Cartan-Khaneja-Glaser decomposition of $\SU(2^n)$ via involutive automorphisms

• URL: http://arxiv.org/abs/2509.05468v1
• Date: Fri, 05 Sep 2025 19:46:50 GMT
• Title: Cartan-Khaneja-Glaser decomposition of $\SU(2^n)$ via involutive automorphisms
• Authors: John A. Mora Rodríguez, Arthur C. R. Dutra, Henrique N. Sá Earp, Marcelo Terra Cunha,
• Abstract summary: We present a novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in $SU(2n)$.<n>We overcome key limitations of their method, such as reliance on ill-defined matrix logarithms and the convergence issues of truncated Baker-Campbell-Hausdorff(BCH) series.
• Score: 0.46664938579243564
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in $\SU(2^n)$, a critical task for efficient quantum circuit design. Building upon the approach introduced by S\'a Earp and Pachos (2005), we overcome key limitations of their method, such as reliance on ill-defined matrix logarithms and the convergence issues of truncated Baker-Campbell-Hausdorff(BCH) series. Our reformulation leverages the algebraic structure of involutive automorphisms and symmetric Lie algebra decompositions to yield a stable and recursive factorization process. We provide a full Python implementation of the algorithm, available in an open-source repository, and validate its performance on matrices in $\SU(8)$ and $\SU(16)$ using random unitary benchmarks. The algorithm produces decompositions that are directly suited to practical quantum hardware, with factors that can be implemented near-optimally using standard gate sets.

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