Qubits, Superposition & The Bloch Sphere

Whereas classical computer bits exist in a deterministic state of either $0$ or $1$, quantum computing leverages the laws of quantum mechanics. A Qubit (Quantum Bit) can exist in a linear combination of both states simultaneously—a phenomenon known as Quantum Superposition.

1. Summary & Key Takeaways

• Quantum Superposition: A single qubit state $|\psi\rangle$ is represented mathematically as: $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ where $\alpha$ and $\beta$ are complex probability amplitudes satisfying $|\alpha|^2 + |\beta|^2 = 1$.
• Hadamard Gate ($H$): Transforms ground state $|0\rangle$ into equal superposition $\frac{|0\rangle + |1\rangle}{\sqrt{2}}$, granting a $50\%$ chance of measuring either $0$ or $1$.
• Measurement Collapse: Observing a qubit forces its wave function to collapse probabilistically into a single classical state ($|0\rangle$ or $|1\rangle$).
• The Bloch Sphere: A geometric representation of single-qubit quantum states as points on the surface of a unit sphere.

2. Interactive Bloch Sphere Playground

Use the interactive Bloch Sphere below to apply Hadamard ($H$), Pauli-X (Bit Flip), and Pauli-Z (Phase) quantum gates to a single qubit state $|\psi\rangle$, then run Quantum Measurement to collapse the state!

# Quantum Circuit Simulation using Qiskit (Python)
from qiskit import QuantumCircuit, Aer, execute

# 1. Create a 1-qubit quantum circuit
circuit = QuantumCircuit(1, 1)

# 2. Apply Hadamard Gate (H) to create 50/50 Superposition
circuit.h(0)

# 3. Apply Pauli-X Gate (Bit Flip)
# circuit.x(0)

# 4. Measure Qubit State (Collapses Superposition)
circuit.measure(0, 0)

# Execute on Quantum Simulator Backend
backend = Aer.get_backend('qasm_simulator')
job = execute(circuit, backend, shots=1000)
result = job.result()
print("Measurement Counts:", result.get_counts())

3. Mathematical Foundations & Quantum Logic Gates

Single-qubit quantum gates are represented as $2 \times 2$ unitary matrices $U$ acting on state vectors:

Hadamard Gate ($H$)

Places ground states into equal superposition: $H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$

Pauli-X Gate (Bit Flip)

Acts as the quantum equivalent of a classical NOT gate: $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad X|0\rangle = |1\rangle, \quad X|1\rangle = |0\rangle$

Pauli-Z Gate (Phase Flip)

Flips the quantum phase angle $\phi$ without altering measurement probabilities: $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad Z\left(\frac{|0\rangle + |1\rangle}{\sqrt{2}}\right) = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$

4. Quantum Computing vs Classical Computing Matrix