Multiple Hadamard gate

We introduce one the most important gates for multiple qubit systems: the multiple Hadamard gate. Note that it may not be its standard name, but surely should.

Recall the properties of tensor product and the notation and consider the action of $\textbf{H}\otimes{\textbf{H}}$ on the state $\ket{0} \otimes \ket{0}$:

$$ (\textbf{H} \otimes \textbf{H})\ket{0} \otimes \ket{0} = (\textbf{H}\ket{0}) \otimes (\textbf{H}\ket{0}) = \left( \frac{1}{\sqrt{2}} (\ket{0} + \ket{1}) \right) \otimes \left( \frac{1}{\sqrt{2}} (\ket{0} + \ket{1}) \right) $$

$$ = \frac{1}{2} (\ket{0} + \ket{1}) \otimes (\ket{0} + \ket{1}) = \frac{1}{2} \left( \ket{00} + \ket{01} + \ket{10} + \ket{11} \right) = \frac{1}{2} \sum_{i=0}^{3} \ket{i}. $$

This can be generalized to :

$$ \textbf{H}^{\otimes n} \ket{0}_n = \frac{1}{\sqrt{2^n}} \sum_{0 \le x < 2^n} \ket{x}_n \tag{$\square$}$$

where

$$ \textbf{H}^{\otimes n} = \underbrace{\textbf{H} \otimes \textbf{H} \otimes \cdots \otimes \textbf{H}}_{n\ \text{times}}. $$

This means that Hadamard gate can be parallelized upon $n$-qubit systems, preserving its main property: creating a balanced superposition of the computational basis’ states (see the relation $\square$).