  We present a cluster expansion method for approximating quantum spin-bath
dynamics in terms of a classical Gaussian stochastic process. The cluster
expansion produces the two-point correlation function of the approximate
classical bath, permitting rapid evaluation of noise-mitigating quantum control
strategies without resorting to computationally intensive dynamical decoupling
models. Our approximation is valid for the wide class of models possessing
negligible back-action and nearly-Gaussian noise. We study several instances of
the central spin decoherence problem in which the central spin and
randomly-located bath spins are alike and dipolarly coupled. For various pulse
sequences, we compare the coherence echo decay computed explicitly quantum
mechanically versus those computed using our approximate classical model, and
obtain agreement in most, but not all, cases. We demonstrate the utility of
these classical noise models by efficiently searching for the 4-pulse sequences
that maximally mitigate decoherence in each of these cases, a computationally
expensive task in the explicit quantum model.
