Quantum Phase Estimation 2D

Hidden Eigenphase φ

φ (true phase) 0.300

Counting Register Size

Estimate

2^n = 32 outcomes | peak at m = 10

Best estimate φ̂

0.3125

|φ̂ − φ| error

0.0125

Peak probability

96.0%

Qubits (2^n)

How it works

Quantum Phase Estimation finds the phase φ hidden in an eigenvalue e^(2πiφ) of a unitary U, given an eigenvector |u⟩ of U. It underlies Shor's algorithm and quantum chemistry solvers.

1. H^⊗n on n counting qubits |0⟩^n
2. Controlled-U^(2^k) kicks phase e^(2πi·2^k·φ) onto qubit k
3. Apply inverse QFT to the counting register
4. Measure → integer m with probability

P(m) = (1/N²) · | sin(πN·δ) / sin(π·δ) |²,  δ = φ − m/N,  N = 2^n

φ slider / presets — the true hidden phase being estimated.
n=3..6 — more counting qubits ⇒ N = 2^n possible outcomes ⇒ the inverse-QFT peak narrows and φ̂ = m/N gets closer to φ.
Bars around the circle — the exact post-inverse-QFT probability P(m) for every possible measured integer m, placed at angle 2π·m/N. Drag to pan, scroll/pinch to zoom.
Pointer — points at the true angle 2π·φ for comparison against the tallest bar.
Measure Register — samples one outcome m from P(m) via the Born rule, exactly as a real quantum computer's readout would.

When φ is an exact multiple of 1/N (try φ=1/4 with n≥2), the distribution collapses to a single bar with probability 1 — QPE returns φ exactly. Otherwise probability leaks into neighboring bins, and the leakage shrinks as n grows.

P(m) probability bars True phase pointer Last measured outcome

Drag to pan · Scroll/pinch to zoom

Quantum Phase Estimation (2D)