---
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.

```mermaid
%%{init: {"theme": "default", "flowchart": {"useMaxWidth": true}}}%%
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]
```