  We have used Ramsey tomography to characterize charge noise in a weakly
charge-sensitive superconducting qubit. We find a charge noise that scales with
frequency as $1/f^\alpha$ over 5 decades with $\alpha = 1.93$ and a magnitude
$S_q(\text{1Hz})= 2.9\times10^{-4}~e^2/\text{Hz}$. The noise exponent and
magnitude of the low-frequency noise are much larger than those seen in prior
work on single electron transistors, yet are consistent with reports of
frequency noise in other superconducting qubits. Moreover, we observe frequent
large-amplitude jumps in offset charge exceeding 0.1$e$; these large discrete
charge jumps are incompatible with a picture of localized dipole-like two-level
fluctuators. The data reveal an unexpected dependence of charge noise on device
scale and suggest models involving either charge drift or fluctuating patch
potentials.
