Lecture Notes on Quantum Algorithms
Andrew Childs (University of Maryland)
The algorithms course. Hidden subgroup problems, quantum walks, Hamiltonian simulation and query complexity, all treated properly. Dense, and worth every hour.
Intermediate · 5 resources
Quantum Fourier transform
The discrete Fourier transform applied to amplitudes, implementable in O(n^2) gates on n qubits — exponentially fewer operations than the classical FFT needs on 2^n numbers.
Learn first:Multi-qubit states & tensor products
Why it matters
The QFT is the engine of the exponential-speedup family: phase estimation, order finding and Shor all reduce to it. Note that you cannot read the transformed amplitudes out — you have to be clever about what you measure.
After this you will be able to
- Write down the QFT circuit for n qubits
- Explain why the exponential gate saving does not immediately give an exponential speedup
- Connect the QFT to period finding
Start here
3 best places to start
Hand-picked and ordered. If you only have time for one, take the first.
Quantum Computing: Lecture Notes
Ronald de Wolf (CWI / QuSoft)
Free, self-contained and rigorous, with exercises. Widely used as the standard course text in Europe and the best single free replacement for a paid textbook.
Quantum Computation and Quantum Information
Nielsen & Chuang (Cambridge University Press)
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.
Also covering this
2 more resources
The paper that created the field as a funded discipline. Read it after you understand phase estimation — it is more approachable than its reputation suggests.
Qiskit Textbook (archived)
Qiskit community
The legendary original Qiskit Textbook, now archived in favour of IBM Quantum Learning. The notebooks still run and its algorithm chapters remain some of the clearest explanations written.
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Topics that list Quantum Fourier transform as a prerequisite.