Mistake of the Week: “X works on paper, but not in the real world”

Time again for Shtetl-Optimized’s Mistake of the Week series! This week my inspiration comes from a paper that’s been heating up the quantum blogosphere (the Blochosphere?): Is Fault-Tolerant Quantum Computation Really Possible? by M. I. Dyakonov. I’ll start by quoting my favorite passages:

The enormous literature devoted to this subject (Google gives 29300 hits for “fault-tolerant quantum computation”) is purely mathematical. It is mostly produced by computer scientists with a limited understanding of physics and a somewhat restricted perception of quantum mechanics as nothing more than unitary transformations in Hilbert space plus “entanglement.”

Whenever there is a complicated issue, whether in many-particle physics, climatology, or economics, one can be almost certain that no theorem will be applicable and/or relevant, because the explicit or implicit assumptions, on which it is based, will never hold in reality.

I’ll leave the detailed critique of Dyakonov’s paper to John Preskill, the Pontiff, and other “computer scientists” who understand the fault-tolerance theorem much better than a mere physicist like me. Here I instead want to take issue with an idea that surfaces again and again in Dyakonov’s paper, is almost universally accepted, but is nevertheless false. The idea is this: that it’s possible for a theory to “work on paper but not in the real world.”

The proponents of this idea go wrong, not in thinking that a theory can fail in the real world, but in thinking that if it fails, then the theory can still “work on paper.” If a theory claims to describe a phenomenon but doesn’t, then the theory doesn’t work, period — neither in the real world nor on paper. In my view, the refrain that something “works on paper but not in the real world” serves mainly as an intellectual crutch: a way for the lazy to voice their opinion that something feels wrong to them, without having to explain how or where it’s wrong.

“Ah,” you say, “but theorists often make assumptions that don’t hold in the real world!” Yes, but you’re sidestepping the key question: did the theorists state their assumptions clearly or not? If they didn’t, then the fault lies with them; if they did, then the fault lies with those practitioners who would milk a nonspherical cow like a spherical one.

To kill a theory (in the absence of direct evidence), you need to pinpoint which of its assumptions are unfounded and why. You don’t become more convincing by merely finding more assumptions to criticize; on the contrary, the “hope something sticks” approach usually smacks of desperation:

There’s no proof that the Earth’s temperature is rising, but even if there was, there’s no proof that humans are causing it, but even if there was, there’s no proof that it’s anything to worry about, but even there was, there’s no proof that we can do anything about it, but even if there was, it’s all just a theory anyway!

As should be clear, “just a theory” is not a criticism: it’s a kvetch.

Marge: I really think this is a bad idea.
Homer: Marge, I agree with you — in theory. In theory, communism works. In theory.

Actually, let’s look at Homer’s example of communism, since nothing could better illustrate my point. When people say that communism works “in theory,” they presumably mean that it works if everyone is altruistic. But regulating selfishness is the whole problem political systems are supposed to solve in the first place! Any political system that defines the problem away doesn’t work on paper, any more than “Call a SAT oracle” works on paper as a way to solve NP-complete problems. Once again, we find the “real world / paper” distinction used as a cover for intellectual laziness.

Let me end this rant by preempting the inevitable cliché that “in theory, there’s no difference between theory and practice; in practice, there is.” Behold my unanswerable retort:

In theory, there’s no difference between theory and practice even in practice.

46 Responses to “Mistake of the Week: “X works on paper, but not in the real world””

  1. Anonymous Says:

    This all seems like little more than semantics. It’s like he’s saying: “Your algorithms for fast factorisation work under these assumptions, but these assumptions don’t hold in the real world. Nevertheless, the algorithms are still valid algorithms, like Euclid’s algorithm is a valid algorithm on the assumptions that certain primitives can be performed, irrespective of whether someone builds a computer to perform those primitives or not…”

    Questions to Scott: do you do any programming, whether as a hobby or as part of your research? If quantum computers couldn’t be built (which would be some what surprising), would you still want to do Quantum complexity research?

  2. Scott Says:

    This all seems like little more than semantics.

