In this paper, we provide a generic version of quantum Wielandt's inequality,
which gives an optimal upper bound on the minimal length k such that
length-k products of elements in a generating system span Mn(C)
with probability one. We show that k generically is of order Θ(logn), as opposed to the general case, in which the best bound to the date is
O(n2logn). 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 Ω(logn) 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.Comment: 13 pages, 6 figure