arXiv:2106.13705v2 [quant-ph] 14 Oct 2021 Quantum Optimal Control of Nuclear Spin Qudecimals in 87 Sr Sivaprasad Omanakuttan, 1, ∗ Anupam Mitra, 1 Michael J. Martin, 2, 1 and Ivan H Deutsch 1, † 1 Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA 2 Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544 (Dated: October 15, 2021) We study the ability to implement unitary maps on states of the I =9/2 nuclear spin in 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 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 inhomogenieties in the light shift. We find that with an rf-Rabi frequency of Ω rf and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time T =4.5π/Ω rf with average fidelity 〈F ψ 〉 =0.9992, and an arbitrary Haar-random SU(10) map in a time T = 24π/Ω rf with average fidelity 〈FU 〉 =0.9923. Ultracold ensembles of alkaline-earth atoms trapped in optical lattices or arrays of optical tweezers are a power- ful platform for quantum information processing (QIP), including atomic clocks and sensors [1–5], simulators of many-body physics [6–11], and general purpose quantum computers [7, 12, 13]. The ability to optically manip- ulate coherence in single-atoms via ultranarrow optical resonances on the intercombination lines, together with the ability to create high-fidelity entangling interactions between atoms when they are excited to high-lying Ryd- berg states [14–16] provides tools that makes this system highly controllable for such applications. In addition, fermionic species have nuclear spin. As the ground state is a closed shell, there is no electron angular momentum, and the nuclear spin with its weak magnetic moment is highly isolated from the environment. Such nuclear spins in alkaline-earth atoms are thus natural carriers of quantum information given their long coherence times and our ability to coherently control them with magnetic and optical fields. Nuclear spins are also seen as excel- lent carriers of quantum information in the solid state as demonstrated in pioneering experiments including in NV-centers [17] and dopants in silicon [18–21]. Using magneto-optical fields, [22] recently demon- strated the control of qubits encoded in two nuclear-spin magnetic sublevels levels in 87 Sr. The nuclear spin in this atomic species, however, it is not a two-level system; the spin is I =9/2 and there are d =2I +1 = 10 nuclear mag- netic sublevels. Such qudits, here “qudecimals,” have po- tential advantage for QIP. First and foremost, one can en- code a D = d n d =2 n2 dimensional Hilbert space associ- ated with n 2 qubits in n d = n 2 / log 2 d qudits. While only a logarithmic saving, this is meaningful for the qudecimal (log 2 d =3.32), especially when trapping and control of each atom is at a premium. This savings extends to algo- rithmic efficiency, in that the number of elementary two- qudit gates necessary to implement a general unitary map scales as O(n 2 d D 2 )= O  n 2 2 D 2 (log 2 d) 2  [23]. Moreover, qudit architectures can show increased resilience to noise [24] and additional routes to quantum error correction [25]. For example, one can protect against dephasing errors by encoding a qubit in a nuclear spin qudit [26]. In addition, fault-tolerant operation of a quantum computer may be more favorable based on qudit vs. qubit codes [27, 28]. While QIP with qudits has great potential, there are substantial hurdles. State preparation and readout are more challenging for systems with d> 2. Moreover, quantum logic with qudits is more complex. Universal quantum logic with qubits can be achieved with a set of logic gates that include the unitary-generators of SU(2) on each qubit, plus one entangling gate between qubits pairwise. In the case of qudits, in addition to the en- tangling gate, we require unitary-generators of SU(d) for each subsystem [23, 29–31]. Unlike qubits, the Lie alge- bra of such gates are not spanned by the native Hamil- tonians, and thus implementation of this generating set is not straightforward. Different approaches have been studied to implement SU(d) gates [32–36]. One approach is to specify an arbitrary SU(d) unitary matrix through a sequence of so-called Givens rotations acting between pairs of levels [37]. In a landmark experiment, the Inns- bruck group employed this construction to experimen- tally demonstrate universal quantum logic with qudits in a trapped ions ion [38], with performance similar to qubit quantum processors. An alternative powerful approach to implementing uni- versal quantum logic is to employ the tools of quantum optimal control . In this paradigm, one numerically searches for a time-dependent waveform that achieves the desired SU(d) unitary map when one has access to