Quantum Fourier transform
The counterpart to the FFT but on a quantum register — exponentially faster, though the result cannot be read out.
Where the classical fast Fourier transform needs on the order of N log N operations, the quantum version manages it in the square of the number of qubits, so exponentially fewer steps measured against the data size. The catch is that the transform leaves amplitudes that cannot be inspected; you can only measure, and then you get a single outcome.
The value appears when you want to learn something about a periodicity rather than about individual values. That is exactly what Shor's algorithm exploits: factoring reduces to finding the period of a function, and there the transform gives the answer with high probability after a handful of runs.