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Wei_Dai comments on The Born Probabilities - Less Wrong
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My guess is that the Born's Rule is related to the Solomonoff Prior. Consider a program P that takes 4 inputs:
What P does is take the boundary conditions, use Schrödinger's equation to compute the wavefunction at time T, then sample the wavefunction using the Born probabilities and the random input string, and finally output the particles in the region R and their relative positions.
Suppose this program, along with the inputs that cause it to output the description of a given human brain, is what makes the largest contribution to the probability mass of the bitstring representing that brain in the Solomonoff Prior. This seems like a plausible conjecture (putting aside the fact that quantum mechanics isn't actually the TOE of this universe).
(Does anyone think this is not true, or if it is true, has nothing to do with the answer to the mystery of "why squared amplitudes"?)
This idea seems fairly obvious, but I don't recall seeing it proposed by anyone yet. One possible direction to explore is to try to prove that any modification to Born's rule would cause a drastic decrease in the probability that P, given random inputs, would output the description of a sentient being. But I have no idea how to go about doing this. I'm also not sure how to develop this observation/conjecture into a full answer of the mystery.
I just read in Scott Aaronson's Quantum Computing, Postselection, and Probabilistic Polynomial-Time that if the exponent in the probability rule was anything other than 2, then we'd be able to do postselection without quantum suicide and solve problems in PP. (See Page 6, Theorem 6.) The same is true if quantum mechanics was non-linear.
Given that, my conjecture is implied by one that says "sentience is unlikely to evolve in a world where problems in PP (which is probably strictly harder than PH, which is probably strictly harder than NP) can be easily solved" (presumably because intelligence wouldn't be useful in such a world).
Interesting. What would such a world look like? I imagine instead of a selection pressure for intelligence there would be a selection pressure for raw memory, so that you could perfectly model any creature with less memory than yourself. It seems that this would be a very intense pressure, since the upper hand is essentially guaranteed superiority, and you would ultimately wind up with galaxy sized computers running through all possible simulations of other galaxy sized computers.
I never put much stock in the simulation hypothesis, because I couldn't see why an entity capable of simulating our universe would derive any value from doing it. This scenario makes me rethink that a little.
In any case, while this is another potential reason why the rule must be 2 in our universe, it still doesn't shed any light on the mechanism by which our subjective experience follows this rule.
I don't know. I don't have a very good understanding of regular quantum computing, much less the non-Born "fantasy" quantum computers that Aaronson used in his paper. But I'm going to guess that your speculation is probably wrong, unless you happen to be an expert in this area. These things tend not to be very intuitive at all.
I honestly can't imagine my evolution story is right. It just seemed like an immensely fun opportunity for speculation.