Given a unitary and one of its eigenstates, how do you read out the eigenvalue?
Simon's algorithm (topic 10) found a hidden XOR-period using n Hadamard gates and ordinary linear algebra, because XOR-periods live in the group (ℤ/2ℤ)ⁿ and the Hadamard transform is exactly the Fourier transform over that group. Real periods (the ones behind factoring) live in ℤ/2tℤ instead, a bigger, different group, and finding them needs the Fourier transform for that group: the quantum Fourier transform. These three topics build it, use it to read out an unknown phase, and use that to recover a period: the exact chain Shor's algorithm runs, derived here instead of asserted there.
Suppose U is a unitary and |u⟩ one of its eigenstates: U|u⟩ = e2πiθ|u⟩ for some unknown θ ∈ [0,1). Phase estimation reads θ out, to t bits of precision, using t extra "counting" qubits and t controlled applications of increasing powers of U.
Each counting qubit, prepared in |+⟩ and used to control U2k applied to |u⟩, gets a phase (−1)-style kickback exactly like topics 08–10: except here the kicked-back phase is e2πiθ·2k, a continuous phase rather than just a sign, because θ isn't restricted to {0,1}. Doing this for k=0,1,..., t−1 writes successive doublings of θ across the t counting qubits: a phase-domain analogue of writing θ's binary digits one at a time. The inverse QFT is exactly the tool that reads a phase ladder like this back out as an ordinary number, the same way topic 11's periodic signal reappeared as sparse peaks.
The honest limit: if θ happens to be exactly representable in t bits, phase estimation returns it with certainty. If not (which is the generic case) the outcome is a random t-bit string, but one whose distribution is sharply peaked around the nearest t-bit approximation to θ, not spread out uselessly. That "sharply peaked, not exact" behaviour is precisely what topic 13 needs and gets.
After the t controlled-U2k operations, the counting register holds:
(1/√2ᵗ) Σⱼ₌₀^(2ᵗ−1) exp(2πi θ j) |j⟩
Applying the inverse QFT (topic 11, run backwards: also unitary, since QFT is) gives the amplitude of measuring outcome k:
amplitude(k) = (1/2ᵗ) Σⱼ exp(2πi j(θ − k/2ᵗ))
If θ = m/2t exactly for some integer m, every term in the sum has the same phase when k=m, so the amplitude has magnitude 1 and outcome m is measured with probability exactly 1. Otherwise, the standard bound (Nielsen & Chuang, §5.2.1) guarantees the probability of measuring within 1 of the nearest t-bit estimate of θ·2t exceeds 4/π² ≈ 0.405: a guaranteed floor, not a typical case. Verified numerically: all 2t exactly-representable phases at t=3,4,5 measured with probability 1.000000 at the correct outcome; 200 random non-exact phases at t=8 gave a minimum near-estimate probability of 0.8558: comfortably above the 4/π² guarantee, as expected since that bound is a worst-case floor.
The general-purpose eigenvalue reader Phase estimation is the subroutine inside more than just factoring: it's also the standard way to estimate ground-state energies in quantum chemistry simulation and the formal core of the HHL linear-systems algorithm referenced on the AI ↔ quantum page. Topic 13 is the one specific choice of U (modular multiplication) that turns "read out an eigenvalue" into "find a hidden period," which is the step Shor's algorithm actually needs.
Check yourself
If the phase theta is exactly representable in t bits, what does quantum phase estimation return?
For exactly representable phases the outcome is returned with probability exactly 1. The 4/pi squared figure is only the guaranteed floor when the phase is not exactly representable.