S Gate

The Phase gate (or S gate) is a single-qubit operation defined by:

 S=(100i)\ S = \begin{pmatrix} 1 & 0 \ 0 & i \end{pmatrix}

The S gate is also known as the phase gate or the Z90 gate, because it represents a 90 degree rotation around the z-axis.

The S gate is related to the T gate by the relationship  S=T2\ S=T^{2}.

The conjugate transpose of the S gate, the  S\ S^{\dagger} gate is defined by:
 S=(100i)\ S^\dagger = \begin{pmatrix} 1 & 0 \ 0 & -i \end{pmatrix}


          S q[0] # execute phase gate on qubit 0
Sdag q[1:2,5] # execute conjugate transpose of the S gate on qubits 1,2 and 5