  We prove a local in time well-posedness result for quasi-linear Hamiltonian
Schr\"odinger equations on $\mathbb{T}^d$ for any $d\geq 1$. For any initial
condition in the Sobolev space $H^s$, with $s$ large, we prove the existence
and unicity of classical solutions of the Cauchy problem associated to the
equation. The lifespan of such a solution depends only on the size of the
initial datum. Moreover we prove the continuity of the solution map.
