I think this is actually not as serious of a problem as you make it out to be, for various reasons.
Finite automatons don’t actually exist in the real world. This was my whole point about (for example) transistors not actually working as binary on/off switches. In the real world there are ways to give agents access to real numbers, just with some measurement error both on the part of the forecaster and on the part of the agent reading the input. The setup of giving a wave-function input to a QTM is where this works best: you can’t get the squared norms of the coefficients exactly right, but you can get them right “in probability”, in the sense that an error of >ε will have a probability of <δ, etc.
Given that you don’t have a continuous function but (say) a convex combination of a continuous function with some random noise, Brouwer’s fixed point theorem still tells you that the continuous part has a fixed point, and then the random noise added on top of it just throws you off by some small distance. In other words, there isn’t a catastrophic failure where a little bit of noise on top of a continuous process totally throws you off; continuous function + some noise on top still has an “almost fixed point” with probability >1−δ.
I agree with you that if you had a finite automaton then this whole trick doesn’t work. The relevant notion of continuity for Turing machines is continuity on the Cantor set, which must hold since a TM that halts can only read finitely many bits of the input and so can only figure out it’s in some nonempty open subset of the Cantor set. However, as you point out the Cantor set is totally disconnected and there’s no way to map that notion of continuity to the one that uses real numbers.
In contrast, quantum Turing machines that take inputs from a Hilbert space are actually continuous in the more traditional real or complex analytic sense of that term, so here we have no such problems.
I think this is actually not as serious of a problem as you make it out to be, for various reasons.
Finite automatons don’t actually exist in the real world. This was my whole point about (for example) transistors not actually working as binary on/off switches. In the real world there are ways to give agents access to real numbers, just with some measurement error both on the part of the forecaster and on the part of the agent reading the input. The setup of giving a wave-function input to a QTM is where this works best: you can’t get the squared norms of the coefficients exactly right, but you can get them right “in probability”, in the sense that an error of >ε will have a probability of <δ, etc.
Given that you don’t have a continuous function but (say) a convex combination of a continuous function with some random noise, Brouwer’s fixed point theorem still tells you that the continuous part has a fixed point, and then the random noise added on top of it just throws you off by some small distance. In other words, there isn’t a catastrophic failure where a little bit of noise on top of a continuous process totally throws you off; continuous function + some noise on top still has an “almost fixed point” with probability >1−δ.
I agree with you that if you had a finite automaton then this whole trick doesn’t work. The relevant notion of continuity for Turing machines is continuity on the Cantor set, which must hold since a TM that halts can only read finitely many bits of the input and so can only figure out it’s in some nonempty open subset of the Cantor set. However, as you point out the Cantor set is totally disconnected and there’s no way to map that notion of continuity to the one that uses real numbers.
In contrast, quantum Turing machines that take inputs from a Hilbert space are actually continuous in the more traditional real or complex analytic sense of that term, so here we have no such problems.