  The Clifford group is the set of gates generated by the controlled not gates,
the Hadamard gate and the P={{1,0},{0,i}} gate. We will say that a n-qubit
state is a Clifford state if it can be prepared using Clifford gates. In this
paper we study the set of all 4-qubit Clifford states. We prove that there are
293760 states and their entanglement entropy must be either 0, 2/3, 1, 4/3 and
5/3. We also show that any pair of these states can be connected using local
gates and at most 3 CNOT gates. We achieve this by splitting the 293760 states
into 18 groups where each pair of states in a group can be connected with a
local Clifford gate. We then study how the different CNOT gates act on the 18
groups. We also study the Clifford states with real entries under the action of
the subgroup C_R of Clifford gates with real entries. This time we show that
every pair of Clifford states with real entries can be connected with at most 5
CNOT gates and local gates in C_R. The link https://youtu.be/42MI6ks2_eU leads
to a YouTube video that explains the most important results in this paper.
