Operator complexity of adiabatic gauge potential

Recently, there has been great interest in studying the growth rates of operator complexity and out-of-time-order correlators in many-body quantum systems. Here we characterize the complexity of the adiabatic gauge potential (AGP), which encodes the geometry of eigenstates when varying a control parameter in a Hamiltonian. For generic systems, the AGP is a highly non-local and entangled operator. We find that its Frobenius norm, which can be explicitly related to operator growth, shows remarkably different scaling with system size for integrable and non-integrable systems: polynomial versus exponential. Using the length of Pauli string operator as a measure of the AGP complexity, we compute operator weight distributions and the Shannon entropy to better understand the norm's system size scaling.