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The Unitarihedron: The Jewel at the Heart of Quantum Computing (scottaaronson.com)
76 points by benjoffe on Sept 20, 2013 | hide | past | web | favorite | 45 comments



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.)


The way I read the joke is that amplituhedron seems to have done very well in giving overnight success to a breakthrough that has taken long painstaking work and has been a field long before it was media-branded, he is pointing out that the same work that spawned the amplituhedron has related work in quantum computing that is similarly important and groundbreaking, so it should have a cool name as well, and n-hedron seems to work.


The construct is a complexity class, meaning that it is a set of problems which can be[1] solved by quantum computers[2] in a reasonable[3] time. Those three aspects correspond to words which get initialized as the name of this complexity class -- BQP, bounded-error quantum polynomial-time.

Aaronson's proof involves an observation that PostBQP = PP, in other words, all of the problems you can solve with an especially powerful quantum computer can also be solved with an even more especially powerful classical computer, and vice versa. Since the quantum computer is easier to reason about, you get a hard result as a freebie.

BQP is not a tongue-in-cheek thing; it's also not a well-understood thing -- just "anything which quantum computers can do in this way quickly with bounded errors" -- but it's certainly better understood than this amplituhedron is. The real problem he's highlighting is that complexity classes like BQP have scary names.

For the sake of this discussion, you can imagine a similar problem with "programmable imperative syntax" and "monads". Really all that monads are, in a computing context, is a way of formalizing a programmable imperative syntax as a purely functional programming construct. The issue is that when you say "functional" and "monads" then people kind of just shut down, they don't want to hear about it. Does that make sense?


> eval(x)

Sort of like the number omega, which contains in each digits proofs to every theorem, but is not computable:

http://en.wikipedia.org/wiki/Chaitin's_constant


Also Greenspun's 10th rule of programming: http://c2.com/cgi/wiki?GreenspunsTenthRuleOfProgramming


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


That's actually where I got the idea most clearly expressed to me in school; I just figured the Javascript thing was a bit more approachable, and conveyed what I was trying to say accurately enough.


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.


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


> It is literally the most powerful Javascript framework ever

It doesn't meet the definition of a framework: http://en.wikipedia.org/wiki/Software_framework

> It trivially encompasses every possible Javascript program

Not every one.


slow golf clap

Wow... yes... you really straightened me out there. Way to go.

Blast. And I had such high hopes that my framework would become the Next Trendy Javascript Thing.


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?


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.


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.


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 -- probably wouldn't be exactly what he described it as.)


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

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



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?


There is an archive of his quotes related to various topics:

http://www.its.caltech.edu/~mason/wisdom/predrag.txt

As he has not objected to this in the ten or so years it's been around, I doubt he'd object to my quoting him either. I thought what he said was illustrative.


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.


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.


You don't need authorization to quote someone.


It is 100% a joke. He seems to be pointing out that you can get the press to over hype a scientific/mathematical advance by giving it a neat name, e.g. dark matter, anti matter, god particle, quantum-anything, electro-anything, super-anything, hyper-anything.


But wasn't the point of the other one that representing it as a multi-dimensional volume was novel? That had nothing to do with the name. What's the point of mocking the name?


He's not mocking the name or the finding, he's mocking the press coverage. This seems clear to me from the opening paragraph, maybe I'm missing something?


I don't see how he's mocking the coverage and not the concept itself. Shrug.


I was actually wondering whether the amplituhedron might possibly have some impact on quantum computing... but I'm not even smart enough with the math to know whether that's a stupid question. Now I don't even know if what I just read was satire.


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.


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


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


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.)


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.


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


I know that now but it was his specific casting that was entertaining and easy to understand. sorry, I'm a chemist - we just talked about and played around with wave functions (schrodinger formulation, not even heisenberg) without a word of referring to them as PDFs except in the psi*psi case (although as a math major in retrospect it's now plainly obvious to me that that's what they were).


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/


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


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.


Out of curiosity, what don't you like about Scott Aaronson? I find him to be really good at making parts of TCS accessible to laypeople, his new book being a good example.


What is QM? Quantum Mechanics?


Correct.


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.


The Simon's Foundation version with discussion: https://news.ycombinator.com/item?id=6403285


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


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.


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!




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