An Introduction to Cartan's KAK Decomposition for QC Programmers

TL;DR

This pedagogical paper explains Cartan's KAK decomposition for 2-qubit quantum operations, providing a constructive proof accessible to QC programmers and including implementation in Matlab/Octave.

Contribution

It offers a complete, rigorous, and constructive proof of a special case of Cartan's Decomposition tailored for quantum computing practitioners.

Findings

Provides a constructive proof using linear algebra

Includes Matlab/Octave implementation files

Clarifies the decomposition's application in quantum compiling

Abstract

This paper presents no new results; its goals are purely pedagogical. A special case of the Cartan Decomposition has found much utility in the field of quantum computing, especially in its sub-field of quantum compiling. This special case allows one to factor a general 2-qubit operation (i.e., an element of U(4)) into local operations applied before and after a three parameter, non-local operation. In this paper, we give a complete and rigorous proof of this special case of Cartan's Decomposition. From the point of view of QC programmers who might not be familiar with the subtleties of Lie Group Theory, the proof given here has the virtues, that it is constructive in nature, and that it uses only Linear Algebra. The constructive proof presented in this paper is implemented in some Octave/Matlab m-files that are included with the paper. Thus, this paper serves as documentation for the attached m-files.