How do I apply a Hadamard gate on a given qubit, in matrix formalism?

Question

Hadamard gate matrix is:

$$\frac{1}{\sqrt2}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$$

The matrix for $|0\rangle$ is:

$$\begin{bmatrix}1 \\ 0\end{bmatrix}$$

I am unable to understand, how can I apply Hadamard gate on $|0\rangle$? The matrix representing $|0\rangle$ is of dimension 2 * 1 and the matrix representing Hadamard gate is of dimension 2 * 2 (so the matrix multiplication is not possible)

Answer 1

Applying quantum gates to quantum states is indeed represented as matrix multiplication. To multiply two matrices, you need only one dimension to match:

$$\frac{1}{\sqrt2}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix} \begin{bmatrix}1 \\ 0\end{bmatrix} = \frac{1}{\sqrt2}\begin{bmatrix}1 \cdot 1 + 1 \cdot 0 \\ 1 \cdot 1 + (-1) \cdot 0 \end{bmatrix} = \frac{1}{\sqrt2}\begin{bmatrix}1 \\ 1\end{bmatrix}$$

Answer 2

|0> is actually a vector, not a matrix. Applying the Hadamard gate (shortly, H) to |0> means computing a matrix vector multiplication:

$H|0\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$

Answer 3

Here's a link to a great matrix multiplier tool:

https://matrixcalc.org/en/