# Simulating complex quantum systems

On a more scientific note, while I enjoyed many of the talks at the workshop, one stood out for me by far: Ignacio Cirac’s.

Briefly, Cirac talked about a bundle of new methods he and others have been developing in order to simulate quantum systems on conventional classical computers.

Doing such simulations is extremely hard in general, and there are very few quantum systems we can simulate effectively.

This failure is important. The ability to simulate classical systems numerically started in the 1930s and 1940s with the advent of computers, and it has completely revolutionized our understanding of those systems. Because of the difficulty in simulating quantum systems, we’re still effectively stuck back in about the mid-40s with those systems, except in a few special cases.

Cirac talked about the new methods he and many others are developing, based on quantum information theory, for simulating such systems. The details are complex, but the idea is pretty simple: that to do such simulations effectively, the programs must take into account both the ordinary (essentially classical) degrees of freedom, as well as the *entangled* degrees of freedom.

This is a simple idea, but by pushing on it, people are starting to get some spectacular results. Cirac described a slew of systems that can now be understood using these techniques, and that were previously inaccessible numerically.

It’s early days yet, but if this success continues, it’ll certainly greatly enhance our understanding of complex quantum systems, and mark what may be the first major contribution of quantum information to another field of physics.

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I keep meaning to write a post about the Cirac/Verstraete/(numerous others) “renormalization” work on my blog. If I could spot trends in the past year in connecting quantum information science to many body quantum systems it would be: (1) Hey, let’s calculate the these pretty entanglement measures in many-body quantum systems because, well, because we can, and (2) Hey, let’s understand why current techniques fail in terms of their inappropriate accounting of entanglement, and let’s see how we can use this to fix certain cases. I yawn when I hear (1), but when I hear (2) I get all excited. Excited enough to start coding up some of these new methods.

Ouch, Dave. I’m pretty sure I did the first such calculation, in my 1998 thesis…

It’s ok.

I’m doing similar calculations in my 2005 thesis.

I think the title will be “How much entanglement can (asymptotically and approximately) fit on the head of a pin?”

Well shit, there I go opening my big fat mouth again. [shovel] But you also did the first work on (2), right? [shovel] And don’t you agree (2) is better than (1)? [shovel] Plust I must admit that one has to do (1) before one even begins to see (2). [exhausted from trying to dig myself out of this hole]

It’s okay, Dave, no need for a shovel; I largely agree with you, and my response was a quick (and not terribly thoughtful) off-the-cuff.

Probably the main difference is that I’m a little less pessimistic than you about the value of calculating entanglement in such systems. If you ask a clever question (i.e., calculate just the right thing), I suspect you may still learn a lot. I agree that most of the papers in this general vein are not asking particularly interesting questions, although they may nonetheless luck upon something interesting.

(As an aside, I note that it’s curious how often in the history of physics simply following one’s nose and calculating has led to interesting general insights. A good example is the universality of the critical exponents in phase transitions, and the fact that the critical exponents are the same on both sides of the critical point.)

But yeah, the DMRG-related stuff _is_ the real deal. I note that Schollwoeck’s recent review of the DMRG (to appear in RMP) gives a lot of play to the ideas coming out of quantum information.

Oops, PS: my work on the DMRG was done jointly with Tobias Osborne. We simply pointed out the connection with quantum information (i.e., that the DMRG optimally preserves entanglement under renormalization), and suggested, without being terribly concrete, that this idea might be useful in improving the DMRG.