You are not far off the mark - quantum computations are all about probabilities.

You may say that a quantum program (i.e, a series of quantum gates) computes $n+m$ (when $n,m$ are inputs), if with "high probability", the end result will be exactly $n+m$.

The role of the quantum mechanics here are represented in the intermediate states - before the end result is "measured". This so-called operation of "measuring", creates a random bit according to probability distribution that the qbit specified.

So even though the input and the output are particular bits - everything in the middle are just probability distributions with the fancy name of "qbits". Therefore, it is natural to define the computation of quantum computers in probabilistic terms.

You are not far off the mark - quantum computations are all about probabilities.

You may say that a quantum program (i.e, a series of quantum gates) computes $n+m$ (when $n,m$ are inputs), if with "high probability", the end result will be exactly $n+m$.

The role of the quantum mechanics here are represented in the intermediate states - before the end result is "measured". This so-called operation of "measuring", creates a random bit according to probability distribution that the qbit specified.

So even though the input and the output are particular bits - everything in the middle are just probability distributions with the fancy name of "qbits". Therefore, it is natural to define the computation of quantum computers in probabilistic terms.

If you are interested, the theoretical definition of "with high probability" is having probability of at least $\frac{2}{3}$. This can be seen in the definition of the BQP complexity class for example.