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Is There Anything Beyond Quantum Computing?

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Re: Is There Anything Beyond Quantum Computing?

#21

Post by trhway »

dkarapetyan wrote: post #16
trhway wrote: post #15

> If so, then much like in Zeno’s paradox, your computer would have completed infinitely many steps in a mere two seconds! The next target isn't infinitely many. The target is infinitely infinitely many. Like something along the lines of forking infinite number of parallel Universes - that is pretty much "many-verse" interpretation of the quantum superposition(and thus computing) - enhanced with forking of infinitely…

Infinitely infinitely many is still infinitely many. N x N has the same cardinality as N. Now if you meant N ^ N then that's actually different. "x" means set product and "^" means function space.

>N x N has the same cardinality as N

until the N is aleph-one if i remember correctly, or do you think it is not?

http://en.wikipedia.org/wiki/Cardinal_number

Re: Is There Anything Beyond Quantum Computing?

#22

Post by Sniffnoy »

dkarapetyan wrote: post #16
trhway wrote: post #15

> If so, then much like in Zeno’s paradox, your computer would have completed infinitely many steps in a mere two seconds! The next target isn't infinitely many. The target is infinitely infinitely many. Like something along the lines of forking infinite number of parallel Universes - that is pretty much "many-verse" interpretation of the quantum superposition(and thus computing) - enhanced with forking of infinitely…

Infinitely infinitely many is still infinitely many. N x N has the same cardinality as N. Now if you meant N ^ N then that's actually different. "x" means set product and "^" means function space.

This isn't about cardinality, though, this is about number of steps, which is more properly measured with ordinal numbers. I'm not familiar with the theory of infinite-time Turing machines, but it's at least plausible that there are things you can do with omega^2 steps (or larger countable ordinals) that you couldn't do with only omega steps.

Re: Is There Anything Beyond Quantum Computing?

#23

Post by Sniffnoy »

trhway wrote: post #21
dkarapetyan wrote: post #16

Earlier quoted context omitted.

Infinitely infinitely many is still infinitely many. N x N has the same cardinality as N. Now if you meant N ^ N then that's actually different. "x" means set product and "^" means function space.

>N x N has the same cardinality as N until the N is aleph-one if i remember correctly, or do you think it is not? http://en.wikipedia.org/wiki/Cardinal_number

By N, dkarapetyan presumably means the set of natural numbers, or aleph_0. So it isn't aleph_1. That said, it is also true that aleph_1 x aleph_1 = aleph_1. More generally, any aleph_alpha x aleph_alpha = aleph_alpha, for any ordinal alpha; and if we assume the axiom of choice, then any infinite cardinal A can be written as some aleph_alpha, and hence satisfies A x A = A.

However, ultimately none of this is relevant; number of steps is properly measured with ordinals, not cardinals.

Re: Is There Anything Beyond Quantum Computing?

#24

Post by calhoun137 »

The most important application of quantum computers seems to actually be to enable scientists who do fundamental research to answer questions from reporters such as "What practical purpose does your research serve?" with a single catch all buzz word. So the answer to the title of this article is clearly: Yes.

Re: Is There Anything Beyond Quantum Computing?

#25

Post by AnimalMuppet »

rhth54656 wrote: post #20
rohansingh wrote: post #18

Earlier quoted context omitted.

An object can accelerate arbitrarily, but in any reference frame it will never be traveling faster than c . Of course, it won't collapse into a black hole, but the amount of energy needed to accelerate further will increase asymptotically.

That is actually false from the frame of the traveler. Using constant acceleration, the traveler will observe constant energy usage. Then length contraction will guarantee that you can reach speeds arbitrarily close to speed of light( and also any location in the universe ) in about ~50 years with 1G acceleration. http://en.wikipedia.org/wiki/Space_travel_using_constant_acc...

Yes, but when do you get there? In 50 years in your time, but it may be a few billion years later in their time...

Re: Is There Anything Beyond Quantum Computing?

#26

Post by rhth54656 »

AnimalMuppet wrote: post #25
rhth54656 wrote: post #20

Earlier quoted context omitted.

That is actually false from the frame of the traveler. Using constant acceleration, the traveler will observe constant energy usage. Then length contraction will guarantee that you can reach speeds arbitrarily close to speed of light( and also any location in the universe ) in about ~50 years with 1G acceleration. http://en.wikipedia.org/wiki/Space_travel_using_constant_acc...

Yes, but when do you get there? In 50 years in your time, but it may be a few billion years later in their time...

I believe that is the point. Even more time for the stationary computer to finish the calculation.

Re: Is There Anything Beyond Quantum Computing?

#27

Post by sp332 »

Sniffnoy wrote: post #22
dkarapetyan wrote: post #16

Earlier quoted context omitted.

Infinitely infinitely many is still infinitely many. N x N has the same cardinality as N. Now if you meant N ^ N then that's actually different. "x" means set product and "^" means function space.

This isn't about cardinality, though, this is about number of steps, which is more properly measured with ordinal numbers. I'm not familiar with the theory of infinite-time Turing machines, but it's at least plausible that there are things you can do with omega^2 steps (or larger countable ordinals) that you couldn't do with only omega steps.

omega == omega^2. They are the same quantity. For every element of omega^2, there is a corresponding unique element of omega.

Re: Is There Anything Beyond Quantum Computing?

#28

Post by Havoc »

forgotprevpass wrote: post #11
Havoc wrote: post #8

Scott is making it really difficult for people to follow along. My initial reaction was "wtf am I looking at" followed shortly by "why wasn't the HN link pointed straight to the PBS article?". And clearly others agree given that the top comment at the moment is a PBS link. No doubt the guy is competent but this just smells like a poorly executed attempt to move traffic to the blog. I fully expect it to get ignored as…

If you're not familiar, Scott is a professor of MIT, specializing in quantum computing. Many readers of his blog are also well known in this sphere, so they would have more technical insight, more in depth than the pbs layman's version.

>>If you're not familiar, Scott is a professor of MIT

As I said, I don't doubt his ability. Its the presentation I object to. The HN blog leads to some random blog that barely touches on the topic that the headline promises and the core of the story is in a link contained in the blog. By any sane definition that is blog spam, MIT professor or not.

>>Many readers of his blog are also well known

So? Its linked to his blog, not to his readers. I don't care if his readers include Albert Einstein himself. The link did not deliver despite the authors (proven) ability to deliver on the content.

Re: Is There Anything Beyond Quantum Computing?

#29

Post by adamtj »

sp332 wrote: post #27
Sniffnoy wrote: post #22

Earlier quoted context omitted.

This isn't about cardinality, though, this is about number of steps, which is more properly measured with ordinal numbers. I'm not familiar with the theory of infinite-time Turing machines, but it's at least plausible that there are things you can do with omega^2 steps (or larger countable ordinals) that you couldn't do with only omega steps.

omega == omega^2. They are the same quantity. For every element of omega^2, there is a corresponding unique element of omega.

Actually, no. I just recently read about this.

1 + omega == omega http://en.wikipedia.org/wiki/Ordinal_number

Re: Is There Anything Beyond Quantum Computing?

#30

Post by sp332 »

adamtj wrote: post #29
sp332 wrote: post #27

Earlier quoted context omitted.

omega == omega^2. They are the same quantity. For every element of omega^2, there is a corresponding unique element of omega.

Actually, no. I just recently read about this. 1 + omega == omega http://en.wikipedia.org/wiki/Ordinal_number

Weird! Thanks for the link!
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