Constructive quantum scaling of unitary matrices

In this work we present a method of decomposition of arbitrary unitary matrix $U\in \mathbf{U}(2^k)$ into a product of single-qubit negator and controlled-$\sqrt{NOT}$ gates. Since the product results with negator matrix, which can be treated as complex analogue if bistochastic matrix, our method can be seen as complex analogue of Sinkhorn-Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively and resulting circuit consists of $O(4^k)$ entangling gates, which is proved to be optimal. An example of such transformation is presented.