Dynamical Lie algebras generated by Pauli strings and quadratic spaces over ${\mathbb{F}}_2$

Dynamical Lie algebras, i.e. Lie subalgebras of $\mathfrak{s}\mathfrak{u}(2^n)$, generated by Pauli strings have recently been studied intensively. They are also called Pauli Lie algebras or Hamiltonian Lie algebras. In this paper we provide a uniform mathematical approach to various recent results on Pauli Lie algebras. Moreover, we present an algorithm that on input of a set of Pauli strings determines the isomorphism type of the dynamical Lie algebra generated by these Pauli’s in time $\mathcal{O}(\max (n,m{)}^3)$ where $m$ is the size of the generating set.