    I agree! But when an invalid distinction (between “working on paper” and “working for real”) is repeatedly invoked by others as a substitute for thought, what choice do I have but to fight bad semantics with good? :)

    do you do any programming, whether as a hobby or as part of your research?

    I used to do quite a bit! (See for example my papers on hypertext, stylometry, Boolean function algorithms, or stabilizer circuits.) And while I haven’t done much coding for years, I’m hoping to implement my algorithm for learning quantum states as soon as I find the time.

    If quantum computers couldn’t be built (which would be some what surprising), would you still want to do Quantum complexity research?

    If a fundamental reason were discovered why quantum computers can’t be built, I’d probably drop everything I’m doing and switch to studying the enormous new questions in CS and physics that such a discovery would open up.

  3. Anonymous Says:

    Scott, notice that Dyakonov used the word “theorem” and you immediately start talking about “theory” as if the two were synonymous. Noo,noo,noo.

    To me, a theory is one or more theorems , with some “alleged” instances/predictions of those theorems. (Much can go wrong with a theory: the theorems might be incorrect or the alleged instances/predictions may be unjustified.)

    So when people compare theory with
    experiment, they are not comparing theorem with practice(which makes no sense), they are comparing predictions with practice. “In theory, there’s no difference between theory and practice”
    Noo, noo, noo.

  4. Jacques Distler Says:

    “Works in the real world” could mean lots of things, not all of which are necessarily incompatible with “works on paper”. (This is not to defend Dyakonov, whose paper I have not read.)

    For instance, say building a quantum computer, of certain specified capabilities, were not strictly impossible, but merely very, very expensive.

    Josephson junction (classical) computers is one such idea which “worked on paper”, but not “in the real world”, in the sense that it never seemed cost-effective to try to build one.

    Also, it’s not clear to me that, in our current state of knowledge, we are necessarily capable of deciding whether the assumption underlying a certain theoretical construct are “unfounded.”

    Say that theory requires properties A, B, and C to hold. We may know of physical systems (candidates for building a quantum computer) in which A and B hold, others in which B and C hold, and yet others in which A and C hold. We don’t know of any system in which A,B and C hold simultaneously. It’s plausible that such a system will eventually be found, but it’s equally possible that none will be.

    What can we say, in that situation? We can’t criticize the individual assumptions A, B or C (after all, they all hold in some known systems). At best, we could grouse that we know of no system in which they all hold simultaneously. But that’s little more than the complaint that no one’s figured out how to build a practical quantum computer yet.

    As a formal theorem (“on paper”) one can hardly complain about this theoretical construct. Certainly, it would be senseless to try single out one of the assumptions as “flawed.” But it is still not necessarily unreasonable to express doubt that this particular construct can be realized “in the real world.”

  5. mick Says:

    Scott, your link to “the Pontiff” in the 1st paragraph after the 1st quote goes back to the paper, not to Dave’s post (whichever one you were going for). Just thought I’d let you know.

    Oh, and nice post. Drives me nuts how people line up to criticize a theory without thinking about the implications of it being wrong. It’s especially pertinient in this case, if the theory of FTQC is wrong for fundamental reasons then a lot of the research done into quantum physics in the last 20 or 30 years is pretty shaky. A lot of critics don’t like to face that point.

  6. Scott Says:

    Thanks, Mick! Fixed.

  7. Mahmed al-Gore Says:

    Although it is out of fashion, it might be helpful to read Kant and Schopenhauer. They made an important and useful distinction between perceptions and concepts. Perceptions are based on real life experience. Concepts are mere thoughts that are derived from those perceptions.

  8. John Sidles Says:

    Another way to say it is: “When you’re younger, you work on theory, when you’re older, you work on practice.” Here “younger” means, “younger than when you were older”.

    As an example, here’s a quote from John von Neumann, who worked on (classical) computational theory when he was younger, and went on to build practical computers when he was older. The latter was harder!

