• Fundamentals
  • Also: controlled-Z gate
  • Also: controlled-phase gate
  • Also: CPHASE

CZ Gate

A symmetric two-qubit gate that applies a phase flip (Z gate) to the target qubit if and only if both qubits are in the |1> state, native to several superconducting and neutral atom platforms.

The CZ (controlled-Z) gate is a two-qubit entangling gate that applies a phase of −1-1 to the ∣11⟩|11\rangle component of the state, leaving all other computational basis states unchanged. Unlike the CNOT gate, the CZ gate is symmetric: it treats both qubits identically, so there is no distinction between control and target.

Matrix representation

In the computational basis {∣00⟩,∣01⟩,∣10⟩,∣11⟩}\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}:

CZ=(100001000010000−1)\text{CZ} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}

Its action on basis states:

∣00⟩→∣00⟩,∣01⟩→∣01⟩,∣10⟩→∣10⟩,∣11⟩→−∣11⟩|00\rangle \to |00\rangle, \quad |01\rangle \to |01\rangle, \quad |10\rangle \to |10\rangle, \quad |11\rangle \to -|11\rangle

The CZ gate is diagonal in the computational basis, meaning it only modifies phases, never swaps populations. It is also Hermitian (CZ†=CZ\text{CZ}^\dagger = \text{CZ}) and its own inverse (CZ2=I\text{CZ}^2 = I).

Relationship to CNOT

The CZ and CNOT gates are equivalent up to single-qubit Hadamard gates:

CNOT01=(I⊗H)⋅CZ⋅(I⊗H)\text{CNOT}_{01} = (I \otimes H) \cdot \text{CZ} \cdot (I \otimes H)

This means any circuit using CNOTs can be rewritten using CZ gates (and vice versa) with at most two additional Hadamard gates per CNOT. On hardware where CZ is native (Google, Rigetti, many neutral atom platforms), this conversion is how CNOT gates are implemented.

The symmetry of CZ is apparent from its matrix: swapping the two qubits leaves the gate unchanged. In circuit diagrams, both qubits are typically shown with a filled dot connected by a vertical line, reflecting this symmetry.

Physical implementations

Superconducting qubits: CZ gates are implemented by tuning two transmon qubits so that the ∣11⟩|11\rangle state transiently interacts with the non-computational ∣02⟩|02\rangle or ∣20⟩|20\rangle state. This conditional phase accumulation produces the CZ gate in 20 to 60 nanoseconds. Google’s Sycamore processor uses this approach.

Neutral atoms: The Rydberg blockade mechanism naturally implements a CZ gate. When two atoms are close enough, exciting both to the Rydberg state is energetically forbidden. A pulse sequence that conditionally excites and de-excites atoms produces a phase of −1-1 on ∣11⟩|11\rangle and +1+1 on all other states. QuEra and Atom Computing use this approach.

Trapped ions: CZ can be constructed from the native Molmer-Sorensen (XX) gate plus single-qubit rotations.

Role in quantum algorithms

The CZ gate appears directly in many quantum algorithms. In Grover’s algorithm, the oracle for searching ∣11⟩|11\rangle is exactly a CZ gate. The diffusion operator also uses CZ as a building block. In the surface code, syndrome extraction circuits commonly use CZ gates on platforms where they are native.

Why it matters for learners

The CZ gate is one of the most common native two-qubit gates. Its symmetry makes it conceptually cleaner than CNOT for many purposes, and its diagonal structure (purely phase-based, no bit flips) connects directly to the phase-based reasoning that underlies much of quantum algorithm design. Understanding the CZ-CNOT equivalence also helps when translating algorithms between different hardware platforms.

See also