Quantum Field Computation: Beyond Quantum Computing?

[Ontoplankton]

Quantum mechanics allows one to compute in a way fundamentally different from any way to compute classically, at least in theory.

Will quantum field theory allow new kinds of computing that go beyond quantum computing? I would have suspected not, but for example this paper argues that it may. (It looks non-crackpot to me, but I have no way to verify its correctness.)

If so, will quantum field computation ever be of practical use? Will making use of it be feasible, or feasible only for Jupiter-sized-brain-with-galaxy-sized-accelerator type entities, or physically impossible, or logically impossible?

Would it have any implications for fundamental physics, computational complexity theory, or the foundations of mathematics?

Any insights, information, or pointers to information on this? Thanks in advance.

 

[selfAdjoint]

The paper is not crackpot at all, although the particular mode it proposes for QFT computation, analytical functions, may not be the one (if any) that finally works. He notes that there is another proposal using topological quantum field theories (TQFTs).

The real eye-opener for me was the discussion of real computation. I have been pushing for years the idea that an end-run around Goedel's theorem is possible using Tarski's proof that geometry is complete. And now I find out that all the time I ws blatting about this, a theory of real computation, computation with real numbers, devised by Smale, and using Tarski, was in existence. Yeni!

I think QFT computation could be the very early stage of an explosive field. The author emphasizes that computation in QFT is very difficult. But there are many physics Ph.D.'s, some of them flipping burgers due to the disasterous physics labor situation, who have mastered those difficulties. They have all these skills and nothing to deploy them on. Here is a field with only a few papers to take you out to the frontier. We could see a feeding frenzy.

 

[Ontoplankton]

That's very interesting -- I hadn't even heard of the possibility until I came across that paper.

A question, though. How would such an end-run around Goedel's theorem work? If you could translate any statement in arithmetic to a statement in geometry, and use the completeness of geometry to prove or disprove any statement in arithmetic, then you'd have a direct violation of Goedel's Theorem, which can't happen, since it's a Theorem. So, that's not how it works, unless I'm mistaken.

Is the idea to implement real computing in physical reality, and then deduce from the results what sort of set theory nature uses, so to speak? If you could do an infinite number of computations, you could just brute-force the Riemann hypothesis, the twin prime conjecture, Goldbach's conjecture, and so on, as well as find out anything Goedel told us was impossible, such as whether arithmetic or ZFC set theory is consistent. Is this sort of thing possible in computation over the real numbers, or am I thinking in the wrong direction completely?

(I know something similar (also trans-Turing computation) can happen in "Malament-Hogarth spacetimes", see e.g. here)

 

[selfAdjoint]

Well I just found out about this, and obviouslky I'm ignorant of the details. I did some googling, and haven't found too much (mathematicians don't all participate in the arxiv). But there's a book called Real Computation and Complexity by several authors including Smale, and with another day or two of talking to myself I may convince myself to buy it.

If I learn anything I'll pass it along.

 

[selfAdjoint]

Here's an update. I still haven't got a good tutorial site, but I have found this out. What Blum, Shub, and Smale (BSS) did in 1989 was introduce a real value replacement for the Turing machine. This machine has real number registers and computations, and instead of a digital tape it has random access to a continuum. Then you can do complexity theory with this device in place of Turing and answer questions that digital complexity theory can't. For expample they show the boundary of the Mandlebrot set is non-recursive, which had been an open question in traditional complexity theory.

I haven't found a site that discusses the Goedel status of BSS computations, but one technical paper did discuss quantifier elimination. It is a fact that proofs of Goedel's theorem require the use of the set theory quantifiers "For Each" and "For All" as well as mathematical induction. If you don't have those tools in a theory you can't (currently) prove a Goedel theorem about the theory.

Induction is not available for continua, and Tarski showed that in real continuum contexts you can transform any statement with a quantifier into an equivalent statement without one. This was the key to his proof that geometry is complete.

So I would assume that BSS exploit this to show that at least some of the computations their abstract machine can do are complete.

I'll keep looking.

 

[selfAdjoint]

Here at last is a http://www.ulb.ac.be/assoc/bms/Bulletin/bul971/meer.pdf on the BSS machine, complexity, and undecidability. As you will see, the computations of the BSS machine are decidable under rather general hypotheses.

It seems to me this has two implications.

1) Science could well be redefined to use only BSS real computations. I don't think this would be difficult at all, though there might be some delicate steps in the cass of infinite dimensional Hilbert spaces and such. Thus worries that a TOE would be undecidable would be moot.

2) Although we know relatively little about the cellular communication in the brain, it would seem to be basically analog, and hence representable as BSS computations. If this is confirmed, then the assertions of Penrose and others that human thinking is necessarily limited by Goedel undecidability are false.