    This is the older von Neumann speaking:

    “Those of you present who have lived with this field, and who have lived with and suffered with computing machines of various sorts, and know what kind of regime it is to invest in one, I’m sure have appreciated the fact that it appears that this machine has been completely assembled less than two months ago, has been run on problems less than two weeks ago, and yesterday already ran for four hours without making a mistake. Those of you who have *not* been exposed to computing machines, and who do not have the desolate feeling which goes with living with their mistakes, will appreciate what it means that a computing machine, after about two weeks of breaking in, has really a faultless run of four hours. It is completely fantastic on an object of this size; I doubt it has ever been achieved before, and it is an enormous reassurance regarding the state of the art and regarding the complexities to which one will be able to go in the future, that this has been achieved.”

    This post wouldn’t be “Aaronsonian” if it didn’t include an actual recording of von Neumann saying these words (click here). Enjoy the Hungarian accent!

    Kinda gives you hope for quantum computing, doesn’t it?

    Details—including the full 20-minute recording of von Neumann’s entire speech—can be found on our QSE Group’s Roadmaps from the 1940s and 1950s web page.

  9. Scott Says:

    John, one could also give plenty of examples of people who became more “theoretical” as they got older: Einstein, Bohr, Bohm, Darwin, E. O. Wilson…

  10. Anonymous Says:

    “people who became more “theoretical” as they got older”
    Schroedinger too. I’m sure he became more theoretical about sex as he got older.

  11. Dave Bacon Says:

    Did you just call me a computer scientist? (Looks up from his computer and the Paul Allen Center) Oh.

    BTW I looked up “work on paper but not in the real world” in my physics translation dictionary and found “you left out some important physics you shmuck!”

  12. John Sidles Says:

    Scott sez: John, one could also give plenty of examples of people who became more “theoretical” as they got older

    Hmmmm … further evidence that “The opposite of a great truth is also true” (a famous Niels Bohr maxim).

    More seriously, Scott, what would you say is the complexity class of the question you posed, Did the theorists state their assumptions clearly, or not?

    Recent research suggests that this question might be intrinsically difficult for human cognition, as evidence accumulates that the boundaries of a person’s knowledge domain is typically a subject of which they are confidently ignorant!

  13. Scott Says:

    Scott, notice that Dyakonov used the word “theorem” and you immediately start talking about “theory” as if the two were synonymous.

    I assumed Dyakonov was taking issue not with the fault-tolerance theorem (which hopefully no one disputes), but with the stronger claim (call it (*)) that the theorem tells us what it’s claimed to about the physical world, given the laws of physics as we currently understand them. My point was a trivial one: that barring a revolutionary discovery in physics, if (*) holds then nothing additional is asserted by saying that (*) is “true in the real world,” and equivalently, if (*) is not true in the real world, then (*) can’t hold even on paper.

  14. Scott Says:

    BTW I looked up “work on paper but not in the real world” in my physics translation dictionary and found “you left out some important physics you shmuck!”

    Thanks, Dave! That entry is every bit as helpful to me as the ones for “must,” “only,” “always,” “certainly,” and “true.”

  15. Nagesh Adluru Says:

    one could also give plenty of examples of people who became more “theoretical” as they got older: Einstein, Bohr, Bohm, Darwin, E. O. Wilson…

    My opinion is philosophical would be apt instead of more “theoretical”.

  16. Nagesh Adluru Says:

    Different people can take fluctuations in perceptions at different levels. Many conceptualize perceptions so as to create atleast some standard grounds.

    Now there are two categories: One trying to discover a completely standard ground (like unified theory) and others who giveup and exploit the current set of standards.

    The first category grows philosophical as they grow older. The others grow more practical. Either way they would be doing what they were not.

    It’s all what one gives importance to discovery or exploitation.

  17. Preston Pennypacker Says:

    There’s a section called “Is P really practical?” in the Wikipedia article “Complexity classes P and NP”. It claims that it is true in theory that P = easy and NP = hard. It also claims that it is not true in practice.

  18. tom s. Says:

    I read through the paper and (as an outsider) it seems to me to make some good points, especially in the later parts (you quote from the introductory portion, which is not quite fair, as it is where people tend to be relatively loose and informal).

    And I agree with him, as Scott knows from our brief discussion after my own post at http://whimsley.typepad.com/whimsley/2006/10/quantum_computi.html.

    You say, Scott, that “To kill a theory (in the absence of direct evidence), you need to pinpoint which of its assumptions are unfounded and why.”

    But QC has made itself indestructible by making very few assumptions. Direct evidence of the impossibility of building a quantum computer is a logical impossibility. And when you work with state vectors |1> that could represent anything from the spin of a carbon-13 nucleus to the polarization of a photon to the enantiomeric state of a water-buffalo — well, proving something impossible is difficult.

    Surely there comes a point where you admit that anything with so little specified about the real world is not physics, but is simply the geometry of Hilbert Space?

    Perhaps you could at least consider narrowing your criteria of proof. Can you prove that you cannot build a quantum computer out of the enantiomeric states of water buffaloes?

    Perhaps more has been done on the consequence of finite relaxation times than the author admits, but my own ignorant opinions, shared with the author, is that as soon as it is done the scope of things you can build quantum computers from will narrow down pretty quickly.

  19. tom s. Says:

    One thing the author is wrong about. If you now try googling “fault-tolerant quantum computation” it claims to find about 51,600 results, but scroll through the pages and you find that there are only 383 (plus the usual “some very similar”).

    I’ve noticed this before. Once you try to observe the results of a google search, large numbers of them disappear. Never trust without experimental observation!

  20. Scott Says:

    There’s a section called “Is P really practical?” in the Wikipedia article “Complexity classes P and NP”. It claims that it is true in theory that P = easy and NP = hard. It also claims that it is not true in practice.

    Preston: That section actually looks OK. It doesn’t say that P=easy is “true in theory”; it says it’s a “common and reasonably accurate assumption in complexity theory,” which is correct. The limitations of polynomial time are extremely well-known, are part of any complexity course, and should really be considered part of the theory itself.

  21. Scott Says:

    Thanks, Tom! A few responses:

    1. I agree that Dyakonov makes several points worth responding to, and I hope and expect that those more versed than I am in quantum error-correction will take him up.

    2. The most obvious way to falsify the assumptions behind quantum computing would be to falsify quantum mechanics itself, as Gerard ‘t Hooft and others want to do. Alternatively, you could discover some fundamental decoherence process (maybe a gravitational one, as Penrose thinks), and prove that it disallows universal QC.

    3. If the enantiomeric states of water buffaloes can be manipulated in coherent superposition, then presumably they can be used for universal quantum computing. (Now there’s a sentence you don’t read every day!) The water buffalo states can certainly be used for classical computing, which doesn’t imply that anyone will be flocking to the stores for Intel Buffalino MMX laptops.

    4. Which quantum computing architectures (if any) will turn out to be practical is obviously a huge and important question. But at least in his latest article, Dyakonov says that he’s interested not in such practicalities, but in whether quantum computing is possible even in principle. This, of course, is also the question that interests me.

  22. John Sidles Says:

    Nagesh Adluru: My opinion is “philosophical” would be apt instead of more “theoretical”.

    Just to contrdict myself (so I contradict myself), Subrahmanyan Chandrasekhar was a great theorist to his last breath.

    I have a paperback copy of Chandrasekhar’s Stellar Structure, which I occasionally read for pure pleasure of his writing style.

  23. wolfgang Says:

    Scott,

    I think the comments (especially jacques’) make it clear that your statement about this ‘mistake of the week’ may be true on paper, but it does not hold up in the real world 8-)

  24. Gil Kalai Says:

    Scott: “the proponents of this idea go wrong, not in thinking that a theory can fail in the real world, but in thinking that if it fails, then the theory can still “work on paper.” If a theory claims to describe a phenomenon but doesn’t, then the theory doesn’t work, period — neither in the real world nor on paper. In my view, the refrain that something “works on paper but not in the real world” serves mainly as an intellectual crutch: a way for the lazy to voice their opinion that something feels wrong to them, without having to explain how or where it’s wrong.”

    Scott, this is an interesting point. I suggest to discuss it by considering various other examples and to move away from the specific issue of quantum computation.

    My conjecture is that when you say “a theory” you have to distinguish between the “written theory” with the formal aspects, and the “spoken theory” with a lot of additional insights/methods/approximations etc. When you say “but doesn’t” you need to explain what tolerance you allow to the theory itself.

    But I agree with you that to say that a theory “fails in the real world” requires (like most statements with 5 words) much more details.

    And what do you have against the lazy?

  25. Paraphrene Says:

    Theories have a bad reputation because they’re usually wrong. In theory, theories aren’t wrong.

  26. John Sidles Says:

    Perhaps a good example of a whole class of “theories that didn’t work” is the 1950-1980 theories of plasma confinement and instability.

    These were beautiful, mathematically sophisticated theories that gave everyone a lot of hope that power-generating plasma fusion machines were just around the corner.

    Even today, after fifty years of work it is not entirely clear why these theoretical assumptions were inadequate, or what was left out.

    But no, we don’t have energy-producing plasma fusion machines … yet!

    Of course, people haven’t given up: these problems are too urgent. But we now recognize that those early plasma theories were, in a certain sense, too beautiful to be true.

    That would be a different, even more controversial “Mistake of the Week: Your theory is too beautiful to be true.”!

  27. Jo Says:

    Hmmmm … examples of physicist’s designs that engineers can’t build would be a good topic.

    Stable thermonuclear power design:

    (1) Purchase 2×10^27 metric tons of 75% hydrogen, 25% helium (by mass, bulk grade)

    (2) Allow to gravitationally self-assemble.

    (3) Verify that core temperature is adiabatically heated to 13.6 MK.

    (4) Profit!

  28. Anonymous Says:

    Mahmed al-Gore is right that “it might be helpful to read Kant” (is this really out of fashion? hadn’t noticed), but not because of his “distinction between perceptions and concepts.” Rather, the tenor of the posting is completely that of an entire independent work of Kant, the celebrated “Über den Gemeinspruch: Das mag in der Theorie richtig sein, taugt aber nicht für die Praxis” (1793), especially the preface – the point is that nothing can work in theory if it doesn’t work in practice, because then the theory is wrong. (Of course, you have this thought already in the Presocratics, but this Kant piece is really the main essay to focus on the fallacy.) And of course, one of the important spillovers of this observation is that modelling that assumes away vital features of a phenomenon (such as, e.g., mainstream economics does not by accident, but as a matter of course) is self-referential and, indeed, frivolous.

    Ebbinghaus

  29. Scott Says:

    See, all you naysayers: in attacking the real world / paper fallacy, I was preceded by none other than the illustrious Immanuel Kant, who made pretty much the same argument I did! Now won’t you listen? :)

  30. Mahmed al-Gore Says:

    Herr Ebbinghaus, In reference to the Kant book: “On the Old Saying: That Might be Right in Theory but it Won’t Work in Practice”, please note the following. Concepts are equated with theory or mere thought. Perceptions are equated with practice, or the real world of experience. His distinction between perceptions and concepts, and his subsequent discussion of their difference, are the basis of the aforementioned book.

  31. Cheshire Cat Says:

    Scott, do you agree with Ebbinghaus that mainstream economics is pseudo-science?

  32. scott Says:

    Only 80% of it.

  33. Anonymous Says:

    Scott, do you agree with Ebbinghaus that mainstream economics is pseudo-science?

    In my book economics is the one of the pseudo-science (aka the humanities) that is closest to making a transition to actual hard core science. It is then surprising that Ebbinghaus singles it out as a target.

    Moreover the primitiveness of economic models does not seem a pernicious practice of the field. That is to say, it’s not like we have much better models ready to be used but economists choose to pass on them.

    Economists are very aware of the limitations of their model, but so far that is the best they got. In fact, several economists have won Nobel prizes by proposing refinements such as bounded rationality, so it is clear that economists are actively seeking to make their models better.

  34. Anonymous Says:

    Kant’s “distinction between perceptions and concepts, and his subsequent discussion of their difference, are the basis of the aforementioned book”? Hardly. This is neither what the preface (in which this is dealt with), nor – indeed let alone – what the book is about.

    As regards mainstream economics (with which I mean STE), the first question is of course what “scientific” yardstick you put to it. Of course, it is very scientific in its physics envy (pre-Heisenberg, that is), but absolutely anti-scientific and not only un-scientific if you agree with a necessary reality connex. Only in economics would you have someone, in his inaugural lecture at Cambridge, say, “I am not interested in the economy, I am interested in economics” – in the sense that no attempt is made to be in touch with “reality” (however defined, and that economics that does not deal with the economy anyway changes the economy, which is largely a matter of perceptions, is also true).

