Cyclotomic Structure and Exact Recurrences in Floquet Many-Body Quantum Dynamics

Periodically driven quantum systems exhibit a wide range of complex dynamics, including chaos. Yet, at special parameter values, their evolution can recur exactly. Several quantum protocols have recently been proposed that exploit exact Floquet recurrences to demonstrate genuine quantum advantage. Such recurrences can be studied through the arithmetic structure of the Floquet unitary. In this work, we establish a necessary and sufficient arithmetic criterion for exact recurrence and determine the minimal recurrence period when one exists. Because recurrence is a global property of the unitary, it holds for every initial state. Applying our framework requires only knowledge of the number field associated with the Hamiltonian parameters. We demonstrate the method using the kicked Ising chain, establishing exact recurrences in both Clifford and non-Clifford regimes. We further show that, in the recurrent non-Clifford regime, magic--a resource for quantum computation--is generated at intermediate times and undergoes exact periodic revivals enforced by the recurrence of the full Floquet unitary. By connecting number theory, quantum many-body dynamics, and quantum information, our work shows that exact recurrence extends beyond Clifford circuits and opens new avenues for magic generation and quantum hardware benchmarking.