This research investigates the possibility of using quantum optimal control techniques to co-optimize the energetic cost and the process fidelity of a quantum unitary gate. The energetic cost is theoretically defined, and thereby, the gradient of the energetic cost for pulse engineering is derived. The Pareto optimality is empirically demonstrated in the trade-off between process fidelity and energetic cost. Thereafter, two novel numerical quantum optimal control approaches are proposed: i) energy-optimized gradient ascent pulse engineering (EO-GRAPE) as an open-loop gradient-based method, and ii) energy-optimized deep reinforcement learning for pulse engineering (EO-DRLPE) as a closed-loop method. The performance of both methods is probed in the presence of increasing noise. It is found that the EO-GRAPE method performs better than the EO-DRLPE methods with and without a warm start for most experimental settings. Additionally, for one qubit unitary gate, the correlation between the Bloch sphere path length and the energetic cost is illustrated.

"Quantum optimal control is a rapidly growing field with diverse methods and applications. In this work, the possibility of using quantum optimal control techniques to co-optimize the energetic cost and the process fidelity of a quantum unitary gate is investigated. The theoretical definition and quantization of quantum unitary gates, as well as the relationship between the process fidelity and the energetic cost of a quantum unitary gate are explored. Two different quantum optimal control methods to co-optimize both fidelity and energetic cost, i.e., the Gradient Ascent Pulse Engineering method and model-free Deep Reinforcement Learning are investigated. The performance of both quantum optimal control techniques in the presence of noise is probed. We find that the energetic cost of a quantum unitary gate can be quantized by integrating the control pulses and norm of the corresponding Hamiltonian operators over the total time duration of the unitary, and for single qubit gates by calculating the arc length of the quantum unitary gate on the Bloch sphere. A Pareto optimal front between the process fidelity and the energetic cost of a quantum gate is identified, where a lower energetic cost yields an inherently lower process fidelity. A python package called ”EUQOC” (Energy Efficient Universal Quantum Optimal Control) has been created to implement energy optimal quantum gate synthesis, both with the Energy Optimal Gradient Ascent Pulse Engineering (EO-GRAPE) method and by model-free Deep Reinforcement Learning. It is found that the EO-GRAPE method performs better than the reinforcement learning methods, for all noise settings and neural network sizes. For future work, the optimization problem could be translated to the frequency domain to increase the computational efficiency. Furthermore, the relationship between information and energy can be investigated by looking at the complexity of the pulse or the decomposition of the quantum unitary gate."