Computing Library › Quantum Logic Gates
Quantum Logic Gates
S Gate (Phase, √Z)
The S gate is a quarter-turn phase gate — the square root of Z — adding a 90° phase to |1⟩.
Matrix
Unitary matrix
100i
Circuit symbol
Action on basis states
| input | output |
|---|---|
| |0⟩ | |0⟩ |
| |1⟩ | i|1⟩ |
| |+⟩ | |+i⟩ |
How it works
S = √Z applies a π/2 phase. It is Clifford but not Hermitian (S† = S³). With H and CNOT it generates the whole Clifford group — the classically simulable backbone that error correction is built on.
In code (Qiskit)
python
qc = QuantumCircuit(1)
qc.s(0) # phase +pi/2
qc.sdg(0) # S-daggersingle-qubitCliffordphase