Fundamentals of Quantum Information

Quantum information is at the heart of quantum computing. This course teaches how quantum information is mathematically represented using quantum circuits, and how to manipulate quantum entanglement with those circuits.

Core to the Quantum 101: Quantum Computing and Quantum Internet professional certificate from Delft University of Technology, taught by QuTech researchers with hands-on expertise in quantum information science.

What you’ll learn

• Dirac notation (bra-ket): the standard mathematical language for quantum states, and why physicists and computer scientists use it universally
• Single-qubit states: the Bloch sphere as a geometric picture, and how unitary operations rotate states on it
• Multi-qubit systems: tensor products and how to describe combined quantum systems mathematically
• Quantum circuit notation: reading and constructing circuits from gate sequences, understanding time ordering
• Single-qubit gates in detail: Hadamard, Pauli X, Y, Z, phase gates S and T, and rotation gates
• Two-qubit gates: CNOT, CZ, SWAP, and controlled-U gates
• How to create entanglement: the Hadamard-CNOT circuit for Bell state preparation and its algebraic analysis
• Bell states: the four maximally entangled two-qubit states, their properties, and their applications
• The no-cloning theorem: why quantum information cannot be copied, with proof
• Quantum teleportation: the complete circuit using shared entanglement and two classical bits to transmit a quantum state
• Superdense coding: using one entangled pair to communicate two classical bits

Course structure

The course runs for approximately eight weeks at six to eight hours per week.

Weeks one and two establish the mathematical language: Dirac notation, state vectors, the rules for combining quantum systems via tensor products, and inner products. These tools are introduced through quantum examples from the start - there is no classical detour.

Weeks three and four cover single-qubit and two-qubit gates in depth, with emphasis on understanding what each gate does geometrically (on the Bloch sphere) and algebraically (as a matrix).

Weeks five and six focus entirely on entanglement: its mathematical characterisation, the Bell inequality as a proof that entanglement is not just classical correlation, and how to construct entangled states using circuits.

Weeks seven and eight cover quantum information protocols: teleportation, superdense coding, and an introduction to entanglement-based cryptography. Weekly quizzes and graded problem sets are included in the certificate track.

Who is this for?

• Learners who have completed an introductory quantum computing course and want rigorous mathematical foundations
• Physics, mathematics, or engineering graduates entering the quantum computing field
• Software engineers who want to understand Qiskit circuits at a deeper theoretical level
• Anyone pursuing the Quantum 101 professional certificate from Delft University

Prerequisites

Linear algebra is essential: comfortable with matrix multiplication, complex numbers, and the concept of a vector space. Basic probability theory (expectation values, probability distributions) is also required. High school physics is helpful but not necessary - the course develops quantum mechanics from mathematical formalism.

Completion of “The Quantum Internet and Quantum Computers: How Will They Change the World?” is recommended as conceptual context, though not strictly required.

Hands-on practice

Problem sets require:

• Calculating quantum states after gate sequences by hand (matrix-vector multiplication)
• Constructing circuits to produce specific entangled states
• Verifying the quantum teleportation protocol step by step algebraically
• Proving simple properties of quantum gates (unitarity, Hermitian conjugates)

Some exercises use Python and circuit simulation tools from the Quantum 101 software stack. The mathematical rigour is higher than most introductory courses - you write out state vectors explicitly and verify circuit equivalences algebraically.

Why take this course?

Fundamentals of Quantum Information is the mathematical backbone of the entire Quantum 101 professional certificate. The rigour here - Dirac notation, tensor products, formal entanglement characterisation - is what separates Delft’s programme from more casual introductions.

If your goal is to work in quantum computing research, software, or hardware engineering, this level of mathematical fluency is non-negotiable. QuTech’s instructors are active researchers who bring precision and depth that you will not find in a Udemy course or popular science book. This is where quantum computing education becomes serious.

Related tutorials

Practise the concepts from this course with these hands-on tutorials:

• What Is a Qubit? - A plain-English explanation of qubits, superposition, and the Bloch sphere
• Quantum Gates Explained - How quantum gates work, with circuit diagrams and Qiskit examples for every common gate
• Quantum Entanglement Explained - What entanglement is, how to create Bell states in Qiskit, and its three main applications