Constructive quantum scaling of unitary matrices

TL;DR

This paper introduces a constructive method to decompose any unitary matrix into a product of specific quantum gates, optimizing the process with a complexity of O(4^k) entangling gates, akin to a complex Sinkhorn-Knopp algorithm.

Contribution

It presents a novel decomposition technique for unitary matrices into single-qubit negator and controlled-$ oot{2} ext{NOT}$ gates, with an optimal entangling gate count.

Findings

Decomposition method is constructive and explicit.

Circuit complexity is proven to be optimal at O(4^k) gates.

Example transformation demonstrates practical application.

Abstract

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-$\sqrt{\mbox{NOT}}$ 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.