Quantum mechanics: Bras, kets, qubits and all that - the state vector formalism for quantum mechanics (in strange physicists’ notation). I won’t cover the more general approach using density matrices.

Quantum circuit model of quantum computing: A pictorial way of representing the unitary operations and measurements carried out in a quantum computer.

Party tricks: Dense coding and teleportation - some (small) quantum circuits doing non-classical things.

Quantum advantage: Deutsch’s (and Deutch-Josza’s) algorithm - the first evidence that a quantum computer could do something a classical one couldn’t.

Finding a needle in a haystack: Grover’s algorithm - a quantum algorithm for unstructured database search that offers a square root speedup over a naive exhaustive classical search.

Breaking the code: Shor’s algorithm. What really got folk interested in the field - the prospect of breaking public key cryptography using the efficiency of quantum period- finding algorithms.

Solving linear systems and PDEs: The HHL algorithm - something that would be a very bad idea for solving a linear system classically turns out to be very effective in the quantum case.

Quantum Simulation: Feynman was one of the first to point out that quantum computers would be good simulators of quantum systems - some simple examples are discussed.