    In any case, the problems of economics in this regard are the subject of a huge body of literature and of a huge fight, in which STE is, well, not yet on the defensive, but getting there. http://www.paecon.net, while occasionally a bit loony and sectarian, gives good links and texts. But contrary to what was said just above, I think that precisely “bounded rationality” is just what has been called the icing on the neo-classical cake, i.e. it is not a serious attempt (overall rather than individual) to change the methodological self-referentiality of STE.

    As for why the latter exists, for the context of the current blog, I think two classic essays that most readers would be familiar with is Einstein’s Geometry lecture from 1924 (I think it was 1924?) and Frege’s review of Mr. Schubert’s numbers.

    Ebbinghaus

  35. John Sidles Says:

    Being tempted to post a serious reply, I first took counsel from Aaronson’s Rule: A serious post to a blog must be either deliberately provocative or wrapped in humor.

    Not being particularly gifted at humor (unlike Scott, who has a deft humorous touch) I will provocatively extend Scott’s “Mistake of the Week” principle to this: “If theory X is taught in engineering courses, then it works.”

    Let’s check some prominent theories to see if they play major roles in practical engineering:

    • Newtonian mechanics: ✔
    • Continuum mechanics: ✔
    • First-quantized mechanics: ✔
    • Second-quantized mechanics: ✔
    • Thermodynamics: ✔
    • Electricity and magnetism: ✔
    • Special relativistic mechanics: ✔
    • General relavistic mechanics: ✔ (GPS)
    • Classical information theory: ✔

    Note: I may not have a deft humorous touch, but hopefully, most of you are admiring my deft touch with HTML dingbat codes (“✔”)!

    Then come our two “mood-breakers” (as they call them in trauma surgery):

    • Quantum information theory: ???
    • String theory: ???

    I will provocatively suggest that there is a wonderful opportunity here. That opportunity being, of course, to invent practical engineering disciplines that are founded upon the fundamental techniques of quantum information theory and string theory. That will answer those pesky critics who claim these theories “don’t work in the real world”!

    More seriously, I chose to call this opportunity “wonderful” because it’s an opportunity create jobs that do good—which is what engineering is all about.

    If you don’t think job-creation deserves to be called “wonderful”, then you’re probably NOT a young scientist, mathematician, or engineer who is looking for a job!

    Of course, now we have to answer that irksome question “How exactly are we going to create new jobs that do good?”

    I will provocatively suggest that Dyanokov’s recent preprint shows the way. Dyanokov is right (as I understand his polemic argument) to ask that algebraically-derived theorems in quantum information theory be implemented as continuum numerical simulations.

    The rationale being, that real-world noise mechanisms can be far more readily and generally embedded in such simulations, than they they can be in purely algebraic quantum formalisms. This real-world flexibility is why such simulations have become an essential tool in every branch of engineering (and quantum chemistry too).

    In consequence, device engineers strongly prefer to conceive their designs in terms of the geometry of continuum processes. This is not wrong and not stupid of them, but rather it is a cognitive strategy that works so well in the nonideal real world, as to be nearly mandatory.

    The word “geometry” brings me to string theory. AFAICT, the most natural geometric language for quantum information theory is the language of Kahler manifolds. Which of course, is the same as the natural language of string theory.

    So my final provacative statement is this: someday seminars on Calabi-Yau manifolds will be conducted in mechanical and electrical engineering departments, and the engineers who attend will grasp the geometry of these manifolds very easily and naturally, in the context of existing engineering techniques for model order reduction.

    Having already given one such seminar (which was pretty well-received), I can pretty confidently predict that once such seminars become common, no-one will ask anymore whether quantum information theory “works”, or whether string theory “works”. Hoorah!

    An even more important consequence will be that these engineering tools will create—especially for you younger folks—plenty of new jobs that do good. Happy Holidays!

  36. Greg Kuperberg Says:

    AFAICT, the most natural geometric language for quantum information theory is the language of Kahler manifolds.

