Como interpretar um circuito quântico como uma matriz?

Se um circuito recebe mais de um qubit como entrada e possui portas quânticas que recebem diferentes números de qubits como entrada, como interpretaríamos esse circuito como uma matriz?

Aqui está um exemplo de brinquedo:

Circuito Específico

O primeiro portão é um portão Hadamard, normalmente representado por

$$\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix}$$

$H\otimes I$$I$

$$\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0\\0 & 1 & 0 & 1\\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end{bmatrix}$$

Em seguida, temos um portão CNOT. Isso é normalmente representado por

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

This is the right size for two qubits, so we don't need to scale using kronecker products. We then have another hadamard gate, which scales the same was as the first. To find the overall matrix for the circuit, then, we multiply them all together:

$$\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0\\0 & 1 & 0 & 1\\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end{bmatrix}\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0\\0 & 1 & 0 & 1\\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end{bmatrix}$$

and get

$$\frac{1}{2}\begin{bmatrix}1&1&1&-1\\1&1&-1&1\\1&-1&1&1\\-1&1&1&1\end{bmatrix}$$

(if python multiplied correctly =) We would then multiply this by our original qubit state, and get our result.

Generalization

So basically, you go through each gate one by one, take the base representation, and scale them appropriately using kronecker products with identity matrices. Then you multiply all the matrices together in the order they are applied. Be sure to do this such that if you wrote out the multiplication, the very first gate is on the far right; as arriopolis points out, this is a common mistake. Matrices are not commutative! If you don't know the base representation of a matrix, check first wikipedia's article on quantum gates which has a lot.