Quantum Country exercise 7

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 7. Find a matrix other than I, X, or H that is unitary.

Answer: Both I and X have two zero entries and two entries with value 1, such that the 1 values end up multiplying each other to produce a 1 value in the resulting matrix I^\dagger I or X^\dagger X.

This raises the possibility of having a matrix where the value i ends up multiplying the value -i, since i \cdot (-i) = -i^2 = -(-1) = 1. Two possible candidate matrices with two entries with value i are \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} and \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}.

We have

\begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}^\dagger \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \left( \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}^T \right)^* \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}^* \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}

= \begin{bmatrix} -i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} -i^2 & 0 \\ 0 & -i^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I

We also have

\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}^\dagger \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} = \begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I

Consider also the matrix \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}. For it we have

\begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}^\dagger \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} = \left( \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}^T \right)^* \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}^* \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}

= \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} = \begin{bmatrix} -i^2 & 0 \\ 0 & -i^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I

Finally, for the matrix \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} we have

\begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}^\dagger \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I

So \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}, \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}, \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}, and \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} are all examples of unitary matrices other than I, X, or H.

This entry was posted in quantum country. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s