# Representing Quantum Gates in Tensor Product Space

## Summary:

I am confused on how to represent the computational (matrix) form of a quantum gate in the tensor product space. For example: how does the computational matrix look for a single qubit gate on in two qubit tensor product vector space?

## Main Question or Discussion Point

Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we apply the Hadamard gate, ##\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}##, on the ##q_0##.

Here is my understanding:

The Hadamard gate on two qubit system only operates on ##q_0##
$$\hat{H}_0(q_0 \otimes q_1) = \hat{H}_0q_0 \otimes q_1$$
$$\hat{H}_0(|0 \rangle \otimes |0 \rangle) = \hat{H}_0|0\rangle \otimes |0\rangle$$
$$\hat{H}_0 (\begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} )= \hat{H}_0 \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$
$$\hat{H}_0 \begin{pmatrix} 1 \\ 0 \\ 0 \\0 \end{pmatrix} = \frac{1}{\sqrt{2}} ( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix})$$
$$\hat{H}_0 \begin{pmatrix} 1 \\ 0 \\ 0 \\0 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}$$

I don't know exactly how to solve this but can give a guess.

$$\hat{H}_0 = \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

How is ##\hat{H}_0## written in the tensor product space? Are there any resources that explain this well? Any help is appreciated!!

Last edited: