Dynamical Lie algebras in quantum computing

A Hamiltonian of a quantum system is a Hermitian operator, which is typically a sum of terms corresponding to certain physical interactions. Taking all real linear combinations and nested commutators of i times these terms generates a subalgebra of the Lie algebra of skew-Hermitian matrices, called the dynamical Lie algebra (DLA). Its significance is that the unitary time evolution of the physical system is given by elements of the associated Lie group. The DLA determines the set of reachable states of the system and its controllability, so it is relevant for designing quantum circuits. In this talk, I will present a classification of DLAs generated by 2-local Pauli interactions on spin chains and on arbitrary interaction graphs. I will also discuss applications of DLAs to variational quantum computing, including the problem of barren plateaus.