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

Distance scaling of electric-field noise in a surface-electrode ion trap