Iterative quantum amplitude estimation

Iterative Amplitude Estimation (IQAE)

Iterative Quantum Amplitude Estimation, proposed by Grinko, Gacon, Zoufal, and Woerner in 2019, reimagines how amplitude estimation can work by removing the need for Quantum Phase Estimation entirely. Instead of the deep QPE circuits used in the canonical algorithm, IQAE applies carefully chosen powers of the Grover operator in sequence, narrowing a confidence interval for the target amplitude with each round.

This is a significant practical advantage: QPE requires many ancilla qubits and controlled operations that are expensive on near-term hardware. IQAE needs only the problem qubits themselves, making it far more hardware-friendly while maintaining the same quadratic query complexity.

How it differs from canonical QAE

In the original Brassard et al. algorithm, all the work happens in one shot — a single deep QPE circuit produces a grid-locked estimate. IQAE takes a fundamentally different approach:

1. Start with a wide confidence interval for the angle (which encodes the amplitude).
2. Choose a Grover power such that the scaled interval fits entirely in one half-circle — this ensures the measurement statistics are unambiguous.
3. Run the circuit , measure, and use Clopper-Pearson (or Chernoff) confidence intervals to tighten the bounds.
4. Repeat until the confidence interval width drops below the target precision .

The result is an estimate that, with probability at least , lies within of the true amplitude.

What the example does

The example estimates a Bernoulli probability with target precision and confidence level 95% (). The algorithm iteratively refines its estimate without ever building a QPE circuit.

Getting started

python -m venv .venv && source .venv/bin/activate
pip install -r requirements.lock
python iterative_amplitude_estimation_example.py

The output reports the estimate, error against the target, 95% confidence interval, number of oracle queries (the complexity metric), and the sequence of Grover powers chosen adaptively at each iteration — illustrating how IQAE refines the interval round by round without ever using QPE.

Dependencies

• Python 3.12
• Qiskit (< 2.0)
• NumPy, SciPy

References

• Grinko, D., Gacon, J., Zoufal, C., & Woerner, S. (2019). Iterative Quantum Amplitude Estimation. arXiv:1912.05559

License

Apache 2.0 — derived from Qiskit Algorithms (C) IBM 2018–2024.