Confusion about whats being processed in a quantum computer

Question

Please correct me if im wrong but this is how I think quantum computers work.

Say we have a q-bit. The q-bit is put through a quantum gate to put it into a superposition and manipulate its probabilities.

We can represent this q-bit just like a normal bit on a vector where a normal bit would, when represented on a circle, be either on the x-axis or on the y-axis.

The q-bit however, being in a superposition between 0 and 1,is between 0 and one on the circle.

Since the quantum state vector has length one and the probabilities determine where the quantum state vector will be, the square-route of the probabilities are how the probabilities are represented on the circle(pythagorean theorem).

The quantum gates act like the matrix of the vector in this case.

But I dont actually understand whats actually being calculated in those steps, what is the actual goal of finding out where the q-bit is on the circle.

Say you wanted to calculate 6 + 2 how would this be implemented in this example?

My assumption is that you'd have to configurate the quantum gate in a way that in the end the probabilities turn out in favor of 8.

But then again when going over this assumption it sounds kind of stupid.

Also note that I didnt actually have vectors in school yet since im only in eight grade and im researching this topic out of pure interest.

I only watched one explaining in detail how quantum computers work since it was the only good I found that didnt assume you already learned this in some way and know about what way this is going.

So dont immediately get angry at me for not using the best terms or phrases and not even using the correct format.

Thanks to everyone who takes the time to explain it to me :D

Answer 1

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.

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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.