Available control in dynamical decoupled quantum systems

Dynamical decoupling (DD) eliminates qubit-bath coupling by applying a sequence of instantaneous pulses. While qubit-bath couplings generally lead to qubit relaxation and dephasing, qubit-qubit couplings are often used to manipulate or control quantum states. We investigate the available control operations in two DD schemes, named periodic DD (PDD) and Uhrig DD (UDD), to see whether universal quantum computation can be realized in these decoupled systems. We find that universal control is possible using Heisenberg interaction in both periodically decoupled system and Uhrig decoupled system, and the available control operations under two kinds of DD sequences obey the same commutation relation. In the UDD case, we also derive a rough bound for control errors.