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 \(\square\)).