Function quantumPhaseEstimation

Applies Quantum Phase Estimation to estimate the phase of a unitary operator.

This is the core implementation of QPE that estimates the phase φ such that U|ψ⟩ = e^(2πiφ)|ψ⟩, where U is a unitary operator and |ψ⟩ is an eigenstate.

Algorithm Steps:

1. Initialize n control qubits in |+⟩ state (superposition)
2. Prepare target qubits in the eigenstate |ψ⟩
3. Apply controlled-U^(2^j) for j = 0, 1, ..., n-1
4. Apply inverse QFT to control register
5. Measure control qubits to read out phase estimate

Precision and Success Probability:

• With n control qubits, achieves precision 2^(-n)
• Success probability depends on how close φ is to a n-bit fraction
• For exact n-bit phases, success probability is 1
• For arbitrary phases, success probability ≥ 4/π² ≈ 40.5%