Constructive quantum scaling of unitary matrices

In this work, we present a method of decomposition of arbitrary unitary matrix (Uin mathbf {U}(2^k)) into a product of single-qubit negator and controlled-(sqrt{text{ NOT }}) gates. Since the product results with negator matrix, which can be treated as complex analogue of 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.