Lie algebra-assisted quantum simulation and quantum optimal control via high-order Magnus expansions

The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.

💡 Research Summary

The paper presents a scalable method for evaluating high‑order Magnus expansions (MEs) of time‑dependent quantum systems, with a focus on Hamiltonians of the form (H(t)=A+d(t)B), where (A) is a static drift term, (B) is a single controllable operator, and the control envelope (d(t)) is expressed as a polynomial of degree (m). By representing the control function analytically, the authors turn the otherwise intractable multiple time‑integrals of nested commutators into closed‑form polynomial expressions in the coefficients ({d_\gamma}) and the evolution time (t).

The key technical insight is to pre‑compute the dynamical Lie algebra (\mathfrak g) generated by ({iA,iB}) and its structure constants (f_{ijk}) defined by (