Quantum Phase Estimation
Quantum phase estimation reads out the eigenphase φ of a quantum operation as a binary fraction. The eigenphase is the hidden turn angle the operation adds to a special state. With t counting qubits you get a t-bit estimate of φ. The answer is exact and certain when φ fits in t binary digits. It is the engine inside Shor's algorithm and quantum chemistry plans. Below you can run a version with 2 counting qubits that reads out φ = 1/4 with a chance of 1.
What question does phase estimation answer?
Why care? Phase estimation is the part of Shor's algorithm that does the real work. It is also the heart of most plans to use quantum computers for chemistry. Learn it once and you understand both.
Every quantum gate or circuit is a unitary. That is an operation you can undo, which keeps the total chance at 1. Call it U.
Some states are special for a given U. They are called eigenstates. When U acts on one of them, nothing you could measure changes. The only effect is that the amplitude gets multiplied by a number. Since unitaries keep the total chance at 1, that number always has size 1. So it is a pure phase, a turn. We write it as e^(2πiφ), with φ between 0 and 1. That number φ is the eigenphase. The multiplying number itself is called the eigenvalue. The eigenphase holds all of its information. (If you have met "eigenvectors" in matrix math, this is the same idea. If not, nothing is missing. The definition above is complete.)
An everyday picture: a bike wheel with a paint dot on the rim. Say U means "turn the wheel by a fixed amount." After the turn it is still the same wheel in the same place. Only the dot's angle changed. φ is how far round the dot moved, as a fraction of a full circle. The picture breaks in one way: you can look at the dot, but a quantum phase is hidden. One measurement cannot see it.
Here is a concrete example we use all chapter. The phase gate P(θ) leaves |0⟩ alone. It 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)). Its eigenphase is φ = 1/4, a quarter turn. And |0⟩ is an eigenstate too, with eigenvalue 1, which means φ = 0.
Quantum phase estimation (QPE) answers this: given a circuit for U and an eigenstate, what is φ, written in binary? Why it matters to an engineer:
- Shor's factoring algorithm is QPE used on a clock-arithmetic unitary (next chapter, reference).
- A molecule's lowest energy is an eigenphase of the unitary that moves it forward in time. That is the basis of quantum chemistry plans (drug discovery reality check).
You cannot measure a phase directly. So QPE has to turn the phase into bits. It does that with two tools you already have: phase kickback and the quantum Fourier transform.
- How does the counting register catch the phase?
- How does the inverse QFT turn phases into bits?
- Run it: estimating φ = 1/4 exactlyINTERACTIVE
- Feed it the other eigenstateINTERACTIVE
- What happens when the phase doesn't fit the register?
- Run it: a phase that needs more bitsINTERACTIVE
- Why is QPE called the engine of quantum algorithms?
- What does QPE look like on real hardware today?
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