Phase estimation

Quantum phase estimation takes a unitary and an eigenstate and estimates the phase of the corresponding eigenvalue. The bounds and the eigenstate assumption are on the algorithm page. Kitaev's 1995 paper on the Abelian stabilizer problem is the reference this library uses for the measurement primitive. Cleve, Ekert, Macchiavello, and Mosca give the circuit that applies controlled powers of the unitary and finishes with a quantum Fourier transform. They note that the transform also has a semiclassical implementation.

Shor's algorithm can be taught as phase estimation applied to modular multiplication: the order appears as a phase. The 1997 paper writes the Fourier transform explicitly rather than citing this packaging. VQE is best understood as the alternative people reach for when those controlled powers are too long for the device. Aspuru-Guzik and coauthors had proposed phase estimation for molecular energies; Peruzzo and coauthors reduced the coherence demand by estimating an expectation value instead of a phase. Phase estimation has precision that scales more gently with the number of bits you want. It spends that advantage in coherent evolution. Naming the routine without naming that cost is how a textbook circuit gets mistaken for a near-term experiment. The HHL algorithm uses the routine to read eigenvalues of a simulated matrix and then invert them. The state it returns is not a classical list of solution entries.