Phase Gate (S Gate) | √Z Clifford Operator & Matrix
Quick Technical Answer:
The Phase gate (S gate) is the square root of the Pauli-Z gate (S² = Z). It introduces a relative phase shift of π/2 (90°) on state |1⟩: S|0⟩ = |0⟩ and S|1⟩ = i|1⟩, rotating the state vector by 90° around the Z-axis.
Formula / Unitary:
S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{pmatrix}
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Clifford Group Hierarchy
The S gate is a core generator of the single-qubit Clifford group alongside the Hadamard gate. According to the Gottesman-Knill theorem, circuits consisting solely of H, S, and CNOT gates can be simulated in polynomial time on classical computers.
Frequently Asked Questions
What is the S-dagger (Sdg) gate?
Sdg is the Hermitian conjugate of the S gate, implementing a -π/2 (-90°) phase shift on |1⟩: Sdg|1⟩ = -i|1⟩. S · Sdg = I.