This is essentially bookwork, because there is only one gate that performs this state change (and the task title already gave it away!) The Toffoli gate is:
$$
 \begin{bmatrix}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
 \end{bmatrix}
$$
and our initial state is:
$$
\begin{bmatrix} \alpha \\ \beta \\ \gamma \\ \delta \\ \epsilon \\ \zeta \\ \eta \\ \theta
\end{bmatrix}
$$

So we have:

$$
 \begin{bmatrix}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
 \end{bmatrix}
 \begin{bmatrix}
 \alpha \\ \beta \\ \gamma \\ \delta \\ \epsilon \\ \zeta \\ \color{blue}\eta \\ \color{blue}\theta
 \end{bmatrix}=
 \begin{bmatrix}
 \alpha \\ \beta \\ \gamma \\ \delta \\ \epsilon \\ \zeta \\ \color{red}\theta \\ \color{red}\eta
 \end{bmatrix}=
\alpha |000\rangle + \beta |001\rangle + \gamma |010\rangle + \delta |011\rangle + \epsilon |100\rangle + \zeta|101\rangle + {\color{red}\theta}|110\rangle + {\color{red}\eta}|111\rangle
$$

@[solution]({
    "id": "multi_qubit_gates__toffoli_gate_solution",
    "codePath": "./Solution.qs"
})
