  The greatest challenge in achieving the high level of control needed for
future technologies based on coherent quantum systems is the decoherence
induced by the environment. Here, we present an analytical approach that yields
explicit constraints on the driving field which are necessary and sufficient to
ensure that the leading-order noise-induced errors in a qubit's evolution
cancel exactly. We derive constraints for two of the most common types of noise
that arise in qubits: slow fluctuations of the qubit energy splitting and
fluctuations in the driving field itself. By theoretically recasting a phase in
the qubit's wavefunction as a topological winding number, we can satisfy the
noise-cancelation conditions by adjusting driving field parameters without
altering the target state or quantum evolution. We demonstrate our method by
constructing robust quantum gates for two types of spin qubit: phosphorous
donors in silicon and nitrogen-vacancy centers in diamond.
