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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

TL;DR

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, that overcome key limitations of the approach introduced by S\'a Earp and Pachos (2005.

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 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 $\SU(8)$ 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.

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

TL;DR

A novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in , a critical task for efficient quantum circuit design, that overcome key limitations of the approach introduced by S\'a Earp and Pachos (2005.

Abstract

We present a novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in , 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 and 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.

Paper Structure

This paper contains 13 sections, 5 theorems, 67 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview of the paper.
  3. KHK decomposition and Khaneja-Glaser bases
  4. The KHK decomposition via involutive automorphisms
  5. The Khaneja-Glaser special unitary bases
  6. Breakdown of the algorithm: step-by-step summary
  7. Implementation analysis and benchmarking
  8. Error management
  9. Comparison with existing approaches
  10. Conclusion and future developments
  11. Detailed example
  12. Phase correction
  13. BCH expansion

Key Result

Lemma 5

Let $(\mathfrak{g}={\mathfrak k}\oplus{\mathfrak m},\theta)$ be a symmetric decomposition of the Lie algebra ${\mathfrak g}$. If ${\mathfrak g}$ is compact and semisimple, then ${\mathfrak g}^*$ is noncompact and semisimple. Additionally ${\mathfrak g}^*=\mathfrak{k}\oplus i\mathfrak{m}$ is a Cartan

Figures (8)

  • Figure 1: Khaneja-Glaser basis for $\mathfrak{su}(4)$. We denote the element $\frac{i}{2}A\otimes B$ by $AB$ (adapted from SaEarp2005).
  • Figure 2: Khaneja-Glaser basis for $\mathfrak{su}(2^n)$, again omitting the factor $\frac{i}{2}$ (adapted from SaEarp2005).
  • Figure 3: Construction of the maximal Abelian subalgebra basis
  • Figure 4: Decomposition of $\mathfrak{su}(2^n)$ into both Cartan pairs. The elements $I^{\otimes n-1}\otimes A$, $A=X, Y, Z$ are multiplied by $\frac{i}{2}$ (adapted from SaEarp2005).
  • Figure 5: Khaneja-Glaser basis for $\mathfrak{su}(8)$, again omitting the factor $\frac{i}{2}$. We also indicate the subspaces that are selected by the involutions we introduce in this paper.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 1: Helgason
  • Remark 2: Helgason
  • Definition 3: Helgason
  • Definition 4: Helgason
  • Lemma 5
  • proof
  • Remark 6
  • Theorem 1: KHK decomposition, Helgason
  • Definition 7
  • Proposition 8
  • ...and 9 more