  We investigate schemes for Hamiltonian parameter estimation of a two-level
system using repeated measurements in a fixed basis. The simplest (Fourier
based) schemes yield an estimate with a mean square error (MSE) that decreases
at best as a power law ~N^{-2} in the number of measurements N. By contrast, we
present numerical simulations indicating that an adaptive Bayesian algorithm,
where the time between measurements can be adjusted based on prior measurement
results, yields a MSE which appears to scale close to \exp(-0.3 N). That is,
measurements in a single fixed basis are sufficient to achieve exponential
scaling in N.
