A Scheme of Cartan Decomposition for su(N)

TL;DR

This paper introduces a new scheme for performing Cartan decomposition on the Lie algebra su(N), utilizing algebraic structures like conjugate partition and quotient algebra, enabling efficient recursive decomposition of unitary transformations.

Contribution

The paper presents a novel algebraic scheme for Cartan decomposition of su(N), applicable to arbitrary finite dimensions, and provides an efficient recursive algorithm for decomposing unitary transformations.

Findings

Efficient recursive algorithm for Cartan decomposition of su(N).

Applicable to su(2^p) and related algebras.

Enables recursive decomposition into local and nonlocal gates.

Abstract

A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.