2. Postulates: Evolution

You have the state. Now you need to move it.

Postulate 2: Evolution (Unitary Dynamics)

The Rule:

The evolution of a closed quantum system is described by a unitary transformation. That is, the state $|\psi\rangle$ of the system at time $t_1$ is related to the state $|\psi'\rangle$ at time $t_2$ by a unitary operator $U$ which depends only on the times $t_1$ and $t_2$.

$$|\psi'\rangle = U|\psi\rangle$$

(Note: In continuous physics, this is derived from the Schrödinger equation. In Quantum Computing, we discretize this into "Gates".)

1. What is "Unitary"?

A matrix $U$ is unitary if its conjugate transpose ($U^\dagger$) is also its inverse ($U^{-1}$).

Mathematically:

$$U^\dagger U = I$$

Where $I$ is the Identity matrix.

2. Why do we care? (Conservation of Probability)

This is not just linear algebra jargon. It is a physical requirement.

If you start with a valid quantum state (total probability = 1), you must end with a valid quantum state (total probability = 1).

If our operators were not unitary, the length of the state vector would stretch or shrink.

Stretch: Total probability > 100%. Impossible.
Shrink: Total probability < 100%. The qubit "disappears."

Unitary matrices are effectively "complex rotations." They rotate the state vector around the Hilbert space without changing its length.

3. Reversibility

Because $U^{-1} = U^\dagger$, quantum mechanics is reversible. If you apply a gate $U$, you can always undo it by applying $U^\dagger$. This is fundamentally different from classical computing, where $x = 0$ destroys the previous value of $x$. In quantum, nothing is ever truly deleted until measurement.

Your Task: verifying the Hadamard Gate

The Hadamard Gate ($H$) is the most important single-qubit gate. It creates superposition.

$$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$

You need to prove it is a valid quantum operator.

Find $H^\dagger$: Calculate the conjugate transpose of $H$. (Note: Since $H$ has only real numbers, the complex conjugate is trivial).
Multiply: Calculate the matrix product $H^\dagger H$.
Verify: Does the result equal the Identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$?

If it does, the gate preserves probability. If not, the math is broken.

2. Postulates: Evolution