Pauli spectrum and nonstabilizerness of typical quantum many-body states

An important question of quantum information is to characterize genuinely quantum (beyond-Clifford) resources necessary for universal quantum computing. Here, we use the Pauli spectrum to quantify how ``magic,'' beyond Clifford, typical many-qubit states are. We first present a phenomenological picture of the Pauli spectrum based on quantum typicality, and then we confirm it for Haar random states. We then introduce filtered stabilizer entropy, a magic measure that can resolve the difference between typical and atypical states. We proceed with the numerical study of the Pauli spectrum of states created by random circuits as well as for eigenstates of chaotic Hamiltonians. We find that in both cases, the Pauli spectrum approaches the one of Haar random states, up to exponentially suppressed tails. We discuss how the Pauli spectrum changes when ergodicity is broken due to disorder. Our results underscore the difference between typical and atypical states from the point of view of quantum information.