Quantum Phase Estimation — Kitaev's Circuit

Estimating eigenphase φ of unitary U|ψ⟩ = e^{2πiφ}|ψ⟩
0.333
6
0.000

Quantum Phase Estimation (Kitaev 1995) uses n ancilla qubits in superposition, controlled-U^{2^k} gates, then inverse QFT to measure the phase φ to n bits. The output probability is sharply peaked at the best n-bit approximation of φ: P(m) = |⟨m|QFT†|φ⟩|² = sin²(πN(φ−m/N)) / N²sin²(π(φ−m/N)). Central to Shor's algorithm, quantum chemistry, and topological quantum computing.