← Back to Quantum Computing 101

Course Syllabus: Quantum Computing 101

Instructor: Dr. Taha Selim (General Director MolKet SAS, Co-founder iQafe)

License: MIT License

Course Description: This course provides a rigorous introduction to quantum computing, taking learners from absolute beginners to a professional understanding of quantum states, multi-qubit systems, and algorithms. It bridges high-level architectural conceptualizations with exact linear algebra, Dirac notation, and unitary matrix transformations used to program real quantum hardware.

Part 1: Foundations of Quantum Information

Module 1: Introduction and Quantum Workflows

Core concepts: Understanding the quantum computer as a machine using quantum mechanics at atomic and subatomic levels to solve complex tasks faster than classical architectures.

  • Quantum state preparation: initialize qubits into coherent states.
  • Quantum gate operations: manipulate qubit states with circuit blocks.
  • Measurement: collapse superpositions to extract classical binary outputs.
  • Classical post-processing: analyze raw quantum output on classical systems.
  • Hybrid workflow loop: problem mapping, Hamiltonian formulation, qubit encoding, QPU execution, and classical refinement.

Module 2: The Quantum Bit (Qubit) and Superposition

Core concepts: Moving from rigid binary registers to dual-level fluid quantum systems.

  • Classical bits (0 or 1) versus qubits with simultaneous state combinations.
  • Parallel computing intuition and system-level speedup framing.
  • Normalization requirement for physical states:
$$|\alpha|^2 + |\beta|^2 = 1$$
  • Qubits as state vectors inside a 2D complex vector space.

Module 3: Mathematical Foundations and Dirac Notation

Core concepts: Formal mathematical frameworks used to model quantum systems.

  • Hilbert space foundations.
  • Ket vector:
$$|\Psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}$$
  • Bra vector (conjugate transpose):
$$\langle\Psi| = \begin{bmatrix} \alpha^* & \beta^* \end{bmatrix}$$
  • Basis states:
$$|0\rangle = \begin{bmatrix}1 \\ 0\end{bmatrix}, \quad |1\rangle = \begin{bmatrix}0 \\ 1\end{bmatrix}$$
  • Inner product and normalization:
$$\langle\Psi|\Psi\rangle = |\alpha|^2 + |\beta|^2 = 1$$
  • Outer products: |Psi>

Module 4: Multi-Qubit Systems and Tensor Products

Core concepts: Scaling from isolated qubits to interconnected composite systems.

  • Tensor product (otimes) for composite Hilbert spaces.
  • Computational basis construction: |00>, |01>, |10>, |11>.
$$\begin{bmatrix}1 \\ 0\end{bmatrix} \otimes \begin{bmatrix}1 \\ 0\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix} = |00\rangle$$
  • Controlled-NOT gate matrix:
$$CX = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{bmatrix}$$
  • Target flips when control qubit is |1>.

Module 5: Quantum Circuits and Unitary Matrices

Core concepts: Sequential quantum operations represented as circuit diagrams and matrix transformations.

  • Circuit anatomy: quantum registers, gate actions, measurement triggers, classical registers.
  • Unitary operator rule:
$$U^{\dagger}U = I$$
  • Tensor ordering and line-aware gate distribution.
  • Example matrix:
$$U = X \otimes I = \begin{bmatrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{bmatrix}$$

Module 6: Visualizing the Hilbert Space

Core concepts: Geometric scaling laws for multi-qubit registers.

  • n-qubit dimension grows as 2^n.
  • 2-qubit systems map into 4D complex vector spaces.
  • Example: apply H otimes I and track amplitude changes in 4D state tables.

Part 2: Measurement, Mechanics, and Advanced Transforms

Module 7: Full Quantum Measurement and Wavefunction Collapse

Core concepts: Extracting classical statistics from quantum probability amplitudes.

  • Collapse from fluid states into deterministic classical vectors.
  • Born rule for probabilities:
$$P = |c_i|^2$$
  • Gate-level tracing with Pauli-X and Hadamard:
$$X = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad H = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$$

Module 8: Partial Measurement and Selective Collapse

Core concepts: Measure targeted qubits while preserving nearby subsystem structure.

  • Localized measurement of q0 with q1 unmeasured.
  • Projection operators over composite spaces:
$$\hat{P}_{q_0}(0) \otimes I$$
  • Post-collapse renormalization enforces sum |c_i|^2 = 1.

Module 9: Visualizing Quantum States - The Bloch Sphere

Core concepts: Geometric model for a single qubit on a 3D unit sphere.

  • State parameterization by polar angle theta and azimuth phi:
$$|q\rangle = \cos\!\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\!\left(\frac{\theta}{2}\right)|1\rangle$$
  • Basis poles: +z corresponds to |0>, -z corresponds to |1>.
  • Superposition transformations rotate vectors toward the equatorial plane.

Module 10: Quantum Entanglement and Bell States

Core concepts: Non-local correlations between qubits with no classical counterpart.

  • Determine when system states cannot be factorized into independent qubits.
  • Generate Bell states with H on control followed by CX on target.
  • Bell state verification example:
$$|B_{00}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) = \begin{bmatrix}\frac{1}{\sqrt{2}} \\ 0 \\ 0 \\ \frac{1}{\sqrt{2}}\end{bmatrix}$$

Part 3: Advanced Algorithm Extensions (New Chapter Additions)

Module 11: The Classical Fourier Transform Bridge

Core concepts: Frequency-domain mathematics before quantum adaptation.

  • Time domain x(t) to frequency domain X(f) transformation.
  • Discrete Fourier Transform:
$$X_k = \sum_{n=0}^{N-1} x_n e^{-i2\pi kn/N}$$
  • Matrix form X = F x and complexity contrast: DFT O(N^2) versus FFT O(N log N).

Module 12: The Quantum Fourier Transform (QFT)

Core concepts: Quantum version of DFT acting directly on register amplitudes.

  • Same Fourier kernel structure with quantum-state execution.
  • Scaling advantage: QFT O(n^2) over n qubits where N = 2^n.
  • Controlled phase shift block:
$$R_d = \begin{bmatrix}1 & 0 \\ 0 & e^{j\pi/2^d}\end{bmatrix}$$
  • Binary fraction phase expansion and circuit mapping with Hadamard plus controlled phase gates.

Final Course Learning Outcomes