    The only Kahler manifold that I have ever seen in quantum information theory is complex projective space. It is just as pleasant as any other Kahler manifold, but it is also among the simples Kahler manifolds and it certainly isn’t Calabi-Yau. Barring some new idea which may or may not be there to be found, this connection is a non-starter.

    What is true is that string theory and quantum information theory are both great topics. So is algebraic geometry. Boasting that they are all the same thing is irresponsible.

  37. John Sidles Says:

    Dear Greg — There’s was surely no intent to boast. Just to be clear, pretty much everything in my post is elementary knowledge to complex geometers (although what these geometric folks know can seem pretty amazing to mere mortals).

    Physical theories often (always?) involve complex submanifolds of Hilbert space—the full Hilbert space being awkwardly large-dimensional—and so it is very natural to ask about the geometry of submanifolds.

    Mathematicians regard as most beautiful those complex manifolds that are without singularities. It is helpful (to me) to regard these most-beautiful complex manifolds as (loosely) the geometric analogs of entire analytic functions (i.e., complex functions with no cuts or poles).

    It then turns out, that a very remarkable geometric property (to us engineers) of the product-state manifolds that arise most naturally in quantum model reduction is that these Hilbert submanifolds have constant scalar Ricci curvature.

    That is, even though these manifolds are very bumpy, their bumps are as smooth as possible under Ricci flow (kinda like the elementary trigonometric functions are the smoothest periodic functions). Formally speaking, they are what is known as Kahler-Einstein manifolds.

    This I know, not by deep geometric understanding, but rather by brute-force symbolic calculation of the curvature tensors.

    These singularity-free complex manifolds are geometrically beautiful, but they are too restrictive for engineering purposes. It turns out, that when we generalize these product subspaces to product-sum subspaces, then we unavoidably introduce geometric singularities (kinda like defining complex functions that have poles and branch cuts).

    The resulting complex manifolds with singularities are less interesting to pure mathematicians, because the geometric singularities are obstructions to, e.g., defining topological Chern classes. For this reason there is not much mathematical literature about them (at least, not much mathematical literature that I have a reasonable hope of understanding).

    But to engineers, complex manifolds become MORE interesting when we introduce singularities, because we can design these singularities to help us generate practical design simulations (this is like moving branch cuts around in complex functions; it is a creative art).

    And one final point: even though these manifolds are very bumpy, they also are richly endowed with paths that are geodesics (rays) of the embedding Hilbert space. The existence of these rays is what makes efficient computing possible—this provides a geometric view of how, e.g., matrix-product state calculations can be both accurate (due to the geometric curvature) and numerically efficient (due to the rays).

    I learned all this during an attempt to program computer-generated pictures of these manifolds. It slowly sank in that I was drawing pictures of manifolds similar to Calabi-Yau manifolds, but of much higher dimension. No wonder it was tough!

    The point I’m making is, there is a huge body of literature on complex geometry that provides very reasonable conceptual foundations for understanding and simulating large quantum systems.

    Algebra and geometry have always enjoyed a very fruitful partnership in mathematics — I do believe that this will happen in quantum information theory too. Neither algebraists nor geometers should feel threatened by this; the prospect is pretty thrilling IMHO.

    If anyone is interested to read further, two good starting points (for physicists) are Weinberg’s book on gravitation for the basic formalism of curvature tensors, followed by:

    @book{Martin:02,
    author = {D. Martin},
    title = {Manifold Theory: An Introduction for Mathematical
    Physicists},
    publisher = {Horwood Publishing Limited},
    year = 2002,
    jasnote = {Good introduction to complex manifolds, draws
    quite heavily on Flaherty:76},
    }

    @book{Flaherty:76,
    author = {E. J. Flaherty},
    title = {Hermitian and K”{a}hlerian Geometry in Relativity},
    publisher = {Springer-Verlag},
    year = 1976,
    jasnote = {Good introduction to complex manifolds},
    }

    For me, Flaherty’s book was best (but it is hard to find).

