# Local decomposition of unitary matrices

In summary, the conversation discusses the decomposition of an arbitrary unitary matrix A into smaller "local" unitary matrices Ai, which only act on a small constant number of vector components. The number of Ai's needed will grow as n, the dimension of the matrix, grows. The question is how to calculate the minimum number of Ai's needed for a specific matrix or class of matrices, which is difficult to prove generically and may require finding a subsystem of values to calculate.

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?

To prove lower bounds is generically difficult. You mentioned a specific matrix, which means this could be done more easily depending on the properties of this matrix. E.g. it could just be ##1##. In the general case one needs to find a subsystems of values which have to be calculated regardless of these properties and count the steps for those calculations. The bigger this subsystem, the better the lower bound. However, it is not clear what such a subsystem must look like.

## 1. What is local decomposition of unitary matrices?

Local decomposition of unitary matrices is a process in which a unitary matrix is decomposed into a product of smaller unitary matrices, each representing a local transformation. This decomposition is useful in quantum information processing and quantum computing.

## 2. Why is local decomposition of unitary matrices important?

Local decomposition of unitary matrices allows for the simplification of complex unitary operations into smaller, more manageable operations. This can make quantum algorithms more efficient and easier to implement on quantum computers.

## 3. What are the applications of local decomposition of unitary matrices?

Local decomposition of unitary matrices has applications in quantum error correction, quantum state preparation, and quantum circuit optimization. It is also used in the design of quantum algorithms and in quantum simulations.

## 4. What are the methods for local decomposition of unitary matrices?

There are several methods for local decomposition of unitary matrices, including the Schmidt decomposition, polar decomposition, and singular value decomposition. Each method has its own advantages and may be more suitable for certain types of unitary matrices.

## 5. Can local decomposition be applied to non-unitary matrices?

No, local decomposition is specific to unitary matrices, which have the property of preserving the norm of a vector. For non-unitary matrices, other decomposition methods such as singular value decomposition can be used.