One way to think about the "dimensions" of Quantum computing.

Think of a bunch of magnetic fillings

When an input is applied, they will align themselves to the magnetic field and their expected place in that field.



Quantum Qubits are able to align in 2 states (in current architectures up spin and down spin, which can also come out as up and down spin simultaneously), and if your input question is in the form of a Hamiltonian (e.g. the equation = 0).

THEN

You can give each Qubit inputs and they will all seek the lowest common combined power level like the magnets. This is due to quantum entanglement.

Then by checking each qubit's spin you know which values were part of the correct answer for that Hamiltonian.

If there are multiple answers you would rerun and see if a different combination of inputs generate a lower (more optimal) power level.

Then classical computers re form the answer into something usable.

What about those dimension?

In classical computing you calculate bit by bit, each "mathematical dimension has to be calculated separately vs another one, in a GPU the smaller calculations happen on many small threads that combined later, which is significantly more efficient, but nothing like quantum.

In quantum, like the magnets, they all snap into place near instantly, you get an instantaneous answer in one go, using physics to combined many degrees of freedom (or mathematical degrees) at once.

Imperfect Example, If I had 100 degrees of temperature, 100 blocks, and 20 materials, a classical way would be to calculate each of the blocks one at a time (or 100 separate calculations on a gpu) and then sum the resulting temperatures as a final step.

A quantum equivalent, I would form a calculation with the 3 degrees of freedom and calculate it in one shot, the qubits would snap to lowest power, and then I would take those values and use them for the answer.

For 3 degrees of freedom it would be faster to do it traditionally, but for 100, no classical computer would be capable.

Caveat, by no means am I a quantum physicist, but I have had exposure to the field and this is my high level analogy to what happens.

Hope this helps when people start talking about "alternate dimensions in quantum"