Source-verifiedLevel 2
Phase (S) Gate
The S gate (phase gate) applies a 90-degree phase rotation to |1⟩, mapping |1⟩ to i|1⟩ while leaving |0⟩ unchanged. It is the square root of the Z gate.
What it means
The S gate, also called the phase gate or the sqrt(Z) gate, applies a relative phase of i (pi/2 radians) to the |1⟩ state.Its matrix is [[1,0],[0,i]].On the Bloch sphere, it corresponds to a 90-degree rotation about the z-axis.Unlike the Pauli gates, S is NOT self-inverse: S-dagger (the conjugate transpose) is a separate gate that applies a -i phase.S-squared equals Z, establishing it as the square root of Z.The S gate is critical in the Clifford group and appears in many quantum error correction codes and stabilizer formalism.Everyday analogy
The S gate is like turning a clock hand 90 degrees -- a quarter turn that, when done twice, gives a half turn (Z gate).
Think of it as adding a quarter-wave plate in optics that shifts the polarization phase by 90 degrees.
Common misconceptions
- The S gate is NOT self-inverse -- applying S twice gives Z, not I. You need S-dagger to undo S.
- The S gate does NOT create superposition -- it only modifies the phase of |1⟩.
Key takeaways
- S|0⟩ = |0⟩, S|1⟩ = i|1⟩ -- a 90-degree phase rotation on the Bloch sphere z-axis.
- S-squared = Z, so S is the square root of Z.
- The S gate is a key element of the Clifford group, essential for stabilizer codes.
Check your understanding
What gate results from applying the S gate twice?
- A.Identity (I)
- B.Pauli-X
- C.Pauli-Z
- D.Hadamard
Show the answer
Answer: C. Pauli-Z
Why: S-squared = Z. The S gate is the square root of Z, so applying it twice gives the Pauli-Z gate.
Builds on
Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457
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