# The Birkhoff theorem for unitary matrices of prime-power dimension

## TL;DR

This paper extends the unitary Birkhoff theorem for matrices of prime-power dimension, showing that a smaller set of permutation matrices suffices for the decomposition, linked to the affine group GA(w,p).

## Contribution

It demonstrates that for prime-power dimensions, the Birkhoff decomposition can be achieved with a subset of permutation matrices related to the affine group, reducing complexity.

## Key findings

- Decomposition uses epicirculant permutation matrices.
- Permutation matrices form a group isomorphic to GA(w,p).
- Reduction from n! to a smaller group of permutation matrices.

## Abstract

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension~$n$ of the unitary matrix equals a power of a prime $p$, i.e.\ if $n=p^w$, then the Birkhoff decomposition does not need all $n!$ possible permutation matrices, as the epicirculant permutation matrices suffice. This group of permutation matrices is isomorphic to the general affine group GA($w,p$) of order only $p^w(p^w-1)(p^w-p)...(p^w-p^{w-1}) \ll \left( p^w \right)!$.