HomeQuantum PhysicsLessonsControlled-Z Gate

Controlled-Z Gate

Entanglement2-qubitctrl-Z

Applies a phase flip to the |11⟩ state.

Intuition

CZ applies a Z gate to the target qubit only when the control qubit is |1⟩. But here is the remarkable thing: unlike CNOT, the CZ gate is completely symmetric between the two qubits. It does not matter which one you call the control and which the target — the effect is the same. CZ flips the phase of the |11⟩ component and leaves everything else unchanged. This makes CZ a natural gate for hardware where qubits interact symmetrically.

Matrix representation

diag (1, 1, 1, - 1)

Action on states

∣11 ⟩ \to - ∣11 ⟩, other basis states unchanged

Circuit

Precise explanation

CZ = diag(1, 1, 1, −1). It applies a −1 phase to the |11⟩ term and leaves |00⟩, |01⟩, |10⟩ unchanged. The symmetry CZ = CZ^T means there is no physical distinction between control and target. CZ is related to CNOT by: CZ = (I⊗H)·CNOT·(I⊗H). It is a native gate on superconducting transmon architectures.

Observable effect

Joint two-qubit phase structure changes.

Hidden effect

Creates entanglement when either qubit is in superposition.

Bloch sphere

Conditional phase flip. Symmetric between qubits — no control/target distinction geometrically.

Worked example

CZ as a symmetric entangling gate

1Start with |+⟩⊗|+⟩ = (|00⟩ + |01⟩ + |10⟩ + |11⟩)/2.
2Apply CZ: only the |11⟩ term picks up a minus sign.
3Result: (|00⟩ + |01⟩ + |10⟩ − |11⟩)/2.
4This state is entangled — it cannot be written as a product of two single-qubit states.
5Notice: swapping which qubit is “control” gives the same result, confirming the symmetry.

Common use cases

Native entangling gate on superconducting transmon processors
Graph state preparation (one CZ per edge)
Measurement-based quantum computation
Alternative to CNOT when symmetric interaction is preferred

Relation to other gates

CZ = (I⊗H)·CNOT·(I⊗H) — CNOT with Hadamards on the target
CZ is symmetric: swapping qubits gives the same gate
CP(π) = CZ — CZ is a special case of controlled-phase
CZ is its own inverse: CZ² = I

Common misconception

The symmetry of CZ is not obvious from the circuit diagram, which usually draws one qubit as control. But mathematically and physically, both qubits play identical roles.

Why this gate matters

CZ is the native entangling gate on many superconducting quantum processors. Understanding CZ and its relationship to CNOT helps you write circuits that map efficiently onto real hardware.

Test your understanding

Why is CZ symmetric between the two qubits while CNOT is not?

Quick reference →↗ Nielsen and Chuang, Quantum Computation and Quantum Information

← Back to all lessons