Quantum Phase Estimation

QPE estimates the phase of unitary time evolution.

(1)\[U = e^{-iHt}\]

For an eigenstate of the Hamiltonian, the unitary in (1) encodes the energy in a measurable phase.

QPE extracts eigenvalues from phase evolution:

\[ U = e^{-iHt} \]

Eigenstate relation:

\[ U|\psi\rangle = e^{-iEt}|\psi\rangle \]

Energy recovered via:

\[ E = -\frac{2\pi \theta}{t} \]

Registers:

• ancilla → phase precision

• system → approximate eigenstate

Tradeoffs:

• ancilla count vs precision

• Trotter depth vs error

• initial state overlap vs success probability

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