Extended Data Fig. 2: Experimental Details.

a, Simplified level scheme of ⁴⁰Ca⁺. Doppler cooling and state detection is performed on the short-lived S_1/2 ↔ P_1/2 transition with a life time of ≈ 9ns. Qubits are encoded in the Zeeman states \(\left\vert 0\right\rangle ={S}_{m = -1/2}\) and \(\left\vert 1\right\rangle ={D}_{m = -1/2}\), while qudits are encoded only in the D_5/2 manifold. Any excitation of a D_5/2 state decays in T₁ ≈ 1.1s to the S_1/2 ground states; this transition is addressed by a narrow-band laser with a coherence time of T₂ = 92(9)ms. b, Qudit circuit for implementing a mixed-dimensional controlled rotation (C-ROT) gate. For each 5-dimensional qudit, we consider a two-level subspace, containing the states \(\left\vert 0\right\rangle\) and \(\left\vert 1\right\rangle\), coupled to an auxiliary ground state \(\left\vert g\right\rangle\). The conditional interaction with the phonon mode (blue) depending on the control state is shown by the red triangles, with its orientation indicating the creation / annihilation of a phonon. If the motional mode is excited, three BSB pulses act locally on the target qudit, realizing the rotation. At the end of the sequence, the qudits are again disentangled from the motion. In the inset on the right, we introduce a symbol for the C-ROT operation CROT(θ, ϕ), which allows us to draw controlled qudit operations in quantum circuits in a similar manner to qubit gates, see for example Fig. 1d. c, AC Stark shifts on spectator levels for an excitation of the blue sideband of the S_−1/2 ↔ D_−1/2 transition. While the AC Stark shift on this transition is directly compensated by a second, off-resonant laser beam, the spectator levels D_−5/2, D_+1/2 and D_+3/2 remain shifted by a few kHz with respect to the S_−1/2 ground state; we measure no shift for the D_−3/2 state. The individual blue dots correspond to different runs over the course of several days and the uncertainty represents one standard deviation of the fit uncertainty. These spectator shifts are compensated in software by storing a phase register for each qudit state and phase-shifting the subsequent operations on the affected qudit transitions by an appropriate amount.