Scrambling and complexity in systems with Pauli dynamical Lie algebras

A key information-theoretic property of quantum systems is the extent to which they scramble information, as quantified - for example - by the out-of-time-order correlator (OTOC).

In this work we characterise, via the commutator graph, the average behaviour of the OTOC for quantum systems whose dynamical Lie algebra (DLA) has a basis consisting of Pauli strings.

The quadratic symmetries of systems with a given Pauli DLA - and therefore the second moment operator over the corresponding ensemble - have recently been linked to properties of the commutator graph, which we can therefore in turn link to the OTOC, the frame potential, the frustration graph of the Hamiltonian of the system, as well as the Krylov complexity of operators evolving under the dynamics. In particular, we connect the Krylov complexity of an operator to the irreducible representation of the adjoint action of the DLA on the space of operators in which it resides, and prove that it is - on average - lower bounded by the average shortest path length between the initial operator and the other operators in the commutator graph.