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arXiv:2509.05468 · Pure and Applied Mathematics Quarterly

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

Published Jan 1, 2026• 1 citation
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Abstract

We present a novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in $\mathrm{SU}\left(2^n\right)$, a critical task for efficient quantum circuit design. Building upon the approach introduced by Sá 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 $\mathrm{SU}(8)$ and $\mathrm{SU}(16)$ 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.

DOI
10.4310/pamq.260224235303

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Cartan-Khaneja-Glaser decomposition of SU(2n)SU(2^n)SU(2n) via involutive automorphisms
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We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G of SU(2^N) in terms of its `Cartan' and `non-Cartan' components. This effectively factors G in terms of group elements that belong in SU(2^n) with n<N, a procedure that can be iterated down to n=2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.
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