Finding generic quantum circuits for k-local Hamiltonians

Question

Let $P_n$ denote the Pauli group on n qubits (think of n as a large number).

Let $G=<g_1,...,g_n> \subset P_n$ be some abelian subgroup such that each $g_i$ acts on at most $k\ll n$ qubits. Assume that the $g_i$ are independent and $−I∉G$.

Does there always exist some unitary $U$ such that $U^ \dagger GU=\langle X_1,...,X_n\rangle$?

If so, what does $U$ look like? Can we ensure that $U$ is "simple" enough? More precisely, can we find such a $U$ when we restrict ourselves to constant depth unitary quantum circuits?

Comments

an interesting link in the context of the Clifford Algebra ( paragraph # 6 ) also for the links it contains : Matchgates and classical simulation of quantum circuits