Proof? Just look at all the replies you got: each one is dozens of pages of complex (imaginary) math, control theory, and statistics.
The hardest part of QC is exactly what you described: how to extract the answer. There is no algorithm, per se. You build the system to solve the problem.
This is why QC is not a general purpose strategy: a quantum computer won't run Ubuntu, but it will be one superfast prime factoring coprocessor, for example (or pathfinder, or root solver). You literally have to build an entire machine to solve just one problem, like factoring.
Look at Shor's algorithm: it has a classical algorithm and then a QC "coprocessor" part (think of that like an FPU looking up a transcendental from a ROM: it appears the FPU is computing sin(), but it is not, it is doing a lookup... just an analogy). The entire QC side is custom built just to do this one task:
https://en.wikipedia.org/wiki/Shor%27s_algorithm
In this example he factors 15 into 5x3, and the QC part requires FFTs and Tensor math. Oy!
Like I said, it will take decades for this to become easier to explain.
For fun, look at the gates we're dealing with, like "square root of not": https://en.wikipedia.org/wiki/Quantum_logic_gate