Simple intuition
Gates in a circuit are like carefully chosen chunks of time evolution. In general quantum mechanics, the Hamiltonian tells the state how to change from one moment to the next.
Precise explanation
The Hamiltonian Ĥ is the energy operator. The Schrödinger equation determines unitary evolution, meaning the total probability stays normalized over time until a measurement is applied.
Example or analogy
Analogy: a gate is like a saved move in a game, while the Schrödinger equation is the deeper physics engine that generates such moves continuously.
Common misconception
The Schrödinger equation does not describe the measurement step itself. It describes the continuous evolution of the state between measurements.