Overparameterization of Realistic Quantum Systems*

In order for quantum computing devices to accomplish preparation of quantum states, or compilation of operators, exceptional control of experimental parameters is required. The optimal parameters, such as time dependent magnetic fields for nuclear magnetic resonance, are found via classical simulation and optimization. Such idealized parameterized quantum systems have been shown to exhibit different regimes of training during optimization, such as overparameterization and lazy training, where global optima may potentially be reached exponentially quickly, while parameters negligibly change (Larocca et al., arXiv:2109.11676, 2021). Here, we study the effects of imposing constraints on the controls, such as bounding or sharing parameters across operators, and relevant noise channels are added after each time step. The constrained system is able to reach the overparameterized regime for certain noise models, however an order of magnitude more time steps are required. Compromises arise between numerical feasibility of exponential convergence, and experimental feasibility depending on the resolution of controls. This realistic approach offers insight into quantum control, as well as quantum learning, and will be tested in physical hardware.