## 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 $H$.

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