  Tensor product states have proved extremely powerful for simulating the
area-law entangled states of many-body systems, such as the ground states of
gapped Hamiltonians in one dimension. The applicability of such methods to the
\emph{dynamics} of many-body systems is less clear: the memory required grows
exponentially in time in most cases, quickly becoming unmanageable. New methods
reduce the memory required by selectively discarding/dissipating parts of the
many-body wavefunction which are expected to have little effect on the
hydrodynamic observables typically of interest: for example, some methods
discard fine-grained correlations associated with $n$-point functions, with $n$
exceeding some cutoff $\ell_*$. In this work, we present a theory for the sizes
of `backflow corrections', i.e., systematic errors due to discarding this
fine-grained information. In particular, we focus on their effect on transport
coefficients. Our results suggest that backflow corrections are exponentially
suppressed in the size of the cutoff $\ell_*$. Moreover, the backflow errors
themselves have a hydrodynamical expansion, which we elucidate. We test our
predictions against numerical simulations run on random unitary circuits and
ergodic spin-chains. These results lead to the conjecture that transport
coefficients in ergodic diffusive systems can be captured to a given precision
$\epsilon$ with an amount of memory scaling as
$\exp[\mathcal{O}(\log(\epsilon)^2)]$, significantly better than the naive
estimate of memory $\exp[\mathcal{O}(\mathrm{poly}(\epsilon^{-1}))]$ required
by more brute-force methods.

Strongly Correlated Electrons