  Accurate control of quantum systems requires precise measurement of the
parameters that govern the dynamics, including control fields and interactions
with the environment. Parameters will drift in time and experiments interleave
protocols that perform parameter estimation with protocols that measure the
dynamics of interest. Here we specialize to a system made of qubits where the
dynamics correspond to a quantum computation. We propose setting aside some
qubits, which we call spectator qubits, to be measured periodically during the
computation, to act as probes of the changing experimental and environmental
parameters. By using control strategies that minimize the sensitivity of the
qubits involved in the computation, we can acquire sufficient information from
the spectator qubits to update our estimates of the parameters and improve our
control. As a result, we can increase the length of experiment where the
dynamics of the data qubits are highly reliable. In particular, we simulate how
spectator qubits can keep the error level of operations on data qubits below a
$10^{-4}$ threshold in two scenarios involving coherent errors: a classical
magnetic field gradient dynamically decoupled with sequences of two or four
$\pi$-pulses, and laser beam instability detected via crosstalk with
neighboring atoms in an ion trap.
