# Local decomposition of unitary matrices

1. May 7, 2005

### chiyosdad

I know that, given an arbitrary unitary matrix A, it can be written as the product of several "local" unitary matrices Ai -- local in the sense that they only act on a small constant number of vector components, in fact it is sufficient to take 2 as that constant, which is best possible. For example,
$$A=\left(\begin{array}{cccc} a & 0 & b & 0 \\0 & 1 & 0 & 0 \\ c & 0 & d & 0\\ 0 & 0 & 0 & 1\end{array}\right)$$
is such a local matrix, because it only acts on the basis element e1 and e3. Another way to think of it is that it's only nontrivial over a subspace of dimension 2.

So the picture is that there are these gigantic linear transformations on C^n(n is very large), being built out of smaller transforms which act on small constant-dimension subspaces. So as n grows, the number of local Ai's needed will grow too. I have an upper bound on the number of Ai needed in the decomposition, in terms of n. But that's not what I'm interested in. What I would like to know, is what is the least number of Ai's needed to decompose a specific matrix, or a class of matrices. I know of no machinery that makes this easy, and looking around online for a bit didn't help.

So the question is, if I have a specific unitary matrix, how do I go about calculating the minimum number of Ais needed in the decomposition, and if I have a class of matrices, how do I figure it out. Is there some reading on this subject I can find online?

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