On the KAK Decomposition and Equivalence Classes

TL;DR

This paper rigorously develops the mathematical foundations of the KAK decomposition for compact semisimple Lie groups, clarifying equivalence classes and correcting misconceptions in quantum gate classification.

Contribution

It provides a complete proof of the KAK decomposition theorem, distinguishes two notions of equivalence, and clarifies their geometric representations for $ ext{SU}(4)$.

Findings

Established a rigorous Lie-theoretic foundation for quantum gates.

Clarified the relationship between different definitions of KAK decomposition.

Showed that local equivalence classes are not represented by the Weyl chamber in $ ext{SU}(4)$.

Abstract

The KAK decomposition is a fundamental tool in Lie theory and quantum computing. Despite its widespread use, the mathematical foundations remain incomplete, particularly regarding the precise conditions for the decomposition and the characterization of equivalence classes under multiplication by elements of $K$. Here, we present a mathematical theory of the KAK decomposition for connected compact semisimple Lie groups and derive the decomposition for $\mathrm{SU}(4)$. In particular, we clarify the relationship between various definitions of a Cartan decomposition in the literature and give a complete proof of a general KAK decomposition theorem. We then distinguish two distinct notions of KAK equivalence classes, double coset equivalence and projective equivalence, thereby addressing mathematical inconsistencies regarding KAK classification in the literature. Specifically, for $\mathrm{SU}(4)$, we show that local equivalence classes under multiplication by $\mathrm{SU}(2)\otimes \mathrm{SU}(2)$ are geometrically represented not by the usual "Weyl chamber" as claimed in the existing literature. Instead, the "Weyl chamber" is only recovered by the projective-local equivalence which disregards global phases. We develop a systematic theory for determining equivalence and uniqueness for both notions of equivalence. Our work establishes a rigorous Lie-theoretic foundation for the theory of quantum gates and circuits.