Let $P_n$ denote the Pauli group on $n$ qubits (think of n as a large number).
Let $G=\left<g_1,...,g_n\right> < P_n$ be some abelian subgroup such that each $g_i$ acts on at most $k\ll n$ qubits. Assume that $g_i$ are independent and $-I\notin G$.
Does there always exist some unitary $U$ such that $U^\dagger G U=\left<X_1,...,X_n\right>$?
If so, what does $U$ look like? Can we ensure $U$ is "simple" enough? More precisely, can we find such a $U$ when we restrict ourselves to constant depth unitary quantum circuits?