Unitary designs from perturbed time evolutions of a chaotic Hamiltonian

Unitary designs provide efficient substitutes for Haar-random unitaries in quantum information processing, randomized measurements, and many-body quantum dynamics. We propose a protocol based on time evolutions of a single chaotic Hamiltonian with intermediate unitary perturbations to generate unitary designs. We derive the frame potential of the resulting ensemble which relies on the frame potentials of the intermediate ensemble. The intermediate ensemble need not itself be Haar random or a unitary design even approximately; it is sufficient that its frame-potential growth remains well suppressed relative to the maximal possible scaling. The nontrivial Pauli set and the Clifford ensemble are simple examples satisfying this criterion, while fixed-size ensembles generally fail. We further show that, at the level of frame potentials, multi-Hamiltonian temporal protocols can be recursively replaced by one-Hamiltonian protocols interleaved with fixed trace-suppressed perturbations.