Quantum Computing Course

Lesson 1 • Entanglement and Correlations

Defines entangled states and Bell pairs, distinguishing them from classical correlations. Motivates entanglement as a computational resource.

Lesson 2 • Superposition and Wave Functions

Introduces quantum state vectors and the principle of superposition. Connects wave function formalism to qubit register behaviour.

Lesson 3 • Quantum Measurement and Collapse

Explains Born rule, measurement bases, and post-measurement state collapse. Grounds students in why measurement is irreversible and probabilistic.

Lesson 4 • Classical versus Quantum Information

Contrasts bits with qubits and probabilistic versus deterministic computation. Establishes why quantum systems offer fundamentally different computational power.

Lesson 1 • Eigenvalues and Spectral Decomposition

Covers eigenvalue problems and diagonalisation of Hermitian matrices. Directly supports understanding of quantum phase estimation and Hamiltonian simulation.

Lesson 2 • Density Matrices and Mixed States

Extends pure-state formalism to density operators for mixed and open systems. Prepares students for noise modelling and quantum error analysis.

Lesson 3 • Vector Spaces and Hilbert Spaces

Defines complex vector spaces and inner products used to represent quantum states. Provides the geometric foundation for all subsequent quantum formalism.

Lesson 4 • Tensor Products and Multi-Qubit Systems

Introduces tensor products to construct multi-qubit state spaces. Enables students to analyse registers and two-qubit gate operations.

Lesson 1 • Universal Gate Sets

Defines universality and proves that small gate sets can approximate any unitary. Connects gate decomposition to practical hardware compilation.

Lesson 2 • Circuit Compilation and Optimisation

Introduces gate cancellation, commutation rules, and depth reduction techniques. Prepares students to translate high-level circuits to hardware-native gates.

Lesson 3 • Quantum Circuit Diagrams

Teaches circuit notation, wire conventions, and reading multi-qubit diagrams. Provides the visual language used throughout algorithm design.

Lesson 4 • Single-Qubit Gates

Introduces Pauli, Hadamard, phase, and rotation gates with Bloch sphere visualisation. Establishes the building blocks for all single-qubit transformations.

Lesson 1 • Deutsch-Jozsa and Bernstein-Vazirani

Analyses the first provable quantum speedups over classical deterministic algorithms. Demonstrates interference-based problem solving with concrete circuits.

Lesson 2 • Quantum Parallelism and Oracle Models

Explains how superposition enables simultaneous function evaluation and defines oracle abstraction. Sets the conceptual stage for query-complexity-based speedups.

Lesson 3 • Quantum Phase Estimation

Builds QPE from QFT and controlled-unitary operations to extract eigenphases. Directly enables factoring, simulation, and linear systems algorithms.

Lesson 4 • Quantum Fourier Transform

Derives the QFT circuit from the discrete Fourier transform and analyses its efficiency. Serves as the core subroutine for phase estimation and Shor's algorithm.

Lesson 1 • Quantum Walk Algorithms

Introduces discrete and continuous quantum walks and their search applications. Provides an alternative algorithmic paradigm beyond oracle and Fourier methods.

Lesson 2 • Complexity Theory for Quantum Algorithms

Defines BQP, QMA, and their relationships to classical complexity classes. Equips students to evaluate quantum advantage claims rigorously.

Lesson 3 • HHL Linear Systems Algorithm

Derives the HHL algorithm for solving sparse linear systems with exponential speedup. Analyses preconditions and practical limitations for real-world use.

Lesson 4 • Shor's Factoring Algorithm

Reduces integer factoring to order-finding and implements it via QPE. Demonstrates exponential speedup over best classical factoring methods.

Lesson 1 • Other Emerging Qubit Technologies

Surveys neutral atoms, topological qubits, and spin qubits in semiconductors. Assesses maturity, error rates, and scalability prospects of each approach.

Lesson 2 • Noise, Decoherence, and Error Sources

Characterises T1, T2 times, gate errors, and crosstalk as primary noise sources. Provides the physical basis for understanding error correction requirements.

Lesson 3 • Trapped Ion and Photonic Platforms

Analyses ion trap gate mechanisms and photonic qubit encoding schemes. Contrasts coherence times, gate speeds, and connectivity with superconducting systems.

Lesson 4 • Superconducting Qubit Systems

Covers transmon qubit design, microwave control, and cryogenic requirements. Explains why superconducting platforms dominate current quantum processors.

Lesson 1 • Fault-Tolerant Gate Operations

Covers transversal gates, magic state distillation, and fault-tolerant gadgets. Explains how to perform universal computation without spreading errors.

Lesson 2 • Classical Error Correction Review

Revisits repetition and Hamming codes to establish correction principles. Motivates why classical techniques cannot be directly applied to quantum states.

Lesson 3 • Threshold Theorems and Overhead

States the threshold theorem and quantifies physical-to-logical qubit overhead. Connects error rates to the feasibility of large-scale fault-tolerant algorithms.

Lesson 4 • CSS and Surface Codes

Derives Calderbank-Shor-Steane codes and the surface code from stabiliser formalism. Focuses on the surface code as the leading near-term error correction candidate.

Lesson 1 • Quantum Cryptography and Security

Examines quantum key distribution, post-quantum cryptography, and Shor's threat to encryption. Guides organisations in assessing cryptographic migration needs.

Lesson 2 • Variational Quantum Algorithms

Covers VQE and QAOA as hybrid classical-quantum approaches for near-term devices. Analyses ansatz design, parameter optimisation, and noise resilience.

Lesson 3 • Optimisation and Finance Applications

Maps combinatorial optimisation and portfolio problems to quantum algorithms. Evaluates realistic timelines for quantum advantage in financial use cases.

Lesson 4 • Quantum Machine Learning Overview

Surveys quantum kernel methods, quantum neural networks, and data encoding strategies. Critically evaluates claimed speedups and practical data-loading bottlenecks.