Quantum Phase Estimation
Quantum phase estimation reads out the eigenphase φ of a unitary operation as a binary fraction: t counting qubits deliver a t-bit estimate of φ, exactly and deterministically when φ is a t-bit binary fraction. It is the engine inside Shor's algorithm and quantum-chemistry proposals, and below you can run a 2-counting-qubit instance that reads out φ = 1/4 with probability 1.
What question does phase estimation answer?
Every quantum gate or circuit is a unitary — a reversible operation that preserves total probability; call it U. From linear algebra you know eigenvectors: inputs a matrix merely rescales. Unitaries have them too, called eigenstates: states that U leaves physically unchanged except for multiplying the amplitude by a factor. Because unitaries preserve probability, that factor always has magnitude 1 — it is a pure phase, writable as e^(2πiφ) with φ between 0 and 1. That number φ is the eigenphase, and it is the entire information content of the eigenvalue — "eigenvalue" being the standard name for that multiplying factor. (If you have never met eigenvectors in matrix algebra, nothing is missing: the definition just given is self-contained.)
Concrete example, used throughout this chapter. The phase gate P(θ) leaves |0⟩ alone and multiplies the |1⟩ amplitude by e^(iθ) (see phase gates and rotations). So |1⟩ is an eigenstate of P(π/2) with eigenvalue e^(iπ/2) = e^(2πi·(1/4)) — eigenphase φ = 1/4. And |0⟩ is an eigenstate too, with eigenvalue 1, i.e. φ = 0.
Quantum phase estimation (QPE) answers: given a circuit for U and an eigenstate, what is φ, in binary? Why an engineer should care: Shor's factoring algorithm is QPE applied to a modular-arithmetic unitary (next chapter, reference), and a molecule's ground-state energy is an eigenphase of its time-evolution unitary — the basis of quantum chemistry proposals (drug discovery reality check). Phase can't be measured directly — a single measurement is blind to it — so QPE has to convert phase into bit values, and it does so with the two tools you already have: phase kickback and the quantum Fourier transform.
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