class MaximumLikelihoodAmplitudeEstimation(evaluation_schedule, minimizer=None, sampler=None)[source]#

Bases: AmplitudeEstimator

The Maximum Likelihood Amplitude Estimation algorithm.

This class implements the quantum amplitude estimation (QAE) algorithm without phase estimation, as introduced in [1]. In comparison to the original QAE algorithm [2], this implementation relies solely on different powers of the Grover operator and does not require additional evaluation qubits. Finally, the estimate is determined via a maximum likelihood estimation, which is why this class in named MaximumLikelihoodAmplitudeEstimation.


[1]: Suzuki, Y., Uno, S., Raymond, R., Tanaka, T., Onodera, T., & Yamamoto, N. (2019).

Amplitude Estimation without Phase Estimation. arXiv:1904.10246.

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

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

  • evaluation_schedule (list[int] | int) – If a list, the powers applied to the Grover operator. The list element must be non-negative. If a non-negative integer, an exponential schedule is used where the highest power is 2 to the integer minus 1: [id, Q^2^0, …, Q^2^(evaluation_schedule-1)].

  • minimizer (MINIMIZER | None) – A minimizer used to find the minimum of the likelihood function. Defaults to a brute search where the number of evaluation points is determined according to evaluation_schedule. The minimizer takes a function as first argument and a list of (float, float) tuples (as bounds) as second argument and returns a single float which is the found minimum.

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


ValueError – If the number of oracle circuits is smaller than 1.



Get the sampler primitive.


The sampler primitive to evaluate the circuits.


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

Compute the alpha confidence interval using the method kind.

The confidence level is (1 - alpha) and supported kinds are ‘fisher’, ‘likelihood_ratio’ and ‘observed_fisher’ with shorthand notations ‘fi’, ‘lr’ and ‘oi’, respectively.

  • result (MaximumLikelihoodAmplitudeEstimationResult) – A maximum likelihood amplitude estimation result.

  • alpha (float) – The confidence level.

  • kind (str) – The method to compute the confidence interval. Defaults to ‘fisher’, which computes the theoretical Fisher information.

  • apply_post_processing (bool) – If True, apply post-processing to the confidence interval.

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


The specified confidence interval.

Return type:

tuple[float, float]

compute_mle(circuit_results, estimation_problem, num_state_qubits=None, return_counts=False)[source]#

Compute the MLE via a grid-search.

This is a stable approach if sufficient grid-points are used.

  • circuit_results (list[dict[str, int]]) – A list of circuit outcomes. Can be counts or statevectors.

  • estimation_problem (EstimationProblem) – The estimation problem containing the evaluation schedule and the number of likelihood function evaluations used to find the minimum.

  • return_counts (bool) – If True, returns the good counts.

  • num_state_qubits (int | None) – [Deprecated]. Number of qubits to process statevector results.


The MLE for the provided result object.

Return type:

float | tuple[float, list[int]]

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

Construct the Amplitude Estimation w/o QPE quantum circuits.

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

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


A list with the QuantumCircuit objects for the algorithm.

Return type:



Run the amplitude estimation algorithm on provided estimation problem.


estimation_problem (EstimationProblem) – The estimation problem.


An amplitude estimation results object.

Return type: