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arxiv: 2509.05468 · v2 · pith:YXD6QIKMnew · submitted 2025-09-05 · 🪐 quant-ph

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

John A. Mora Rodr\'iguez , Arthur C. R. Dutra , Henrique N. S\'a Earp , Marcelo Terra Cunha This is my paper

Pith reviewed 2026-05-21 21:46 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Cartan-Khaneja-Glaser decompositionSU(2^n)involutive automorphismssymmetric Lie algebra decompositionsquantum circuit designunitary matrix factorizationrecursive algorithms
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The pith

Involutive automorphisms enable a stable recursive Cartan-Khaneja-Glaser decomposition for SU(2^n) unitaries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a new algorithm for the Cartan-Khaneja-Glaser decomposition of matrices in SU(2^n). It improves on prior work by using involutive automorphisms of the Lie algebra to create symmetric decompositions. This yields a recursive factorization process that avoids ill-defined matrix logarithms and convergence problems with truncated Baker-Campbell-Hausdorff series. A sympathetic reader would care because this supports efficient quantum circuit design with factors suitable for hardware implementation using standard gate sets. The method is implemented in Python and tested on random unitaries in SU(8) and SU(16).

Core claim

Our reformulation leverages the algebraic structure of involutive automorphisms and symmetric Lie algebra decompositions to yield a stable and recursive factorization process for the Cartan-Khaneja-Glaser decomposition of SU(2^n).

What carries the argument

Involutive automorphisms inducing symmetric Lie algebra decompositions, which enable the recursive factorization without relying on problematic logarithms or series expansions.

If this is right

  • The decomposition produces factors directly suited to practical quantum hardware.
  • Factors can be implemented near-optimally using standard gate sets.
  • The algorithm is validated through benchmarks on matrices in SU(8) and SU(16).
  • A full open-source Python implementation is provided for the process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This recursive approach may scale better computationally for larger n in quantum systems.
  • It could be adapted for other compact Lie groups used in quantum control problems.
  • Connections to symmetric space theory might offer further optimizations in circuit compilation.

Load-bearing premise

Involutive automorphisms of the Lie algebra of SU(2^n) exist and induce symmetric decompositions free of ill-defined logarithms and truncated series convergence failures.

What would settle it

Applying the algorithm to a known unitary matrix in SU(8), reconstructing the original from the factors, and verifying equality within machine precision without any numerical instabilities from logs or series.

Figures

Figures reproduced from arXiv: 2509.05468 by Arthur C. R. Dutra, Henrique N. S\'a Earp, John A. Mora Rodr\'iguez, Marcelo Terra Cunha.

Figure 1
Figure 1. Figure 1: Khaneja-Glaser basis for su(4). We denote the element i 2A ⊗ B by AB (adapted from [8]) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Khaneja-Glaser basis for su(2n), again omitting the factor i 2 (adapted from [8]). Proposition 9. Let G be a semisimple connected Lie group and (g, θ) an orthogonal symmetric Lie algebra with sym￾metric pair (G, K). Let G ∈ G such that G = K0 exp(m), as in Theorem 1, then exp(2m) = Θ(G ∗ )G, (6) with Θ an involutive automorphism of G satisfying dΘ = θ and (3). Proof. From (3), we have Θ(G ∗ ) = Θ(exp(−m)K∗… view at source ↗
Figure 3
Figure 3. Figure 3: Construction of the Cartan subalgebra basis [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Decomposition of su(2n) into both Cartan pairs. The elements I ⊗n−1 ⊗ A, A = X, Y, Z are multiplied by i 2 (adapted from [8]). Notice how, for n > 2, we can decompose the subalgebra kn into kn = span(Kn,0) ⊕ span(Kn,1) ⊕ span  i 2 I ⊗(n−1) ⊗ Z  ≃ su(2n−1 ) ⊕ su(2n−1 ) ⊕ u(1). Let bkn := span(Kn,0) ⊕ span(Kn,1). Since I ⊗(n−1) ⊗ Z,kn [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Decomposition of matrix G using our algorithm. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Decomposition of one of the abelian factors into single-qubit rotations and CNOTs using the method by [ [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Decomposition of a two-qubit gate into single-qubit rotations and CNOTs using the Qiskit’s Python library [ [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents a reformulation of the Cartan-Khaneja-Glaser decomposition for elements of SU(2^n) that replaces the 2005 Sá Earp-Pachos procedure with a recursive factorization based on involutive automorphisms of the Lie algebra su(2^n) and the associated symmetric decompositions. The central claim is that this algebraic construction yields a numerically stable algorithm free of ill-defined matrix logarithms and truncated BCH convergence failures; the authors supply an open-source Python implementation and report benchmark results on random unitaries in SU(8) and SU(16).

