Nonstabilizerness of Permutationally Invariant Systems

Typical measures of nonstabilizerness of a system of N qubits requirecomputing 4^N expectation values, one for each Pauli string in the Pauligroup, over a state of dimension 2^N. For permutationally invariant systems,this exponential overhead can be reduced to just O(N^3) expectation values ona state with a dimension O(N). We exploit this simplification to study thenonstabilizerness phase transitions of systems with hundreds of qubits.