Isolated quantum-state networks in ultracold molecules

Precise rotational-state control is at the heart of recent advances in quantum chemistry, quantum simulation, and quantum computation with bialkali molecules. Each rotational state comprises a rich manifold of hyperfine states arising from combinations of rotation and nuclear spins; this often yields hundreds of transitions available between a given pair of rotational states, and the efficient navigation of this complex space is a current challenge for experiments. Here, we describe an approach based on a heuristic and graph theory to quickly identify optimal sets of states in bialkali molecules. We find pathways to prepare the molecule in a specific state with maximum speed for any desired fidelity. We then examine networks of states where multiple couplings are present at the same time. As example applications, we identify a closed loop of 4 states where there is minimal population leakage out of the loop during simultaneous microwave coupling; we then extend the optimisation procedure to account for decoherence induced by magnetic-field noise and obtain an optimal set of 3 states for quantum computation applications.