Constructive quantum scaling of unitary matrices

Abstract

In this work, we present a method of decomposition of arbitrary unitary matrix U ∈ U(2 k ) into a product of single-qubit negator and controlled-√ 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.

QPCF: Higher-Order Languages and Quantum Circuits

“…i=0 α i is the average of all amplitudes (see [59]). Some interesting decomposition of D in terms of smaller quantum circuits can be found in [15,21].…”