Between measurements, a quantum state evolves according to a precise rule: the Schrodinger equation. The key ingredient is the Hamiltonian (H-hat), which encodes the system's energy. The Hamiltonian determines how the state changes from moment to moment -- it is the engine of quantum time evolution. In circuit language, each gate is like a carefully chosen chunk of this continuous evolution, applied in one step.

A gate in a circuit is like a saved move in a game -- it applies a specific transformation in one step. The Schrodinger equation is the deeper physics engine that generates such moves continuously. The gate model discretizes what is really a smooth process.

The Hamiltonian H-hat is the energy operator. The Schrodinger equation generates unitary evolution, meaning the total probability stays at 1 and the evolution is reversible. Phase accumulates during this evolution even when measurement probabilities appear unchanged at an intermediate step. This is why a gate like Z can change the state without changing the immediate measurement statistics.