
The Unitarihedron: The Jewel at the Heart of Quantum Computing - benjoffe
http://www.scottaaronson.com/blog/?p=1537
======
jerf
For a while I thought he was making a joke about how if you continue to
abstract a given theory, you end up "encompassing" everything eventually, but
not in a useful way. To put what I mean in programmer terms, if you take a
Javascript framework, and just keep abstracting and abstracting and
abstracting, to a pathological degree, you ultimately can end up with your
"Javascript framework" consisting of:

    
    
        eval(x)
    

where the user supplies x. It is literally the most powerful Javascript
framework ever!... and yet, obviously, also not useful. It trivially
encompasses every possible Javascript program, at the price of not saying
anything useful about any of them.

But then he threw me when he linked to what appears to be a real proof. I read
through it, and found no signs it was tongue-in-cheek itself.

So I'll admit I'm at an impasse here; if he's got a real mathematical
construct that is usefully better than what was presented, I don't get the
tongue-in-cheek tone; if it really is tongue-in-cheek, I don't get what the
real proofs are doing there, unless the proof is itself the best-disguised
joke I've seen in the math world. (Generally I've got a pretty good eye for
mathematician humor, even in fields I know little about.)

~~~
polynomial
This is the machine learning principle that the more a model explains the less
accurate it is.

~~~
troymc
That principle might be a good rule of thumb, but it's not universally true.
Sometimes a new model comes along that both explains more _and_ is more
accurate. General Relativity is an example. Fundamental physics is funny that
way.

~~~
polynomial
Einstein always seems to be the exception that proves the rule.

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powertower
I don't quite get it.

After looking at the linked to papers, I'm assuming this is not a joke, and he
is not putting down (making fun) of the Amplituhedron, nor making up the
Unitarihedron (just abstracting his work with a media-friendly name to itit).

What he is joking about is simply that mathematicians should named their work
as a (insert-something-here)-hedron, as that would cause the media to pick it
up and report on it.

Is this correct?

~~~
killerpopiller
I don't know either, what a brick. I really hate expressions where you need
context to understand it's intentions, or die stupid.

He could have given context but prefered being condescending.

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msie
Sadly, the joke's lost on me. Is he making fun of the other "hedron" and this
Unitarihedron is a joke too? Which is too bad. I was having high hopes for the
future and I wasted time reading an inside joke.

~~~
derefr
I think he's genuinely suggesting that if journalists are getting so excited
about the amplituhedron, they should get even more excited about this, and
probably can only do that if they have a fun word for it. No joke.

(If it was a joke, the paper he linked to --
[http://www.scottaaronson.com/papers/pp.pdf](http://www.scottaaronson.com/papers/pp.pdf)
\-- probably wouldn't be exactly what he described it as.)

~~~
scythe
If you read some of the links in the first paragraph you'll get a feel for the
intent of the article. Aaronson probably expects his audience to follow
traditional physics sources. See for example:

[http://www.math.columbia.edu/~woit/wordpress/?p=6260](http://www.math.columbia.edu/~woit/wordpress/?p=6260)

Yesterday in QFT class the professor [0] explained the name by saying that the
guy who came up with it was a "very good salesman". The reality is that
probabilities can in a great deal of cases be expressed as the volume of
something in some phase space, for a trivial example, consider the area under
the curve between two x-values on the normal distribution's probability
density function -- it's the 2-volume of a section of the Gaussiohedron.

So after branding quantum information science as the study of the
"unitarihedron", Aaronson dragged out one of his more miraculous results,
which is that PP is closed under union and intersection as a direct result of
PP = PostBQP, where PostBQP is a complexity class derived from BQP using an
idea called _postselection_. That proof was actually in his dissertation (if I
recall correctly), written in 2006, and it was an extremely impressive proof
which did not use a word like "unitarihedron" to describe its methods.

So, while it was a joke, if you read the article in the context of the physics
community's reaction to the amplituhedron story, the included proof really
just helps to drive the point home.

The methods now called "amplituhedron" have been under continuous development
for the past ten years or so. My QFT professor[0] seemed sort of enthusiastic
about the development. They do seem to represent a step forward in our
understanding of quantum field theory, though the reaction of the media has
been, in the eyes of some commentators, characteristically buzzword-driven.

[0]:
[http://en.wikipedia.org/wiki/Predrag_Cvitanovic](http://en.wikipedia.org/wiki/Predrag_Cvitanovic)

~~~
carlob
Are you sure your professor authorised you to quote him on calling one of the
most powerful persons in his field: 'a very good salesman' on a public forum
of sorts?

