I aks because I don’t think it is correct; the argument, at least, has some holes which I do not know how to fill out… even for the simplest case k=3 is not clear how to build the example… ]]>

“Quantum mechanics is a system for tracking the potential observations of atoms, and other phenomena on that scale. In particular it allows calculating probabilities of multiple possibilities, even though single outcomes are observed.

A quantum computer is a conjectural machine to do computations from interpreting those multiple possibilities as separate realities that can each contribute to what appears to be an almost-magic parallel computation. Despite almost a billion dollars in investment, no such speedups have been achieved.”

]]>The so-called “negative probabilities” are in a quasiprobability distribution where the anomaly is created at less than hbar and there are no final negative probabilities. By the way, the wave function of an electron is a complex-coefficient spinor function, which isn’t just a simple amplitude. The wave function can be positive, negative, complex, spinor or vector. Note that mathematicians have found quantum probability to be useless for modeling anything but atomic particles.

Just because simulating quantum states requires a high Turing complexity does not mean the argument runs backwards because the exponential computation on the wave function may well have nothing to do with the physical system of something like an dumb electron. I believe many in the QC community subscribe to the Schrödinger’s cat fallacy. The fallacies here are as persistent as those about EPR non-locality. QCs may not be possible at all given that they go beyond the current accepted theory of QM. People need to be honest about that. People making money building them are not honest, however.

]]>http://www.sci-prew.inf.ua/v120/1/S0305004100074600.pdf

This is very similar to what we see in the Ulam sequences except that there are rare outliers which form the unique sums. Even the periodic cases fall into this pattern with the parameter equal to 2 so that most numbers are odd with a finite number of even outliers.

The question is, does this pattern always emerge for any given starting point, and why? ]]>

A word about the colourful breakdown of the distribution: Every Ulam number a can be written uniquely as a sum of two smaller Ulam numbers b and c (say b less than c), the summands of a. It turns out that almost all Ulam numbers have as a summand something from a limited set, which includes 2, 3, 47, 69, 102, 13, 36, and some more. If we take any x from that set and plot only Ulam numbers that have x as a summand, we get an interestingly clean peak. As x ranges over the set, we get further peaks that seem to comprise the original distribution.

If we instead plot, for any x, the set of Ulam numbers y such that x+y is an Ulam number (that is, we plot the “complements” of x, rather than the things with x as a summand), these peaks seem to line up. See https://github.com/daniel3735928559/wip-ulam/raw/master/figs/shifted_summands_mod_5422.png

Gabriel: The phenomenon seems to exist for other pairs of initial values u, v as well, as is mentioned at the end of the post. However! For certain families of initial values u, v (e.g. 2, 2n+3 or 4, 4n+1) it is known that there are actually only finitely many even terms in the ensuing sequence! This entails, among other things, that the sequence of consecutive differences is periodic, so these sequences are entirely predictable. The phomenon here could maybe be seen as a generalisation of this for other starting conditions? For e.g. the sequence starting 2, 5, there would be a massive spike at theta = pi, corresponding to a non-uniform distribution mod 2.

]]>I think one thing makes this particularly interesting is a comparison with the harmonic structure in the sequence of logarithms of primes which is of course a matter of intense interest. Both the log primes and the Ulam sequence are defined by additive rules. Perhaps there is something in common in the way harmonic structure emerges from these pseudo random processes.

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