Source-verifiedLevel 2

Controlled-Z Gate

The Controlled-Z (CZ) gate applies a phase flip to the |11⟩ state. It is symmetric: either qubit can be considered the control.

What it means

The CZ gate applies a Z gate to the target qubit only when the control qubit is |1⟩, adding a phase of -1 to the |11⟩ component.Unlike CNOT, the CZ gate is symmetric between its two qubits.Its matrix is diag(1,1,1,-1).CZ is native to many hardware platforms including superconducting and neutral atom architectures, and is locally equivalent to CNOT up to single-qubit rotations.

Everyday analogy

CZ is like two people who only react when both raise their hands simultaneously.
Think of it as a conditional phase stamp: only the |11⟩ ticket gets stamped.

Common misconceptions

  • CZ does NOT flip any qubit — it only applies a phase.
  • CZ is symmetric, unlike CNOT which has a distinct control and target.

Key takeaways

  • CZ applies a -1 phase to |11⟩ only.
  • CZ is symmetric between both qubits.
  • Equivalent to CNOT with Hadamards on the target.

Check your understanding

What does the CZ gate do to |11⟩?

  1. A.Flips to |00⟩
  2. B.Applies -1 phase
  3. C.No change
  4. D.Creates entanglement
Show the answer

Answer: B. Applies -1 phase

Why: CZ applies a -1 phase factor to the |11⟩ state only.

Builds on

CNOT GatePauli-Z Gate

Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457

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