  We investigate anomalous ion-motional heating, a limitation to multi-qubit
quantum-logic gate fidelity in trapped-ion systems, as a function of
ion-electrode separation. Using a multi-zone surface-electrode trap in which
ions can be held at five discrete distances from the metal electrodes, we
measure power-law dependencies of the electric-field noise experienced by the
ion on the ion-electrode distance $d$. We find a scaling of approximately
$d^{-4}$ regardless of whether the electrodes are at room temperature or
cryogenic temperature, despite the fact that the heating rates are
approximately two orders of magnitude smaller in the latter case. Through
auxiliary measurements using application of noise to the electrodes, we rule
out technical limitations to the measured heating rates and scalings. We also
measure frequency scaling of the inherent electric-field noise close to $1/f$
at both temperatures. These measurements eliminate from consideration
anomalous-heating models which do not have a $d^{-4}$ distance dependence,
including several microscopic models of current interest.
