Actually, *all* the essays on Cohn’s website are well worth reading, and I wish to specially recommend the essay Why symplectic geometry is the natural setting for classical mechanics.

Cohn’s essay can be translated into a parallel essay Why symplectic geometry is the natural setting for quantum mechanics pretty much automatically … it is only necessary to replace the natural almost complex structure that is associated to cotangent bundles manifolds with the natural complex structure that is associated to Kähler manifolds.

We then appreciate why (as mentioned above) the Bloch equation dynamics are a Hamiltonian flow, yet the Bloch equations cannot be derived via a Langrangian variation (try it!) … it’s because the Riemann sphere has the wrong topology to be a tangent bundle manifold.

There’s one minor glitch in Henry’s essay … it incorrectly asserts that the “laws of physics” require that the alternating forms of Hamiltonian mechanics be closed. This is not the case … rattleblocks and Chaplygin sleighs are two well-known examples of classical dynamical systems whose dynamical flow is governed by alternating forms that are not closed.

It’s true, though, that rattleblocks and Chaplygin sleighs violate the Second Law … and there is a considerable literature on the remarkable consequences of this fact.

These various mathematical writings reflect a shared theme that is familiar from complexity theory. As Scott’s writings often emphasize, in complexity theory one has a formal tool, namely reduction theory, that is so broadly effective and elegant relative to all other tools, that by its mere existence it largely determined the course of 20th century research.

Similarly, in the theory of dynamical systems we have a formal tool, namely group theory, that is so broadly effective and elegant relative to all other tools, that by its mere existence it largely determined the course of 20th century research.

The great promise of Lindblad theory as a formal tool for 21st century CS/CT/QIT/QSE is that it is comparably effective and elegant to group theory in reducing the effective dimensionality of dynamical state-spaces, thus rendering them amenable to analysis and simulation.

Moreover, from a purely practical point-of-view, it is “a truth universally acknowledged” that computationally reducible systems are ubiquitous … and symmetric systems are ubiquitous … and that noisy, thermalized, open dynamical systems are ubiquitous too.

Thus we see the outlines of a partial answer to Jeannette Wing’s concluding question in her Five deep questions in computing (2008):

More ambitiously (or crazily), is there a complexity theory that spans both the theory and practice of computing?

That answer is along the lines of “A great many computations are associated to dynamical systems that are computationally reducible and/or symmetric and/or open, and generically these systems can be simulated efficiently.”

How efficiently? By what algorithms? By what hardware? By what software? As described in what texts? Enabling what great enterprises? These are key questions for 21st century CS/CT/QIT/QSE.

Thanks for the tip, I will try to catch up on Lindbald theory when I have get a couple free hours!

More seriously, it was good to hear about the views of Arnol’d and Kolmogorov. I have studied 2.5 of Arnol’d's books, and 0.10 of Kolmogorov’s.

Long ago, I was working on a Ph.D. in math, and despised the Bourbaki-ization of math then going on. It caused me to switch to physics, and then applied math. I wound up in CS. Funny how those things go.

Of course, everything is nonlinear. I highly recommend “Foundations of Modern Analysis” by Dieudonne, who is perhaps the best known member of Bourbaki. As I recall, this book starts with some wisecracks about how things have changed since a book of the same title by Whittaker and Watson was published. Both of these books are excellent expositions of huge parts of math, with zero intersection.

LifeIsChill, learning Lindblad theory—or more broadly, learning the theory of open dynamical systems—is pretty much guaranteed to be enjoyable adventure, provided that one begins with realistic expectations regarding the immense size of the territory to be explored.

One good entry point is a thorough familiarity with Chapters 2 and 8 of Nielsen and Chuang’s Quantum Computation and Quantum Information, augmented by Carlton Caves’ Quantum error correction and reversible operations (1998) and by Caves’ excellent on-line notes Completely positive maps, positive maps, and the Lindblad form (2000, revised 2002 and 2008).

At this point it will be evident that the simplest generator of Lindbladian dynamics is simply the Hamiltonian itself; thus a reasonable course of study is to review what is known about Hamiltonian dynamics from an algebraic, informatic, and geometric point-of-view.

For algebraic dynamics see Dirac’s Principles of Quantum Mechanics (1930). For informatic dynamics see Nielsen and Chuang (op cit). For geometric dynamics—which has by far the largest and most diverse literature—a solid foundation is supplied by Jack Lee’s Introduction to Smooth Manifolds (2003), augmented by Arnold’s Mathematical Methods of Classical Mechanics (1978) and by Mac Lane’s Chauvenet Lecture Hamiltonian mechanics and geometry (1970).

At this point you will be prepared to read, with good understanding and enjoyment, Ashtekar and Schilling’s classic Geometrical formulation of quantum mechanics (1999, arXiv:gr-qc/9706069), and in particular, you will be able to translate freely between algebraic, informatic, and geometric descriptions of dynamical systems. For example, simple answers will be evident to questions that are not commonly discussed in quantum mechanical textbooks, like “Why does Bloch dynamics (and Landau-Lifshitz-Gilbert dynamics too) have a natural Hamiltonian description, but not a natural Lagrangian description?”

