Kronecker Product of Quantum Operations

When dealing with single-qubit quantum gates, we often use $2 \times 2$ matrices.

The Kronecker product allows us to combine these operations when considering systems of multiple qubits. Let's first focus on how to compute the Kronecker product for two general $2 \times 2$ matrices.

If $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$, their Kronecker product $A \otimes B$ is a $4 \times 4$ matrix.

Note, we could index the rows and columns starting at $0$ instead of $1$ (i.e. $a_{00}$ instead of $a_{11}$ for example) as we have done previously, however, for the purposes of this lesson, where we are not referring to the computational basis at all, there is no real clarity to be gained either way, so we'll stick with this.

The rule is to multiply each element of $A$ by the entire matrix $B$, forming a block matrix:

$$A \otimes B = \begin{bmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{bmatrix} = \begin{bmatrix} a_{11}\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} & a_{12}\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \\ a_{21}\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} & a_{22}\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \end{bmatrix}$$

This expands to:

$$A \otimes B = ...$$