Two Papers

Sometimes you slog through tons of papers and wonder how much the whole huge mess really matters. But then there are days like today where I found two papers which I think completely and totally rock. Maybe they don’t really matter, but they are really interesting. The first paper appear on the arXiv today, so I really didn’t have to dig for it, but the other paper I just stumbled upon and somehow missed it when it came out in 2002.
Paper 1: quant-ph/0401137
“Fast simulation of a quantum phase transition in an ion-trap realisable unitary map” by J.P. Barjaktarevic, G.J. Milburn, Ross H. McKenzie. The idea in this paper is very beatiful. Consider a system with a Hamiltonian which posses a quantum phase transition. On a quantum computer it is possible to simulate the dynamics of this Hamiltonian. Suppose that your Hamiltonian is a sum of two noncomuting terms H_1 and H_2 and that you can easily implement evolution according to each of these terms separately, i.e. you can do exp(iH_1t) and exp(iH_2t). One way to then simulate the full Hamiltonian is to “trotterize” the evolution and perform alternating infinitesimal exp(iH_1 dt) exp(iH_2dt) exp(iH_1 dt) exp(iH_2 dt)… =exp(i(H_1+H_2)t)+small error. But suppose that you don’t do this (because, for example you can’t really do good infinitesimal evolutions in the real world!) So instead you use “big” steps exp(iH_1T)exp(iH_2T)… Now you can ask, does this system have a quantum phase tranisition! So in what sense does the “big” evolution model have the same properties as the “infinitesimal” evolution mode? In this paper the authors address this issue for the ising model with a transverse field. And indeed, the authors present strong evidence that there is a quantum phase transition in the behavior of this “big” model! A summer student and I worked a bit on this problem for a different decomposition of the same Hamiltonian. As a nice summer project the summer student, Jaime Valle, wrote code to simulate this evolution. In this model we indeed did see evidence of the phase transition. And now we see that for the decomposition choosen by these authors there is direct analytic evidence of the quantum phase transition!
The second paper that I discovered which I loved was quant-ph/0206016, “The Dirac Equation in Classical Statistical Mechanics” by G.N. Ord. Now this paper, and a series of other papers by this author and coworkers, rocks! What they show is that there is a microscopic statistical mechanical model for the Dirac equation in one dimension! There is a famous prescription for obtainin the Dirac equation in one dimension which is due to Feynman. Basically this prescription works as follows. Consider a particle which moves either forwards or backwards at the speed of light. If you want to calculate the amplitude for the particle to go from spacetime point A to spacetime point B, you simply take all paths for such a particle and associate with it an amplitude which is (im)^(# corners) where m is an infinitesimal parameter, i is the square root of minus 1 and the # corners it the number of times the particle switches directions in the path. If you use this to calculate the amplitudes for all of the paths between A and B and add up all of these amplitudes, you get the kernel for the Dirac equation in one dimension!
What Ord talks about is similar to Feynman’s prescription but what Ord shows how it is possible to construct a model where the statistics of the dirac equation fully explained by a microscopic classical model. One of his version of this model has some very nice properties, like being a beautiful nonlocal hidden variable model of the Dirac equation (it is interested that even for one particle, one gets a nonlocal hidden variable model)