| name | Qubits & Gates |
| description | Fundamental operations of quantum computing, including the Bloch Sphere, Hadamard gate, Pauli-X/Y/Z, and CNOT gate. |
Qubit Architecture and Unitary Operations
A qubit is a two-level quantum system defined in a complex vector space $\mathbb{C}^2$, represented geometrically on the Bloch Sphere by spherical coordinates ($\theta, \phi$). Operations are reversible unitary matrices $U$ where $U^{\dagger}U = I$.
- Pauli Group:
- $X$: Bit-flip ($\sigma_x$), $\pi$-rotation around x-axis.
- $Y$: Bit-phase-flip ($\sigma_y$), $\pi$-rotation around y-axis.
- $Z$: Phase-flip ($\sigma_z$), $\pi$-rotation around z-axis.
- Hadamard ($H$): Maps $|0\rangle$ and $|1\rangle$ to mutually unbiased superposition states $(|0\rangle \pm |1\rangle)/\sqrt{2}$. Crucial for initiating phase interference.
- CNOT: The fundamental two-qubit entangling gate. Flips the target qubit conditionally based on the control qubit's state.
flowchart TD
Q[Qubit State in Bloch Sphere] --> U[Apply Unitary Gate]
U --> H[Hadamard Gate]
U --> P[Pauli Gates]
P --> X[Pauli-X]
P --> Y[Pauli-Y]
P --> Z[Pauli-Z]
H --> S[Superposition]
U --> C[CNOT Gate]
C --> E[Multiqubit Interference]
