Course Structure

1. Core Quantum Computing

   • Quantum circuits
   • Algorithms (Grover, Shor)
   • Error correction

2. Simulation & Tools

   • Qiskit tutorials
   • Hardware implementations
   • Analogue quantum simulation

3. Advanced Topics

   • Quantum annealing
   • Variational algorithms (VQA, QAOA)
   • Quantum chemistry (VQE)

4. Special Topics

   • Quantum advantage
   • Machine learning
   • Error mitigation

Historical & Technical Motivation

Evolution of Classical Computing

• Pre-1940s: Mechanical calculators (Pascal, Babbage)
• 1940s: Vacuum tubes (ENIAC, Colossus)
• 1950s: Transistors → Integrated circuits (Moore's Law)
• 2000s: Clock speed plateau (~3-6 GHz due to heat limits)

Limitations:

• Moore's Law: Impossible to continue forever.
• Fundamental physical limitation (quantum tunneling)

Quantum Advantages

• Exponential Speedup:
  • Shor’s algorithm: Factors integers in polynomial time.
  • Grover’s search: Quadratically faster unstructured search.
• NISQ Applications:
  • Quantum simulation (e.g., molecular dynamics).
  • Hybrid classical-quantum algorithms (e.g., QAOA for optimization).

Limitations:

• No proven speedup for NISQ algorithms yet.
• Error rates require mitigation techniques.

Quantum Mechanics Foundations

Key Principles

1. Superposition:

   • Qubit state: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$.
   • Double-slit experiment: Interference patterns demonstrate wave-particle duality.

2. Entanglement:

   • Bell state: $|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$.
   • "Spooky action at a distance" (Einstein) → Verified via Bell tests.

3. Measurement:

   • Projective collapse: $|\psi\rangle \rightarrow |i\rangle$ with probability $|\langle i|\psi\rangle|^2$.
   • Heisenberg Uncertainty: $\Delta x \cdot \Delta p \geq \hbar/2$.

Open Quantum Systems

• Decoherence:

  • $T_1$: Energy relaxation (e.g., $|1\rangle \rightarrow |0\rangle$).
  • $T_2$: Phase damping (loss of superposition).

• Schrödinger’s Cat: Macroscopic superposition → Decoherence in large systems.

References & Tools

Textbooks

1. Applied Quantum Algorithms (Leiden)
2. EPiQC Lecture Notes

Software

• Qiskit: Install via pip install qiskit.
• IBM Quantum: Free access to quantum hardware.