Energetic cost as a consequence of parallel transporting speed limit

Abstract Time-optimal unitary operating quantum systems and energy-efficient quantum gates are vital in quantum engineering. Here, we establish the connection between the energetic cost and the quantum speed limit for parallel evolving of a quantum system. Our results show that the energetic cost is the consequence of parallel transporting speed limit, which opens up the critical role of parallel transport of the quantum state for the quantum technologies such as the quantum simulation, the quantum information processing. Our investigations present a route to construct the energetic-efficient quantum gate. In addition, our theoretical framework provides an alternative method for calculating quantum speed limit. Two typical relevant systems are employed to illustrate our results: the spin-1/2 particle in the rotating magnetic field and the entanglement swap gate.