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Pauli-Z Gate
The Pauli-Z gate is the phase-flip gate that leaves |0⟩ unchanged and maps |1⟩ to -|1⟩, corresponding to a 180-degree rotation about the z-axis of the Bloch sphere.
What it means
The Pauli-Z gate, also known as the phase-flip gate, applies a phase of -1 to the |1⟩ component while leaving |0⟩ unchanged.Its matrix representation is sigma_z = [[1,0],[0,-1]].On the Bloch sphere, it corresponds to a pi rotation about the z-axis, which flips the sign of the equatorial components without swapping the poles.The Z gate is diagonal in the computational basis, meaning |0⟩ and |1⟩ are its eigenstates with eigenvalues +1 and -1 respectively.It is Hermitian and self-inverse (Z squared = I).The Z gate is fundamental in quantum computing: it defines the computational basis, appears in phase kickback, and is essential for quantum error correction (correcting phase-flip errors).Everyday analogy
The Pauli-Z gate is like adding a minus sign to the 'tails' side of a coin without flipping it -- the coin stays the same way up, but its 'tails' identity is inverted.
Think of it as reversing the direction of a spinning top without changing whether it points up or down -- only the phase of rotation changes.
Common misconceptions
- The Pauli-Z gate does NOT flip |0⟩ and |1⟩ -- it only changes the phase of |1⟩ to -|1⟩ while leaving |0⟩ completely unchanged.
- The phase flip is NOT observable on a single Z-basis measurement -- you need interference (e.g., by applying H gates before and after) to detect it.
Key takeaways
- The Pauli-Z gate applies a phase flip: Z|0⟩ = |0⟩, Z|1⟩ = -|1⟩.
- On the Bloch sphere, it is a 180-degree rotation about the z-axis.
- The computational basis states |0⟩ and |1⟩ are eigenstates of Z with eigenvalues +1 and -1.
Check your understanding
What is Z|+⟩ where |+⟩ = (|0⟩+|1⟩)/sqrt(2)?
- A.|+⟩
- B.|-⟩
- C.|0⟩
- D.|1⟩
Show the answer
Answer: B. |-⟩
Why: Z|+⟩ = Z(|0⟩+|1⟩)/sqrt(2) = (|0⟩-|1⟩)/sqrt(2) = |-⟩. The Z gate converts |+⟩ to |-⟩ by flipping the relative phase.
Builds on
Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457
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