So what’s going on at QIP 2011? Anyone? Bueller? Bueller?
update: It looks like pdfs of the talk slides are available. Were the talks videotaped (err, I guess I’m showing my age: were the talks recorded in video format?)
more update: John Baez has a post on a few talks.
O brave new quantum world!
There is apparently no arxiv version (yet) of the QIP 2011 talk by David Poulin, Angie Qarry, Rolando Somma and Frank Verstraete, titled Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. Comments on their presentation would be very welcome …. this line of research is closely coupled to practical applications and thus is immensely interesting to engineers.
Thank you for the pointers to the literature, Aram … Boixo & Somma’s Necessary Condition for the Quantum Adiabatic Approximation (arXiv:0911.1362) had in it precisely the physical background that I was looking for.
One wonders if this line-of-reasoning could be reversed. That is, we suppose that a certain adiabatic quantum algorithm is observed to work very efficiently, and we further suppose that it is known to have a fixed-scale energy gap. Then by the Boixo & Somma result (and by physical reasoning) we infer that the algorithm is traversing a short-path quantum trajectory.
Under what circumstances (if any) could knowledge that a short-path problem-solving quantum trajectory exists help in deducing classical simulation algorithms? More broadly, we are led to wonder, under what circumstances (if any) can quantum computers (adiabatic or otherwise) exponentially speed-up the search for efficient classical simulation algorithms?
Without knowing the answer to the above (hard) question, I will note that there is older work by Ford, Lewis & O’Connell, relating to quantum Langevin models of heat baths, that embodies many of these same physical ideas.
I was just at this talk. Basically it gives a way to simulate a time-dependent Hamiltonian H(t) even when H(t) is changing arbitrarily quickly. The idea is that very high-frequency oscillations can be smoothed away without affecting the evolution very much. Here “high-frequency” means that the frequency is >> the average energy of the state.
It’s in some ways related to the nice line of work that Rolando Somma and collaborators have been doing on rigorous version of the adiabatic theorem, as well as quantum algorithms for achieving the goals of the adiabatic theorem. So if you’re interested in this work, you might contact the authors to ask for a preprint, but could also like at Rolando’s previous papers.
Thanks for the alert! Especially interesting (to us engineers) were the video and slides for David Poulin’s talk on Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space … in his talk David called it “the delusion of Hilbert space” … with its conclusion
There was a concluding question from the audience that I thought was excellent, asking whether these considerations apply if the initial state (arising for exampl from the Big Bang) is outside the manifold of physical states.
It is folklore among physicists that this question has an answer that is both geometrically and physically natural. From a geometric point-of-view, the dynamical flow assumed by Poulin, Quarry, Somma, and Verstraete is purely symplectic. Suppose we generalize this flow to the mixed symplectic and metric flow that is generically associated to Lindblad dynamics.
Then the folklore of physics is that Lindbladian flows generically compress arbitrary initially states in Hilbert space onto the doubly exponentially small, and hence algorithmically compressible, submanifold of physical states. This is the geometric view of the physical process that Zurek calls einselection.
It would be very enjoyable to see a rigorous mathematical proof of an einselection theorem, via Poulin’s methods, in some toy model … perhaps we can look forward to further work along these lines from Poulin and his collaborators.
More broadly, our QSE Group’s experience has been that is very useful for engineering students to appreciate that effectively the state-space of quantum systems is “hollow” (as Poulin’s slides call it) and/or “foamy” (a description that for engineering students works better pedagogically)—such descriptions are increasingly useful, as the engineering curriculum puts increasing emphasis on geometrically natural descriptions of classical and quantum dynamics.
Of course, in practical quantum simulation codes the state-space of physical states is not only geometrically foamy, but dynamical in its own right, as the simulation codes continuously adapt their basis states to the physics of the system.
It remains to be see whether Nature exploits this same computational trick, to render herself easy to simulate; thus Poulin’s line of research is thought-provoking from both a practical and a fundamental point-of-view.
Yes, videos of the talks are available here.
In cross-referencing the results that David Poulin presented, relating to what Poulin, Qarry, Somma and Verstraete call “the convenient illusion of Hilbert space”, there are natural links to two recent works by Johnson, Biamonte, Clark, and Jaksch, namely Dynamical simulations of classical stochastic systems using matrix product states (arXiv:1006.2639) and Categorical Tensor Network States (arXiv:1012.0531).
It is great fun to see what amounts to the same physical ideas presented alternatively in the language of traditional quantum mechanics, computer science, linear algebra, information theory, Lindbladian dynamical flow, and now. category theory.