Four-qubit states generated by Clifford gates

The Clifford group is the set of gates generated by controlled not (CNOT) gates and the two local gates [Formula: see text] and [Formula: see text]. We will say that an [Formula: see text]-qubit state is a Clifford state if it can be prepared using Clifford gates, this is, [Formula: see text] is Clifford if [Formula: see text] where [Formula: see text] is a Clifford gate. In this paper, we study the set of all 4-qubit Clifford states. We prove that there are 293760 states, each of which has entanglement entropy equal to 0, [Formula: see text], 1, [Formula: see text], or [Formula: see text]. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We also study the Clifford states with real entries under the action of the subgroup [Formula: see text] 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 [Formula: see text]. To understand the action of the 12 different CNOT gates, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We label each orbit in such a way that it is easy to see the effect of the CNOT gates. Diagrams and tables explaining the action of the CNOT gates on all the orbits are presented in the paper. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.