We present exact analytic formulae which describe the interaction of multistate quantum systems possessing the Wigner-Majorana and Morris-Shore dynamic symmetries with a train of pulses. The pulse train field can be viewed as repeated interactions of the quantum system with the same field and hence the overall propagator is expressed as the matrix power of the single-pulse propagator. Because of the symmetries the multistate dynamics is characterised by intrinsic two-state features, described by one or more pairs of complex-valued Cayley-Klein parameters. This facilitates the derivation of explicit formulae linking the single-step and multi-step propagators. The availability of such analytic relations opens the prospects for a variety of applications with ensembles of qubits, qutrits and generally qudits, e.g., analytic description of coherent pulse-train interactions, coherent amplification of quantum gate errors for accurate quantum gate tomography, dynamical rephasing of inhomogeneously broadened ensembles, quantum sensing of small electric or magnetic fields, etc.

We present exact analytic formulae which describe the interaction of multistate quantum systems possessing the Majorana and Morris-Shore dynamic symmetries with a train of pulses. The pulse train field can be viewed as repeated interactions of the quantum system with the same field and hence the overall propagator is expressed as the matrix power of the single-pulse propagator. Because of the Majorana and Morris-Shore symmetries the multistate dynamics is characterised by intrinsic two-state features, described by one or more pairs of complex-valued Cayley-Klein parameters. This facilitates the derivation of explicit formulae linking the single-step and multi-step propagators. The availability of such analytic relations opens the prospects for a variety of applications, e.g., analytic description of coherent pulse train interactions, or coherent amplification of quantum gate errors for accurate quantum gate tomography for ensembles of qubits, qutrits and generally qudits.