Constructive Quantum Shannon Decomposition from Cartan Involutions

TL;DR

This paper introduces a constructive method for quantum circuit decomposition based on Cartan involutions, providing a systematic approach to implement the Quantum Shannon Decomposition for unitary matrices.

Contribution

It presents a new algorithm leveraging Cartan decompositions and involutions to construct quantum circuits, extending the existing Quantum Shannon Decomposition framework.

Findings

Provides a simple constructive algorithm for quantum circuit synthesis.

Utilizes Cartan decompositions to improve circuit design methodology.

Enhances the understanding of quantum circuit structure through involutions.

Abstract

The work presented here extends upon the best known universal quantum circuit, the Quantum Shannon Decomposition proposed in [Vivek V. Shende, Stephen S. Bullock and Igor Markov, Synthesis of Quantum Logic Circuits, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 25 (6): 1000-1010 (2006)]. We obtain the basis of the circuit's design in a pair of Cartan decompositions. This insight gives a simple constructive algorithm for obtaining the Quantum Shannon Decomposition of a given unitary matrix in terms of the corresponding Cartan involutions.