Learning finitely correlated states: stability of the spectral
  reconstruction
Marco Fanizza Niklas Galke Josep Lumbreras Cambyse Rouzé Andreas Winter
We show that marginals of subchains of length $t$ 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
a multiple of 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.