Fig. 4: Fidelity decay measured using multiple gates.

a Schematic of the measurement sequence. Two ions initialized in \(\left|00\right\rangle\) are rotated into an equal superposition of all states by applying the operator P with the 729 nm laser. After applying the gate operator G(θ) a variable number of times n a reversed preparation pulse P^† is applied. The populations of the resulting state are measured by a set of transfer pulses \({T}_{0}^{j}\), which are resonant π pulses between \(\left|0\right\rangle \leftrightarrow \left|\,j\right\rangle\) to transfer the state \(\left|\,j\right\rangle\) to the S_1/2 manifold, allowing us to distinguish the qudit states. An analysis pulse \({A}_{0,\phi }^{j}\) consisting of a resonant π/2 pulse between \(\left|0\right\rangle \leftrightarrow \left|\,j\right\rangle\) with variable phase ϕ is used to measure the coherence between the \(\left|0\right\rangle\) and \(\left|\,j\right\rangle\) levels. Combined with the transfer pulses, all pairwise coherences can be measured. b A plot of qudit gate fidelity as a function of dimension. The average gate fidelities, shown as red circles, are extracted from fits to the fidelity decay when applying multiple gates G(θ) between P and P^†. The error bars correspond to 1 standard deviation in the fit parameters. A quadratic curve has been fitted to the data to highlight the empirically observed scaling of the fidelity with dimension. The simulated fidelities from a detailed noise model are shown as blue diamonds, see supplementary note 2 for details.