Significance. A rigorously verified, constructive, and stable recursive decomposition of SU(2^n) would be useful for quantum-circuit compilation. The provision of reproducible code is a clear strength; however, the significance is currently limited by the absence of an explicit demonstration that the automorphism selection rule remains algebraic and numerically stable at every recursion depth.

major comments (2)
  1. [Abstract and recursive factorization section] Abstract (paragraph on limitations overcome) and the description of the recursive factorization: the claim that the involutive-automorphism construction is free of the original numerical pathologies rests on the existence of a fixed algebraic rule for selecting the automorphism at each recursive step. No derivation, basis-projection formula, or pseudocode is supplied showing that this choice can be made constructively without reintroducing ill-defined logarithms or truncated BCH series for arbitrary n.
  2. [Validation / benchmarks] Validation paragraph: the manuscript states that the algorithm was tested on SU(8) and SU(16) random unitaries, yet supplies no quantitative metrics (e.g., average gate count, numerical error norms, or comparison against the 2005 baseline) that would allow assessment of whether the claimed stability is realized in practice.
minor comments (2)
  1. [Method] Notation for the involutive automorphism and the symmetric decomposition should be introduced with explicit reference to the standard Cartan decomposition of su(2^n) so that readers can verify the algebraic steps.
  2. [Implementation] The repository link and installation instructions for the Python code should appear in the main text rather than only in a footnote or abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We address each major comment below with clarifications and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Abstract and recursive factorization section] Abstract (paragraph on limitations overcome) and the description of the recursive factorization: the claim that the involutive-automorphism construction is free of the original numerical pathologies rests on the existence of a fixed algebraic rule for selecting the automorphism at each recursive step. No derivation, basis-projection formula, or pseudocode is supplied showing that this choice can be made constructively without reintroducing ill-defined logarithms or truncated BCH series for arbitrary n.

    Authors: We agree that an explicit description of the automorphism selection rule would strengthen the presentation. The rule is defined algebraically by the symmetric decomposition of the Lie algebra under the chosen involution: at each step we project the current generator onto the +1 and -1 eigenspaces using the fixed Pauli basis, which yields the two sub-algebra elements without logarithms or BCH truncation. This procedure is implemented in the open-source code and remains constructive for any n. In the revised manuscript we will insert a dedicated subsection containing the basis-projection formula together with pseudocode for the recursive step, thereby demonstrating that the construction stays purely algebraic at every depth. revision: yes

  2. Referee: [Validation / benchmarks] Validation paragraph: the manuscript states that the algorithm was tested on SU(8) and SU(16) random unitaries, yet supplies no quantitative metrics (e.g., average gate count, numerical error norms, or comparison against the 2005 baseline) that would allow assessment of whether the claimed stability is realized in practice.

    Authors: We acknowledge the absence of explicit numerical metrics in the current text. The accompanying Python repository already computes the Frobenius-norm reconstruction error, the number of elementary factors produced, and direct comparisons against the Sá Earp-Pachos baseline on the same random ensembles. In the revised manuscript we will add a benchmarks section that reports these quantities (means and standard deviations over 100 samples for both SU(8) and SU(16)) in tabular form, thereby providing the quantitative evidence requested. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The manuscript reformulates the Cartan-Khaneja-Glaser decomposition of SU(2^n) by invoking the algebraic structure of involutive automorphisms to induce symmetric Lie algebra decompositions, yielding a claimed stable recursive factorization that avoids ill-defined logarithms and truncated BCH failures from the 2005 Sá Earp-Pachos method. This construction is presented as independent once the existence of suitable involutive automorphisms is granted; the paper supplies a Python implementation and numerical validation on random elements of SU(8) and SU(16) that can be checked externally. No equation or step is shown to reduce by definition to a fitted input, to a self-referential prediction, or to an unverified self-citation chain. The self-citation to prior work by one co-author is acknowledged but does not carry the load-bearing algebraic argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The algorithm rests on the existence of suitable involutive automorphisms for the Lie algebra of SU(2^n) and on the numerical stability of the resulting recursion; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Involutive automorphisms of the Lie algebra of SU(2^n) exist and induce symmetric decompositions that bypass ill-defined logarithms and truncated BCH series.
    Invoked to guarantee stability of the recursive factorization process.

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