Protection of quantum systems by nested Uhrig dynamical decoupling

Based on a theorem we establish on dynamical decoupling of time-dependent systems, we present a scheme of nested Uhrig dynamical decoupling (NUDD) to protect multi-qubit systems in generic quantum baths to arbitrary decoupling orders. This scheme uses only single-qubit operations. Higher order decoupling is achieved at the cost of a polynomial increase in pulse number. For general multi-level systems, this scheme protects a set of unitary Hermitian system operators which mutually either commute or anti-commute, and hence all operators in the Lie algebra generated from this set of operators, generating an effective symmetry group for the system up to a given order of precision. We also show how to implement NUDD with pulses of finite amplitude, up to an error in the second order of the pulse durations.