Probing quantum scars and weak ergodicity breaking through quantum complexity

Scar states are special many-body eigenstates that weakly violate the eigenstate thermalization hypothesis (ETH). Using explicit formalism of Lanczos algorithm, usually known as forward scattering approximation in this context, we compute Krylov state (spread) complexity typical generated by time evolution PXP Hamiltonian, hosting such states. We show for N\'eel revives an approximate sense, while generic ETH-obeying always increases. This can be attributed to SU(2) structure corresponding generators Hamiltonian. quantify ``closeness'' $q$-deformed algebra and provide analytic expression coefficients within subspace. intuitively explain results terms a tight-binding model. further consider deformation Hamiltonian complexity. find shows nearly perfect revival same does not hold state.