David Edward Bruschi - Universität des Saarlandes & Forschungszentrum Jülich

Toward Algebraic Quantum Control

Determining the exact dynamics of a given system is paramount in most areas of physics. In classical systems the problem usually reduces to solving a set of coupled nonlinear differential equations, while in quantum mechanics the complexity of finding a solution increases dramatically due to the non-commutative character of operators. A well-known method for systematically solving quantum dynamics by factorizing the time-evolution operator into a finite product of exponentials is the Wei-Norman factorization algorithm, which has enjoyed an increasing central role in the development of new protocols for quantum computing and control in recent years. Unfortunately, this method is well developed in the case of finite-dimensional systems while its applicability to continuous-variables systems remains outstanding

Recently, a new approach has been proposed to investigate the classes of Hamiltonians of continuous-variables systems for which the Wei-Norman factorization method is applicable. This new approach involves analyzing the dimensionality of Hamiltonian Lie algebras by appropriately characterizing their generating terms. In our work, we generalize existing results by significantly extending their applicability to a broader class of physically relevant bosonic Hamiltonians. We reduce the complexity of verifying finiteness conditions from quadratic to linear, and we also introduce a visual algorithm to implement the corresponding procedure. Furthermore, we identify a universal Lie algebraic structure encompassing all finite-dimensional algebras within this framework. Our contributions represent a substantial step toward a comprehensive classification of Hamiltonian Lie algebras, with potential impact for practical applications ranging from quantum control and computing to the development of efficient algorithms for new quantum technologies.