The Cliffords

 In quantum computing, the so-called Clifford gates are the hard workers, transforming

gates into other gates.

These are : the Hadamard, the phase S, and the CNOT. Only the Hadamard is hermitian.

The Hadamard as an outer product:

                                                             

The Hadamard snadwiching a Pauli gate. Here it transforms an X-gate into a z-gate;

 and a z-gate into an X-gate.

                                                                      

https://www.perplexity.ai/search/Quantum-computing-how-RNsm9Kd9QDuPZc2P5nRNsw