  We show that marginals of blocks of $t$ systems of any finitely correlated
translation invariant state on a chain can be learned, in trace distance, with
$O(t^2)$ copies -- with an explicit dependence on local dimension, memory
dimension and spectral properties of a certain map constructed from the state
-- and computational complexity polynomial in $t$. The algorithm requires only
the estimation of a marginal of a controlled size, in the worst case bounded by
the minimum bond dimension, from which it reconstructs a translation invariant
matrix product operator. In the analysis, a central role is played by the
theory of operator systems. A refined error bound can be proven for
$C^*$-finitely correlated states, which have an operational interpretation in
terms of sequential quantum channels applied to the memory system. We can also
obtain an analogous error bound for a class of matrix product density operators
reconstructible by local marginals. In this case, a linear number of marginals
must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The
learning algorithm also works for states that are only close to a finitely
correlated state, with the potential of providing competitive algorithms for
other interesting families of states.
