavior in time-varying Hamiltonian dynamics in the presence of
external fields. We also discuss the potential implications of the findings for
machine learning tools to efficiently solve the many-body or other high
dimensional Schr{\"o}dinger equations.
