Knowledge base

Hadamard gate

Definition

The Hadamard gate is a single-qubit operation that maps the basis state ∣0⟩|0\rangle to ∣0⟩+∣1⟩2\frac{\vert 0 \rangle + \vert 1 \rangle}{\sqrt{2}} and ∣1⟩|1\rangle to ∣0⟩−∣1⟩2\frac{|0\rangle - |1\rangle}{\sqrt{2}}, thus creating an equal superposition of the two basis states.

H=12(111−1) H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

Examples:

Decompositions

The Hadamard gate can also be expressed as a 90º rotation around the Y-axis, followed by a 180º rotation around the X-axis. So, H=XY1/2H = X Y^{1/2}.

Useful XY-decompositions (also visualized below) are given by: H=XY1/2H = X Y^{1/2} H=Y−1/2XH = Y^{-1/2} X

Useful YZ-decompositions are: H=ZY−1/2H = Z Y^{-1/2} H=Y1/2ZH = Y^{1/2} Z