First of all, you can now watch the talks, all the links are on the QIP'10 programme webpage.

Second of all, Pablo Arrighi is giving the talk Unitarity + causality => localizability one more time here at QuPa. I did not understand anything the first time. I'm hapy to have a second chance.

Unitarity plus causality implies localizability

Pablo Arrighi, Universiy of Grenoble

« I already explained this a couple of times. So I redo… »

We are considering only finite dimensional Hilbert spaces because « we cannot say much on circuit representation on unitaries in infinite dimensional Hilbert spaces » (huh???) and we consider only discrete time evolution.

Consider a graph. Each vertex is a Hilbert space. The "global" HS is the tensor product of all of them. U: unitary operator between time t and t+1. There is an edge between a vertex v and all of his neighbors if the states in the Hilbert space of v depends on U and the states in the HS of its neighbors. (Is that understandable?) — That's causality.

I'm almost lost again. He tries to decompose U into small "local" unitaries. To do that, he double his graph with ancillas.

Basically, the result is that time t, the evolution of the state in the HS of a node does depend only of states of his neighbors (and their neighbors, and their neighbors, etc. but not too many of them) — That's localizability.

If the graph is regular grid and the size of all the Hilbert spaces are the same, this thing is a cellular automata ; and QCA are universal.

(It's all I got, and and it can be quite wrong. Hey people, it's not because something is on the Internet that it's right, does it?)