At this point you will have achieved only the barest beginning of an integrated mathematical understanding of Kraus/Lindblad dynamics … in the sense that you will not yet be able to fluently translate between algebraic, informatic, and geometric descriptions of open system dynamics … but since nobody at present knows how to do this … why … that’s why they call it “research”!

For sure, though, you will have read a lot of great articles and books, and you will have acquired in integrated mathematical toolset that will serve you well in analyzing pretty much any dynamical system, whether classical or quantum, that naturally ariss in any branch of mathematics, science, or engineering.

So enjoy the great STEM adventure that awaits you!

Prof Aaronson: I think this kind of gets what I am asking from a practical perspective. May be this is what I should have asked. Whether any ‘classical TCS Moore’ law still holds?

http://agtb.wordpress.com/2010/12/23/progress-in-algorithms-beats-moore%E2%80%99s-law/

Your question is a good excuse to search my quotation database …

In the ACM’s Interview with Vladimir Arnol’d we read:

The Bourbakists claimed that all the great mathematicians were, using the words of Dirichlet, replacing blind calculations by clear ideas. The Bourbaki manifesto containing these words was translated into Russian as “All clear ideas were replaced by blind calculations.”

The editor of the translation was Kolmogorov. His French was excellent. I was shocked to find such a mistake in the translation and discussed it with Kolmogorov. His answer was: “I had not realized that something was wrong in the translation since the translator described the Bourbaki style much better than the Bourbakists did.”

What we really want in complexity theory is (of course) the best of both worlds: clear ideas accompanied by rigorous proofs … we are at present quite far from that ideal (obviously) … but (IMHO) there is no reason to be downcast about the rate of progress overall.

LifeIsChill, perhaps what you have in mind to ask is more aligned with Jeannette Wing’s concluding question in her recent Five deep questions in computing (2008):

More ambitiously (or crazily), is there a complexity theory that spans both the theory and practice of computing?

My own opinion (or rather, hope) is that at least one key element of Wing’s envisioned theory already exists in embryonic form … it is Lindblad theory.

If I understand your viewpoint: we are beginning to scratch the surface of interaction between what mother nature + TCS.

But if one considers only ‘blind math’ based TCS research, apart from the question of infeasible practical computations(NP problems), for purposes of practical utility, is it safe to say that the stage of research in ‘blind math’ approach to TCS is probably close to saturation just as some topics in pure Mathematics are close to complete and satisfactory results and essentially fully classified? By that I mean the amount of investment that could be made to get even lower $/bit or $/joule or joule/bit is probably not worth it just as for practical communications systems the limits of efficient compression and communication in the Shannon sense is for all sane reasons of engineering essentially achieved.

LifeIsChill, the short answer to your question is “no,” and  to appreciate why, let’s review some literature.

In Redner’s Citation statistics from more than a century of Physical Review (2004) we find abundant evidence that “breakthrough algorithms” often require much more than a decade to be recognized.

Consider as a paradigmatic example the most-cited article in the literature of physics and chemistry: Kohn and Sham’s Self-consistent equations including exchange and correlation effects (1965). As Redner’s review establishes, only sparse references to the Kohn-Sham article appeared during its first decade … yet nowadays the frequency of references to this work is still accelerating (forty-six years later).

To appreciate the roots of the Kohn-Sham algorithm in complexity theory and quantum information theory, a recommended starting reference is Kohn’s Nobel lecture Electronic structure of matter-wave functions and density functionals (1999), and in particular two short articles referenced therein: van Vleck’s Nonorthogonality and ferromagnetism (1936) and Kohn’s own Density functional and density matrix method scaling linearly with the number of atoms (1996).

All of these articles—extending back seventy-five year—bear directly upon a subject of central interest to Shtetl Optimized readers, namely, the feasibility and scalability of experimentally preparing and/or computationally simulating highly entangled quantum states, both as inputs to and outputs from quantum computing devices. These are matters regarding which, even today, our mathematical and physical understanding is uncertain and unsatisfactory, and our engineering ability to produce and/or measure these states is deplorably limited.

The preceding examples apply mainly to quantum aspects of theoretical computer science, but similar considerations apply broadly to all aspects of computational math, science, and engineering … Robert Bixby’s Solving real-world linear programs: a decade and more of progress (2002) is a terrific example.

The overall point, LifeIsChill, is that we are still today in the process of appreciating the informatic and computational significance of articles that were written many decades ago; thus a short-sighted focus on “breakthroughs in TCS in the last decade” is more likely to impair one’s overall understanding, than to help it.

In the end, there’s no substitute for reading and reflecting upon the literature oneself … because a strictly uniform consensus as to what work(s) qualify as “TCS breakthroughs” is neither desirable nor feasible at the present time.

I think that is a terrible oversimplification and misrepresentation of Taoist philosophy, though I’m sure you’re not the first to come up with that interpretation.

I think the more useful Taoist lesson to draw for science is something more like this:

Having enough knowledge to understand the system is not the same thing as having enough knowledge to beat the system.