  Quantum Wielandt's inequality gives an optimal upper bound on the minimal
length $k$ such that length-$k$ products of elements in a generating system
span $M_n(\mathbb{C})$. It is conjectured that $k$ should be of order
$\mathcal{O}(n^2)$ in general. In this paper, we give an overview of how the
question has been studied in the literature so far and its relation to a
classical question in linear algebra, namely the length of the algebra
$M_n(\mathbb{C})$. We provide a generic version of quantum Wielandt's
inequality, which gives the optimal length with probability one. More
specifically, we prove based on [KS16] that $k$ generically is of order
$\Theta(\log n)$, as opposed to the general case, in which the best bound to
date is $\mathcal O(n^2 \log n)$. Our result implies a new bound on the
primitivity index of a random quantum channel. Furthermore, we shed new light
on a long-standing open problem for Projected Entangled Pair State, by
concluding that almost any translation-invariant PEPS (in particular, Matrix
Product State) with periodic boundary conditions on a grid with side length of
order $\Omega( \log n )$ is the unique ground state of a local Hamiltonian. We
observe similar characteristics for matrix Lie algebras and provide numerical
results for random Lie-generating systems.
