The average energy and energy spread of a quantum system are two important factors that affect a system's evolution speed, which is usually defined as the changing rate of the distance between a quantum state and the state during its evolution. Alternatively, the evolution speed of a quantum system could also be studied based on its evolution path. It is found that if we define the evolution speed of a quantum unitary system as the changing rate of the path distance between a quantum state and the state during its evolution, the instantaneous evolution speed is determined by the Hamiltonian's eigenvalues, rather than its expectation value or variance imposed on any quantum state. More specifically, the range of the Hamiltonian's eigenvalues determines the evolution speed of a quantum system through the relation ${\omega }_H=\frac{1}{2}({\lambda }_{max}-{\lambda }_{min})$, which is independent of the quantum state. Given an initial state ${\rho }_0$ in a quantum unitary system, the minimum time required to evolve it to a target state ${\rho }_{\tau }$ is ${\tau }_{min}=\frac{{\mathcal{L}}_{min}({\rho }_0,{\rho }_{\tau })}{{\omega }_H}$, where ${\mathcal{L}}_{min}({\rho }_0,{\rho }_{\tau })$ is the path distance contained in the shortest evolution from ${\rho }_0$ to ${\rho }_{\tau }$. This lower bound of the evolution time is tight and achievable in a finite-energy system.

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• Quantum control
• Quantum foundations
• Quantum state transfer