# Computing with qutrits

I'm doing some calculations with qutrits and I need a unitary matrix $$U$$ that does the following:

$$U|00\rangle = |12 \rangle - | 21\rangle$$

$$U|11\rangle = |20 \rangle - | 02\rangle$$

$$U|22\rangle = |01 \rangle - | 10\rangle$$

where $$|0\rangle$$,$$|1\rangle$$, $$|2\rangle$$ are the basis states. I don't care how it acts on the other states as long as it acts in the way prescribed earlier. Until now I've been using ad-hoc methods for finding such unitaries. In summary, I require two things: the unitary that fulfills the conditions mentioned above and if possible some general method to find unitaries that perform certain transformations over qutrits or qudits.