Source-verifiedLevel 2
Rz Gate
The Rz gate performs a rotation by angle theta about the z-axis of the Bloch sphere, applying a relative phase between |0⟩ and |1⟩.
What it means
The Rz(theta) gate rotates the qubit state by angle theta about the z-axis.Its matrix is [[e^(-i*theta/2), 0], [0, e^(i*theta/2)]].Being diagonal in the computational basis, Rz does not change measurement probabilities -- it only modifies the relative phase.Rz(pi) = -iZ, Rz(pi/2) = e^(-i*pi/4)*S.The Rz gate is natively implementable on most quantum hardware platforms (superconducting, trapped ion) as it corresponds to a frequency detuning.Any single-qubit gate can be decomposed as Rz(alpha)*Ry(beta)*Rz(gamma) up to global phase (ZYZ decomposition).Everyday analogy
The Rz gate is like adjusting the phase of an oscillating wave without changing its amplitude -- the wave height stays the same, but the peaks shift in time.
Think of it as rotating a compass needle around the vertical axis -- the tilt angle stays the same, but the pointing direction changes.
Common misconceptions
- The Rz gate does NOT change measurement probabilities in the computational basis -- it only affects the relative phase, which is invisible without interference.
- Rz is NOT the same as the Z gate -- Z = Rz(pi) up to a global phase.
Key takeaways
- Rz(theta) is diagonal and applies relative phase without changing measurement probabilities.
- Most hardware platforms can implement Rz natively and with high fidelity.
- The ZYZ decomposition Rz*Ry*Rz can represent any single-qubit unitary.
Check your understanding
Does the Rz gate change the probability of measuring |0⟩ or |1⟩?
- A.Yes, it swaps probabilities
- B.Yes, it increases P(|1⟩)
- C.No, it only changes the relative phase
- D.No, it does nothing
Show the answer
Answer: C. No, it only changes the relative phase
Why: The Rz gate is diagonal in the computational basis, so it only modifies the relative phase between |0⟩ and |1⟩ without changing measurement probabilities.
Builds on
Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457
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