Steps to apply Hadamard gate to $n$ qubits

Can someone shows me, step by step, how to apply Hadamard and output the result?

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First, note that the Hadamard gate has the matrix representation as $$H = \dfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$. When you apply Hadamard gates to all the qubits,

what you essentially doing is applying the operation $$U = H \otimes H \otimes H$$ to the state $$|\psi_2\rangle$$. Now,

$$H \otimes H = \dfrac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} \otimes \dfrac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} = \dfrac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end{pmatrix}$$

To get $$H \otimes H \otimes H$$ you have to do another tensor. Thus, applying Hadamard gates to the state $$|\psi_2\rangle$$ can be written in term of matrix multiplication as

$$$$\begin{bmatrix} \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} \\ \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & \tfrac{1}{\sqrt{8}} & -\tfrac{1}{\sqrt{8}} \\ \end{bmatrix} \begin{bmatrix}\dfrac{1}{\sqrt{8}} \\ \dfrac{1}{\sqrt{8}} \\ \dfrac{1}{\sqrt{8}}\\ \dfrac{1}{\sqrt{8}}\\ \dfrac{1}{\sqrt{8}}\\ -\dfrac{1}{\sqrt{8}} \\ -\dfrac{1}{\sqrt{8}} \\ \dfrac{1}{\sqrt{8}} \end{bmatrix} = \dfrac{1}{2} \begin{bmatrix} 1 \\ 0 \\ 0\\ 1 \\ 1 \\ 0 \\ 0 \\ -1 \end{bmatrix}$$$$

Now, the resulting vector above can be written in term of the ket representation as your formula $$|\psi_{3a} \rangle$$. That is, $$$$\dfrac{1}{2} \begin{bmatrix} 1 \\ 0 \\ 0\\ 1 \\ 1 \\ 0 \\ 0 \\ -1 \end{bmatrix} = \dfrac{1}{2} \big( |000\rangle + |011\rangle + |100\rangle - |111\rangle \big)$$$$

• How do you know there are 3 qubits? – pyb yesterday
• @pyb There are three qubits in your system because if you look at the basis states, there are 3 qubits within it. To see this more clearly, note that the state $|\psi \rangle = \dfrac{|000\rangle + |111\rangle }{\sqrt{2} } = \dfrac{|0\rangle^{\otimes 3} + |1\rangle^{\otimes 3} }{\sqrt{2} }$ is another three qubit systems. But $|\psi \rangle = \dfrac{|0000\rangle + |1111\rangle }{\sqrt{2} } = \dfrac{|0\rangle^{\otimes 4} + |1\rangle^{\otimes 4} }{\sqrt{2} }$ is a four qubit system. Now, another 4 qubits state could be: $|\psi \rangle = |0101\rangle$. – KAJ226 yesterday
• You can just look at the position of the element in the vector and convert it to binary. For instance, there is a $1$ in the first (0th) position of the vector, so this corresponds to $|000\rangle$. Then there is also a $1$ in the 3rd position of the vector. Now $3$ is binary is $011$ so this corresponds to $|011\rangle$. There is a $1$ in the 4th position, and $4$ in binary is $100$ so this corresponds to $|100\rangle$. Then there is a $1$ in the last spot, the $7th$ position, and $7$ in binary is $111$ thus the state is $|111\rangle$. put them together you have the last line :) – KAJ226 yesterday
• wow, thank you so much! It will make reviewing intro courses much easier. I miss why the 1st $1$ is encoded to |000⟩ and not |001⟩, but I will figure it out. – pyb 19 hours ago
• This is because we start counting from $0$. $0$ corresponds to $000$. Then $1$ corresponds to $001$. – KAJ226 18 hours ago