~~~
gajomi
This particular professor, as far as I am aware, would probably not be
considered as active in QFT. He has been working on turbulence and chaos
recent (albeit, employing techniques from field theory). At any rate, I don't
really see the quip as taking a knock on these guys (he is known for his witty
banter). This is really good salesmanship and the media are having a filed day
with it.

~~~
carlob
Just checking, sometimes people banter, but they don't want to see such banter
spread in print.

FWIW I have met Arkani-Hamed and he didn't strike me for his salesmanship.

------
gtr32x
Can someone help enlighten me a bit here. From a layman's standpoint after
reading all the comments associated with the Amplituhedron article, it seems
both from a semi-professional standpoint and layman standpoint it was an
unique discovery. Now that a true professional has responded, is Scott here
simply mocking at the marketting success of the Amplituhedron or is actually
making a point about the existence of such mechanics prior to the discovery of
Amplituhedron? Thanks.

~~~
anigbrowl
Both, but since he hasn't mnaged to come up with any pictures of how elegant
BQP/unitarihedron is, his point is lost.

~~~
gtr32x
If that's the case, he just sounds sour ಠ_ಠ

------
maxander
What I had taken away as the big impact of the original "amplituhedron"
article was that the researchers had found a way to replace long calculations
with a shorter, _geometrical_ calculation. Which, unless there's some profound
connection between quantum complexity classes and geometry that I'm not aware
of, wouldn't be the case with the "unitarihedron." So perhaps Aaronson isn't
giving the amplituhedron concept the quite credit it deserves?

(Disclaimer- I haven't the background in complexity nor physics to have any
idea, obviously. But geometrical calculations playing a key role in a theory
that "did away with space" seemed like a wonderfully cool paradox.)

------
dnautics
I don't usually like scott aaronson, (although I like his casting of QM as
complex-valued probability space) - but this blog post was a gem.

~~~
gonnakillme
Casting QM as a complex-valued probability space is how it's usually done.

~~~
loup-vaillant
Calling "probability" something that's out there in the world. What a
confusing thing to do…

[http://lesswrong.com/lw/oj/probability_is_in_the_mind/](http://lesswrong.com/lw/oj/probability_is_in_the_mind/)

~~~
gonnakillme
I'm not really sure what to make of this response.

> The Bayesian says, "Uncertainty exists in the map, not in the territory. In
> the real world, the coin has either come up heads, or come up tails. Any
> talk of 'probability' must refer to the information that I have about the
> coin—my state of partial ignorance and partial knowledge—not just the coin
> itself.

See, QM doesn't actually work like this. There is no hidden variable. This has
been proven[1].

[1]:
[http://en.wikipedia.org/wiki/Bell%27s_theorem](http://en.wikipedia.org/wiki/Bell%27s_theorem)

~~~
loup-vaillant
Generally, when you know everything there is to know about an experiment, it
makes you quite certain about the outcome you will observe.

Yet this is not the case when you send a single photon through a half-sieved
mirror. Even when you know everything there is to know about this experiment,
you just don't know which sensor will go off, and if you repeat the
experiment, you will find that previous results tells you _nothing_ about the
next one. Perfect independence.

When total knowledge of the setup yields only partial knowledge about future
observations, one is very tempted to look for "probabilities" in the setup
itself. But it's a cop-out, a non-explanation. The fact that there is no
hidden variable does not mean amplitudes (or their squared norm) are
probabilities.

So, what to make about our uncertainty about the perceived outcome of quantum
experiments? Well, it's simple: the laws of physics as we know them are
deterministic. When you send a photon through a half-sieved mirror, the actual
result is always the same: the world splits in half. There will be one blob
where the photon hit the first sensor, and one blob where the photon hit the
second one. Subjectively, the result is the same: we still observe the Born
statistics. But step further from the amplitudes as probabilities cop-out: the
Born statistics are now more of an anthropic problem.

------
teeja
Baj. Jumbug. Their hedrons are only special cases of my 42-dimensional
Universahedron, which certifiably and undeniably contains everything,
including all possible universes. The proof is too small to publish in this
post.

------
Groxx
The Simon's Foundation version with discussion:
[https://news.ycombinator.com/item?id=6403285](https://news.ycombinator.com/item?id=6403285)

------
pintglass
I had a unitarihedron hanging from my rearview mirror in high school. It was
pretty awesome.

------
bsullivan01
_Though, in recent months, my research has focused even more on the
diaperhedron_

Damn, we should compare notes, we're researching the same thing :-)

I need to make a mental note that if the mathematician or physicist tries too
hard to get on TV /news and sell something, he might not be as great as
others.

------
damianknz
The article is definitely satirical and points out the futility of any -Hedron
based device regardless of its theoretical truth. You can't calculate the
volume of a fractal in fact some fractals can be proved to have zero volume!

