The ZZ-gate reflects the state of a qubit on the ZZ-axis. It has a similar effect to the XX-gate that reflects the state on the XX-axis. A reflection on the XX-axis affects the resulting measurement probabilities because it changes the proximities to the ends of the Z-axis (∣0⟩|0\rangle and ∣1⟩|1\rangle). However, it leaves the phase untouched. On the other hand, a reflection on the ZZ-axis flips the phase but leaves the measurement probabilities unchanged.

The following equation denotes the transformation matrix of the ZZ-gate.

Z=[100−1]Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}

The ZZ-gate turns a qubit in-state ∣+⟩|+\rangle into state ∣−⟩|-\rangle. The states ∣+⟩|+\rangle and ∣−⟩|-\rangle reside on the XX-axis. Mathematically, the following equation describes this transformation.

HZ∣0⟩=12[111−1][100−1][10]=12[11−11][10]=12[1−1]=∣0⟩−∣1⟩2=∣−⟩HZ|0\rangle=\tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\tfrac{1}{\sqrt{2}}\begin{bmatrix} 1\\ -1 \end{bmatrix}=\frac{|0\rangle - |1\rangle}{\sqrt{2}}=|-\rangle

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