# Is there a method to implement qutrit using qubits?

I want to implement 3 level system on a qubit quantum computer.

Currently I feel I can use 2 qubits and encode the states of a 3-level system as such

|0>=|00>, |1>=|01>, |2>=|10>

The co-efficient of the |11> will always be zero and we will ignore it. However the gates that would be implemented on such states will be 3x3 matrices, but the system of 2 qubits is of dimension 4x4.

How to implement unitary gates ($$U$$) on such systems ?

One method I thought is using a direct sum $$U_{3 \text{ x }3} \oplus 0_{1 \text{ x } 1}$$, but the resultant matrix does not stay unitary.

Given a unitary $$3 \times 3$$ matrix $$V$$ that performs your desired transformation in the $$|00\rangle, |01\rangle,$$ and $$|10\rangle$$ states, the following matrix performs that transformation and leaves $$|11\rangle$$ intact.
$$U = \begin{bmatrix} V & 0\\ 0 & 1 \end{bmatrix}$$
\begin{align} U^*U &= \begin{bmatrix} v_{1,1}^* & v_{2,1}^* & v_{3,1}^* & 0 \\ v_{1,2}^* & v_{2,2}^* & v_{2,3}^* & 0 \\ v_{1,3}^* & v_{2,3}^* & v_{3,3}^* & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} v_{1,1} & v_{1,2} & v_{1,3} & 0 \\ v_{2, 1} & v_{2,2} & v_{2,3} & 0 \\ v_{3,1} & v_{2,3} & v_{3,3} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} V^* & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} V & 0 \\ 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} V^*V & 0 \\ 0 & 1 \end{bmatrix} = I \end{align} Where the last follows since we are given that $$V$$ is unitary. You can see the same is true for $$UU^*$$, so we have that $$U$$ is unitary.