  Quantum operations, or quantum channels cannot be inverted in general. An
arbitrary state passing through a quantum channel looses its fidelity with the
input. Given a quantum channel ${\cal E}$, we introduce the concept of its
quasi-inverse as a map ${\cal E}^{qi}$ which when composed with ${\cal E}$
increases its average input-output fidelity in an optimal way. The channel
${\cal E}^{qi}$ comes as close as possible to the inverse of a quantum channel.
We give a complete classification of such maps for qubit channels and provide
quite a few illustrative examples.
