✦Quantum phase estimation

Read an eigenphase using controlled powers.

1. Core idea

If an eigenstate |u⟩ of U is available, controlled U^(2^j) operations encode the unknown phase θ into a counting register. Inverse QFT then converts relative phases into a binary approximation. Accuracy depends on the number of counting qubits and eigenstate preparation.

2. Mathematical foundation

The notation describes a precise relationship among states, transformations or probabilities. Identify each symbol, check the dimensions of the objects, and distinguish amplitudes from the probabilities produced by a measurement.

3. Worked example

4. Reasoning flow

1. Define the initial register and its basis states.
2. Apply the transformations in their intended order.
3. Track amplitude magnitudes and relative phases.
4. Measure in the basis that reveals the desired property.
5. Interpret the result and check the resource assumptions.

Qiskit implementation

Run this standalone Python script with Qiskit installed. It uses exact statevector simulation rather than noisy hardware shots.

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.circuit.library import QFTGate
from qiskit.circuit.library import PhaseGate
qc = QuantumCircuit(3)  # counting qubits 0,1; eigenstate qubit 2
qc.x(2)                 # |1> is eigenstate of P(pi/2)
qc.h([0, 1])            # counting register |++>
qc.cp(3.141592653589793 / 2, 0, 2)  # controlled U
qc.cp(3.141592653589793, 1, 2)      # controlled U^2
qc.append(QFTGate(2).inverse(), [0, 1])
print(qc.draw())
print(Statevector.from_instruction(qc).probabilities_dict())
# The counting register encodes phase 1/4; inspect bit ordering.

5. Check your understanding

6. What to watch out for

Ideal-state calculations assume perfect gates and measurements. When assessing an algorithm, separate its mathematical guarantee from the cost of building oracles, preparing inputs, repeating experiments and correcting physical noise.