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Commutator-free Cayley methods

Sofya Maslovskaya, Christian Offen, Sina Ober-Blöbaum, Pranav Singh, Boris Wembe

TL;DR

The paper addresses high-order numerical integration of non-autonomous linear systems evolving on quadratic Lie groups, where preserving group structure and invariants is crucial in applications like quantum dynamics. It introduces a fourth-order commutator-free Lie-group integrator built from products of Cayley transforms (CFCT) by deriving a Cayley–BCH-type matching and enforcing real, structure-preserving coefficients. The method relies on Legendre–Gauss quadrature to form accurate, low-cost approximations $A_1,A_2$ that feed the Cayley factors, avoiding nested commutators and expensive matrix exponentials. Numerical experiments on a driven two-level system and a time-dependent Schrödinger equation demonstrate norm preservation and competitive accuracy relative to commutator-based exponential methods, with favorable stability and efficiency. The approach advances geometric numerical integration for quantum control and other physics problems by combining Cayley transforms with commutator-free design on quadratic Lie groups.

Abstract

Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley-Magnus methods the method is also free from nested matrix commutators.

Commutator-free Cayley methods

TL;DR

The paper addresses high-order numerical integration of non-autonomous linear systems evolving on quadratic Lie groups, where preserving group structure and invariants is crucial in applications like quantum dynamics. It introduces a fourth-order commutator-free Lie-group integrator built from products of Cayley transforms (CFCT) by deriving a Cayley–BCH-type matching and enforcing real, structure-preserving coefficients. The method relies on Legendre–Gauss quadrature to form accurate, low-cost approximations that feed the Cayley factors, avoiding nested commutators and expensive matrix exponentials. Numerical experiments on a driven two-level system and a time-dependent Schrödinger equation demonstrate norm preservation and competitive accuracy relative to commutator-based exponential methods, with favorable stability and efficiency. The approach advances geometric numerical integration for quantum control and other physics problems by combining Cayley transforms with commutator-free design on quadratic Lie groups.

Abstract

Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley-Magnus methods the method is also free from nested matrix commutators.
Paper Structure (14 sections, 8 theorems, 68 equations, 3 figures)

This paper contains 14 sections, 8 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preparation
  3. Cayley transforms for quadratic Lie groups
  4. The Cayley--Magnus expansion
  5. Cayley--Magnus and Legendre expansion
  6. Commutator-free Cayley scheme
  7. BCH-formula for Cayley transform
  8. Commutator-Free Cayley Time-propagator scheme (CFCT)
  9. Examples
  10. A driven two-level system
  11. Linear time-dependent Schrödinger equation
  12. Numerical simulations.
  13. Details on the computation of coefficients
  14. Details on the proof of Proposition \ref{['prop:Cayley-BCH']}

Key Result

Lemma 2.1

Let $Y(t)$ be the solution of system eq:initial_system, with $-1 \notin \sigma(Y(t)Y_0^{-1})$ for any $t \in [t_0,t_f]$, then $Y(t)$ can be written in the form $Y(t) = \,{\rm Cay}(\Omega(t))Y_0$, where the matrix $\Omega \in \tilde{\mathfrak g}$ satisfies with the Lie bracket (commutator) of two matrices $A$ and $B$ defined by $[A,B] = A\cdot B - B\cdot A$.

Figures (3)

  • Figure 1: (Left) Projection of the solution of system \ref{['eq:syst-1']} along the first axis, together with error obtained for CFET4:2, CMT, CFCT taking $T = 20\pi/\omega$, $\omega = 1$ and $\Delta = V = 0.5$. (Right) Illustration of the norm conservation during the propagation for CFET4:2, CMT and CFCT. We can clearly see the loss of this property by using the classical integrator as RK45.
  • Figure 2: (Left) Solution of system \ref{['eq:SE']} together with error obtained for CMT, CFCT (taking CFET4:2 as reference solution), propagated until $T = 2$. (Right) Illustration of the norm conservation and change of the energy during the propagation for CFET4:2, CMT and CFCT. Again, there is no conservation of the norm along the propagation when using the classical integrator RK45.
  • Figure 3: Illustration of the energy blow-up when using RK45 to solve example 2.

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Remark 1
  • Lemma 2.2
  • Remark 2
  • Proposition 2.1
  • Remark 3
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • ...and 8 more