We explore reachable sets of open $n$-qubit quantum systems, the coherent parts of which are under full unitary control and that have just one qubit whose Markovian noise amplitude can be modulated in time such as to provide an additional degree of incoherent control. In particular, adding bang-bang control of amplitude damping noise (non-unital) allows the dynamic system to act transitively on the entire set of density operators. This means one can transform any initial quantum state into any desired target state. Adding switchable bit-flip noise (unital), on the other hand, suffices to explore all states majorised by the initial state. We have extended our open-loop optimal control algorithm (DYNAMO package) by such degrees of incoherent control so that these unprecedented reachable sets can systematically be exploited in experiments. As illustrated for an ion trap experimental setting, open-loop control with noise switching can accomplish all state transfers one can get by the more complicated measurement-based closed-loop feedback schemes.
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