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arXiv · 2509.05468

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

John A. Mora Rodríguez·Arthur C. R. Dutra·Henrique N. Sá Earp·Marcelo Terra Cunha
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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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John A. Mora Rodríguez, Arthur C. R. Dutra, Henrique N. Sá Earp, Marcelo Terra Cunha. 2025-09-05. Cartan-Khaneja-Glaser decomposition of $SU(2^n)$ via involutive automorphisms. https://doi.org/10.4310/pamq.260224235303

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