2.3. Time-evolution

The state of the classical particles change according to the Newton equation, which is a set of ordinary differential equations. Equivalently, the trajectory can be computed from the Hamilton equations of motion, by measurement

\[ \dot{q} = \frac{\partial H}{\partial p}, \qquad \dot{p} = -\frac{\partial H}{\partial q} \]

where \(H\) is a Hamiltonian.

The state vector of a quantum system evolves in time according to the Schrödinger equation

\[ i \frac{\partial}{\partial t} |\psi\rangle = H |\psi\rangle \]

where \(H\) is Hamiltonian operator. Equivalently, the solution can be written as

\[ |\psi(t)\rangle = U(t) |\psi(0)\rangle \]

where \(U(t)\) is a unitary operator and \(|\psi(0)\rangle\) an initial state. We call this type of evolution unitary evolution. Quantum computation solves this equation. We will solve it using Qiskis.

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(version 0.0.1:   Lat modified on April 7, 2022)