A small finite set of gates — for example Clifford+T, or CNOT plus arbitrary single-qubit rotations — suffices to approximate any unitary to arbitrary precision.
Why it matters
Hardware only implements a handful of gates natively. Universality is why that is enough, and the Solovay-Kitaev theorem tells you how expensive the translation is.
After this you will be able to
Hand-picked and ordered. If you only have time for one, take the first.
Universally called "Mike & Ike" and still the field's reference text after two decades. Comprehensive rather than gentle — use it as the book you look things up in, not the one you read cover to cover first.
The algorithms course. Hidden subgroup problems, quantum walks, Hamiltonian simulation and query complexity, all treated properly. Dense, and worth every hour.
Topics that list Universal gate sets as a prerequisite.
Encoding one logical qubit across many physical qubits so that errors can be detected and corrected by measuring stabilisers — without ever measuring, and thus destroying, the logical state.
AdvancedTurning an abstract circuit into one a specific device can execute: decomposing into native gates, mapping logical to physical qubits, and inserting SWAPs to satisfy connectivity — while minimising depth.
Advanced