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This is not really about quantum computing. A classical probabilistic Turing samples from a random distribution:

"probabilistic Turing machines can be defined as deterministic Turing machines having an additional "write" instruction where the value of the write is uniformly distributed"

I remember that probabilistic Turing machines are not more powerful than deterministic Turing machines, though Wikipedia is more optimistic:

"suggests that randomness may add power."

https://en.wikipedia.org/wiki/Probabilistic_Turing_machine

1 comments

Power is not the point. The point is just that probabilistic TM's (i.e. TMs with a random oracle) are different. For example, the usual proof of the uncomputability of the halting problem does not apply to PTMs. The proof can be generalized to PTMs, but the point is that this generalization is necessary. You can't simply reduce a PTM to a DTM.

The problem is about physics, not Turing machines. You don't need to make a random choice as part of your physical model, the model only makes predictions about the distribution. You can't represent the continuous dynamical manifolds of classical or quantum mechanics on a TM either, but that's ok, because we have discrete models that work well.

I am asking myself:

Does a probabilistic Turing machines needs aleatory uncertainty? (would have called this ontological but (1) disagrees)

Epistemic uncertainty would mean her:

We don't know which deterministic Turing machine we are running. Right now, I see no way to use this in algorithms.

(1) https://dictionary.helmholtz-uq.de/content/types_of_uncertai...

The whole point of Turing Machines is to eliminate all of these different kinds of uncertainty. There is in point of actual physical fact no such thing as a Turing Machine. Digital computers are really analog under the hood, but they are constructed in such a way that their behavior corresponds to a deterministic model with extremely high fidelity. It turns out that this deterministic behavior can in turn be tweaked to correspond to the behavior of a wide range of real physical systems. Indeed, there is only one known exception: individual quantum measurements, which are non-deterministic at a very deep fundamental level. And that in turn also turns out to be useful in its own way, which is why quantum computing is a thing.

Right, the point is that we don't need a solution to the 'measurement problem' to have a quantum computer.

Well, yeah, obviously. But my point is that you do need a solution to the measurement problem in order to model measurements in any way other than simply punting and introducing randomness as a postulate.

And is that solution required to be deterministic? If so, that is another postulate.

> You don't need to make a random choice as part of your physical model

You do if you want to model individual quantum measurements.

Interaction in quantum physics is something that remains abstract at a certain level. So long as conservation principles are satisfied (include probability summing to one), interactions are permitted (i.e., what is permitted is required).

Yes. So? What does that have to do with modeling measurements, i.e. the macroscopic process of humans doing experiments and observing the results?

Would you agree that measurement is considered an interaction?

Sure. So?