    I am writing a review article on quantum model order reduction (with plenty of numerical examples) which is centered on this geometric point of view. The numerical examples turn out to work amazingly well — much better than I can presently account for (kinda like matrix product state calculations). So I am strongly motivated to present these geometric ideas as simply as possible … this simplicity turns out to be not so easy to achieve.

    I’ll close with a Fermi quote (from memory): “For me, there are two kinds of mathematics books. The ones where I don’t understand the first page, and the ones where I don’t understand the first sentence.” I will freely admit that my present geometric understanding is about page 10 or so … there are hundreds of pages of known geometric theory still to master.

  38. Bram Cohen Says:

    Scott, could you diagram that last sentence for us?

  39. Scott Says:

    In theory, ((there’s no difference between theory and practice) even in practice).

  40. Douglas Knight Says:

    In fact, several economists have won Nobel prizes by proposing refinements such as bounded rationality,

    Actually, a psychologist won the prize for bounded rationality.

  41. John Sidles Says:

    Several economists have won Nobel prizes by proposing refinements such as bounded rationality …

    … a concept for which strong support is emerging from evolutionary biology and neurophysiology. Today’s NYT Science Section is full of examples:

    ——

    Evolutionary Theories of Right and Wrong

    “People are generally unaware of [moral bias] because the mind is adept at coming up with plausible rationalizations for why it arrived at a decision generated subconsciously.”

    Certain Areas of the Brain Size Up Your Competition

    “Humans spend a lot of time sizing each other up — a fact long known to social scientists. But a new study has pinpointed the brain areas that appear to be involved in this process of social comparison. … The study, led by Caroline Zink, a postdoctoral fellow at the National Institute of Mental Health, found that several brain regions showed increased activity when people were evaluating their standings in a social hierarchy.”

    ——-

    Being highly intelligent and highly trained, isn’t it pretty much hard-wired that scientists and mathematicians will be much more “adept at coming up with plausible rationalizations” than most people? While paradoxically, remaining resolutely unconscious of it? Especially when scientists and mathematicians are “evaluating their standings in a social hierarchy”?

    This is what makes the study of the agnotology of science—and being human in general—so richly rewarding and intricate (see, e.g., reactions to Dyanokov’s article).

    More humorously, this subject is the theme of a a classic Simpsons episode.

    Like it or not, these built-in mechanisms of human cognition are as central to the practice of science as are theory, algorithms, and hardware.

    We might as well acknowledge them gracefully, and especially, be grateful that both peer review and participatory democracy work as well as they do!

  42. John Sidles Says:

    WHoops! Overlooked another link between cognition, biology, and information theory in that same NYT edition:

    Computing, 2016: What Won’t Be Possible?

    Algorithms are small but beautiful,” Dr. Karp observed. And algorithms are good at describing dynamic processes, while scientific formulas or equations are more suited to static phenomena. Increasingly, scientific research seeks to understand dynamic processes, and computer science, he said, is the systematic study of algorithms.

    Biology, Dr. Karp said, is now understood as an information science. And scientists seek to describe biological processes, like protein production, as algorithms. “In other words, nature is computing,” he said.

    —–

    Seems to me, there are very few bounds to the applicability of this principle.

  43. Bram Cohen Says:

    Thank you Scott. I wasn’t sure if you meant

    (In theory, (there’s no difference between theory and practice)) even in practice.

    Because in theory there are differences and in practice which doesn’t happen in theory.

  44. destiny's manifold Says:

    j. sidles,

    there are also Chern classes for singular varieties.
    arxiv or mathscinet search for Chern-MacPherson-Schwarz (in various spellings) or Brasselet.
    No idea if it’s useful for QM.

  45. John Sidles Says:

    Destiny’s Manifold sez: there are also Chern classes for singular varieties …

    I was afraid of that.

  46. later-remark Says:

    It is possible that there is a gap between theory and practice.

    In such a case we have to look for the theoretical reasons for this gap.
    In such a case we also have to look for the practical reasons for this gap.

    It is possible that there is a gap between the theoretical reasons and practical reasons.

    In such a case we have to look for the theoretical reasons for this gap.
    In such a case we also have to look for the practical reasons for this gap.

    It is possible that there is a gap between the theoretical reasons and practical reasons.

    Very late remarks – the ultimate weblog experience