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class MaximumLikelihoodAmplitudeEstimation(evaluation_schedule, minimizer=None, quantum_instance=None, sampler=None)

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Bases: qiskit.algorithms.amplitude_estimators.amplitude_estimator.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(opens in a new tab).

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

Quantum Amplitude Amplification and Estimation. arXiv:quant-ph/0005055(opens in a new tab).


  • evaluation_schedule – 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 – 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.
  • quantum_instance – Pending deprecation: Quantum Instance or Backend
  • sampler – A sampler primitive to evaluate the circuits.


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



static MaximumLikelihoodAmplitudeEstimation.compute_confidence_interval(result, alpha, kind='fisher', apply_post_processing=False)

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 – A maximum likelihood amplitude estimation result.
  • alpha – The confidence level.
  • kind – The method to compute the confidence interval. Defaults to ‘fisher’, which computes the theoretical Fisher information.
  • apply_post_processing – If True, apply post-processing to the confidence interval.


The specified confidence interval.


  • AlgorithmError – If run() hasn’t been called yet.
  • NotImplementedError – If the method kind is not supported.


MaximumLikelihoodAmplitudeEstimation.compute_mle(circuit_results, estimation_problem, num_state_qubits=None, return_counts=False)

Compute the MLE via a grid-search.

This is a stable approach if sufficient gridpoints are used.


  • circuit_results – A list of circuit outcomes. Can be counts or statevectors.
  • estimation_problem – The estimation problem containing the evaluation schedule and the number of likelihood function evaluations used to find the minimum.
  • num_state_qubits – The number of state qubits, required for statevector simulations.
  • return_counts – If True, returns the good counts.


The MLE for the provided result object.


MaximumLikelihoodAmplitudeEstimation.construct_circuits(estimation_problem, measurement=False)

Construct the Amplitude Estimation w/o QPE quantum circuits.


  • estimation_problem – The estimation problem for which to construct the QAE circuit.
  • measurement – Boolean flag to indicate if measurement should be included in the circuits.


A list with the QuantumCircuit objects for the algorithm.



Run the amplitude estimation algorithm on provided estimation problem.


estimation_problem (EstimationProblem) – The estimation problem.

Return type



An amplitude estimation results object.


  • ValueError – A quantum instance or Sampler must be provided.
  • AlgorithmError – If state_preparation is not set in estimation_problem.
  • AlgorithmError – Sampler job run error



Pending deprecation; Get the quantum instance.

Return type

QuantumInstance | None


The quantum instance used to run this algorithm.


Get the sampler primitive.

Return type

BaseSampler | None


The sampler primitive to evaluate the circuits.

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