
NP-Complete Problems and Physical Reality (2005) - andyjohnson0
http://arxiv.org/abs/quant-ph/0502072
======
kenny-log_ins
If you enjoyed this I highly recommend both his book, "Quantum Computing Since
Democritus", and blog
([http://www.scottaaronson.com/blog/](http://www.scottaaronson.com/blog/)).
Tagline of the blog is "If you take just one piece of information from this
blog: Quantum computers would not solve hard search problems instantaneously
by simply trying all the possible solutions at once."

~~~
_asummers
Wholeheartedly recommend this book as well. He gives a brief overview of
classical complexity, how quantum matrices work, then how quantum complexity
works. It's really accessible to those with some complexity background.

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FeepingCreature
On the point of anthropic computing: it is important to note that you have not
accomplished any _improvement_ to your condition by killing yourself; before
as after, you had found a correct solution with probability 2^-n. Anthropic
computing is just "fancy guessing".

What is interesting about it to me is that the "true computational nature of
reality" must be such that sufficient computation is available for you to run
the _validation_ of your solution in logarithmic time an exponential number of
times ("in parallel"), because otherwise you could not find yourself _at all_
in a universe where you knew you had found the correct computation. Which does
make it strange that quantum physics does not seem to give us any way to "get
at" all this computation that must be going on behind the scenes.

(Unless the "true nature" of reality is such that a subset of the wavefunction
is being computed "for cheap". Collapse exonerated?)

~~~
andyjohnson0
By running the validation "in parallel", do you mean running it in each of the
possible quantum "many worlds" in which validation takes place?

If so, aren't you implicitly assuming that these other parallel worlds somehow
draw on a shared pool of computing resource? This might be true [1], but
equally there may not be a "behind the scenes" of reality at all.

[1] How would we know? Run a timing attack on the underlying compute
infrastructure?

~~~
kordless
I'm leaning toward the parallel universe hypothesis as well, except I'm
thinking it's more like there are part-time processes that run in parallel, do
their work, then exit after they come to some type of consensus of state. It
would end up looking something like a bunch of containers running a game on
your computer, processing all the other player's moves, arriving at consensus,
sending that consensus to a central 'compute' spot where a global consensus
took place, wait for a reply, then take what the global central thing said was
a valid state of things. Rinse and repeat. Stick it all in blockchain in case
there's a disagreement.

I keep pointing out to people I can't make their beers (including glass)
disappear while we're talking at a table. That this universe doesn't allow me
to hack it (and presumably keeps others from hacking it) is a nice feature.

~~~
andyjohnson0
If the multiverse / physical reality has a "global central thing" then it has
a scaling bottleneck, and we have a problem.

~~~
adrianN
We wouldn't really notice if the simulation keeps getting slower though.

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dsfsdfd
"But another reason we believe P≠NP is that otherwise mathematical creativity
could be automated!"

I have a problem with this. Just because the thought is unpleasant does not
make it true or false. Besides, perhaps this is the very difference between
automation and intelligence, perhaps the point at which you can ask this of a
computer is the point at which you should no longer consider it a computer.

~~~
nabla9
There is zero reason to believe that human level mathematical creativity is
computationally significantly more expensive than normal day to day thinking.

~~~
SilasX
In terms of the computations performed in the brain by the mathematician, at
the time the insight is achieved, it's not.

But appropriately accounted for, I think it is.

The relevant metric would be, "what do I have to increase in order to get more
[interesting, new] theorems proved?" Is it as easy has having mathematicians
work more hours? Having more people start working on them?

Intuitively, it is not so easy -- each insights need exponentially more work
as time progresses. An exponentially-small fraction of humans is capable of
producing novel results, and so on.

This, I think, is what Aaronson is getting at: if P = NP, if proving is as
easy as verifying, then coming up with hard mathematical insights should be as
easy as following a cookbook, or any other routine, mundane mental task.

But that doesn't seem to describe the world we live in, where getting
mathematical insights has rapidly diminishing returns per unit resource
invested.

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sqrt17
A point that has become more obvious over the years is this: there are "soft"
instances of NP-complete problems where you can find good approximations in
reasonable time, and there are "crunchy" instances of the same class of
problem where you don't find a good approximation.

Commercial ILP solvers (e.g. Gurobi, CPlex) profit from the fact that quite
often it is possible to formulate NP-complete problems and find good solutions
in acceptable time. Similarly, many of the "physical" ways of solving NP-
complete problems work ok for easy instances and get unwieldly fast if
confronted with difficult instances.

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nafets
I remember a TED talk on the failed attempts to find cases in which nature
solves a NP-complete problem. Paper must be an interesting read.

~~~
seiji
This one is always fun:
[https://www.youtube.com/watch?v=mvBSkt6LhJE](https://www.youtube.com/watch?v=mvBSkt6LhJE)

Related: [http://www.wired.com/2010/01/slime-mold-grows-network-
just-l...](http://www.wired.com/2010/01/slime-mold-grows-network-just-like-
tokyo-rail-system/)

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andyjohnson0
I posted this link because (a) it seemed like a good survey of the subject,
and (b) I felt like I almost understood most of it. Does anyone have any
suggestions for accessible further reading at the intersection of computer
science and physics?

~~~
giech
His Philosophy-Complexity paper is also fantastic, in case you have not read
it already:
[http://www.scottaaronson.com/papers/philos.pdf](http://www.scottaaronson.com/papers/philos.pdf)

It has also been discussed most recently here:
[https://news.ycombinator.com/item?id=9061744](https://news.ycombinator.com/item?id=9061744)

~~~
andyjohnson0
Thanks. I'll definitely be reading those.

Aaronson's talk on Computational Complexity and the Anthropic Principle [1]
also looks interesting.

[1]
[http://www.scottaaronson.com/talks/anthropic.html](http://www.scottaaronson.com/talks/anthropic.html)

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amatus
"If some physicist wants to continue the tradition of naming quantum gravity
theories using monosyllabic words for elongated objects that mean something
completely different in computer science, then I propose the most
revolutionary advance yet: thread theory."

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sudioStudio64
I'm looking forward to his take on the recent thing about memristor machines
being able to solve NP-complete problems...it was on HN a couple of days ago.

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a1b2c3
From 2005

