class AmplitudeEstimation(num_eval_qubits, phase_estimation_circuit=None, iqft=None, sampler=None)[source]#

Bases: AmplitudeEstimator

The Quantum Phase Estimation-based Amplitude Estimation algorithm.

This class implements the original Quantum Amplitude Estimation (QAE) algorithm, introduced by [1]. This canonical version uses quantum phase estimation along with a set of \(m\) additional evaluation qubits to find an estimate \(\tilde{a}\), that is restricted to the grid

\[\tilde{a} \in \{\sin^2(\pi y / 2^m) : y = 0, ..., 2^{m-1}\}\]

More evaluation qubits produce a finer sampling grid, therefore the accuracy of the algorithm increases with \(m\).

Using a maximum likelihood post processing, this grid constraint can be circumvented. This improved estimator is implemented as well, see [2] Appendix A for more detail.


This class does not support the EstimationProblem.is_good_state property, as for phase estimation-based QAE, the oracle that identifies the good states must be encoded in the Grover operator. To set custom oracles, the EstimationProblem.grover_operator attribute can be set directly.


[1]: Brassard, G., Hoyer, P., Mosca, M., & Tapp, A. (2000).

Quantum Amplitude Amplification and Estimation. arXiv:quant-ph/0005055.

[2]: Grinko, D., Gacon, J., Zoufal, C., & Woerner, S. (2019).

Iterative Quantum Amplitude Estimation. arXiv:1912.05559.

  • num_eval_qubits (int) – The number of evaluation qubits.

  • phase_estimation_circuit (QuantumCircuit | None) – The phase estimation circuit used to run the algorithm. Defaults to the standard phase estimation circuit from the circuit library, qiskit.circuit.library.PhaseEstimation when None.

  • iqft (QuantumCircuit | None) – The inverse quantum Fourier transform component, defaults to using a standard implementation from qiskit.circuit.library.QFT when None.

  • sampler (BaseSampler | None) – A sampler primitive to evaluate the circuits.


ValueError – If the number of evaluation qubits is smaller than 1.



Get the sampler primitive.


The sampler primitive to evaluate the circuits.


static compute_confidence_interval(result, alpha=0.05, kind='likelihood_ratio', exact=False)[source]#

Compute the (1 - alpha) confidence interval.

  • result (AmplitudeEstimationResult) – An amplitude estimation result for which to compute the confidence interval.

  • alpha (float) – Confidence level: compute the (1 - alpha) confidence interval.

  • kind (str) – The method to compute the confidence interval, can be ‘fisher’, ‘observed_fisher’ or ‘likelihood_ratio’ (default)

  • exact (bool) – Whether the result comes from a statevector simulation or not


The (1 - alpha) confidence interval of the specified kind.


NotImplementedError – If the confidence interval method kind is not implemented.

Return type:

tuple[float, float]

static compute_mle(result, apply_post_processing=False)[source]#

Compute the Maximum Likelihood Estimator (MLE).

  • result (AmplitudeEstimationResult) – An amplitude estimation result object.

  • apply_post_processing (bool) – If True, apply the post processing to the MLE before returning it.


The MLE for the provided result object.

Return type:


construct_circuit(estimation_problem, measurement=False)[source]#

Construct the Amplitude Estimation quantum circuit.

  • estimation_problem (EstimationProblem) – The estimation problem for which to construct the QAE circuit.

  • measurement (bool) – Boolean flag to indicate if measurements should be included in the circuit.


The QuantumCircuit object for the constructed circuit.

Return type:



Run the amplitude estimation algorithm on provided estimation problem.


estimation_problem (EstimationProblem) – The estimation problem.


An amplitude estimation results object.

  • ValueError – If state_preparation or objective_qubits are not set in the estimation_problem.

  • AlgorithmError – Sampler job run error.

Return type:


evaluate_measurements(circuit_results, threshold=1e-06)[source]#

Evaluate the results from the circuit simulation.

Given the probabilities from statevector simulation of the QAE circuit, compute the probabilities that the measurements y/grid-points a are the best estimate.

  • circuit_results (dict[str, int]) – The circuit result from the QAE circuit. Can be either a counts dict or a statevector or a quasi-probabilities dict.

  • threshold (float) – Measurements with probabilities below the threshold are discarded.


Dictionaries containing the a grid-points with respective probabilities and

y measurements with respective probabilities, in this order.

Return type:

tuple[dict[float, float], dict[int, float]]