  Classical simulation of quantum dynamics plays an important role in our
understanding of quantum complexity, and in the development of quantum
technologies. Compared to other techniques for efficient classical simulations,
methods relying on the Lie-algebraic structure of quantum dynamics have
received relatively little attention. At their core, these simulations leverage
the underlying Lie algebra - and the associated Lie group - of a dynamical
process. As such, rather than keeping track of the individual entries of large
matrices, one instead keeps track of how its algebraic decomposition changes
during the evolution. When the dimension of the algebra is small (e.g., growing
at most polynomially in the system size), one can leverage efficient simulation
techniques. In this work, we review the basis for such methods, presenting a
framework that we call "$\mathfrak{g}$-sim", and showcase their efficient
implementation in several paradigmatic variational quantum computing tasks.
Specifically, we perform Lie-algebraic simulations to train and optimize
parametrized quantum circuits, design enhanced parameter initialization
strategies, solve tasks of quantum circuit synthesis, and train a quantum-phase
classifier.
