Quantum Optimal Control of Nuclear Spin Qudecimals in ${}^{87}\text{Sr}$

We study the ability to implement unitary maps on states of the $I=9\mathrm{/}2$ nuclear spin in \textsuperscript{87}Sr, a $d=10$ dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin-resonance and a tensor AC-Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline earth atoms, such as \textsuperscript{87}Sr, have a very favorable figure-of-merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed-limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf-Rabi frequency of ${\mathrm{\Omega }}_{\text{rf}}$ and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T=4.5\pi \mathrm{/}{\mathrm{\Omega }}_{\text{rf}}$ with average fidelity $\langle $\mathcal{F}$_\psi \rangle =0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\pi \mathrm{/}{\mathrm{\Omega }}_{\text{rf}}$ with average fidelity $\langle $\mathcal{F}$_U \rangle = 0.9923$.