# MaximumLikelihoodAmplitudeEstimation¶

class MaximumLikelihoodAmplitudeEstimation(num_oracle_circuits, state_preparation=None, grover_operator=None, objective_qubits=None, post_processing=None, a_factory=None, q_factory=None, i_objective=None, likelihood_evals=None, quantum_instance=None)[source]

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.

References

[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.

Parameters
• num_oracle_circuits (int) – The number of circuits applying different powers of the Grover oracle Q. The (num_oracle_circuits + 1) executed circuits will be [id, Q^2^0, …, Q^2^{num_oracle_circuits-1}] A |0>, where A is the problem unitary encoded in the argument a_factory. Has a minimum value of 1.

• state_preparation (Union[QuantumCircuit, CircuitFactory, None]) – A circuit preparing the input state, referred to as $$\mathcal{A}$$.

• grover_operator (Union[QuantumCircuit, CircuitFactory, None]) – The Grover operator $$\mathcal{Q}$$ used as unitary in the phase estimation circuit.

• objective_qubits (Optional[List[int]]) – A list of qubit indices. A measurement outcome is classified as ‘good’ state if all objective qubits are in state $$|1\rangle$$, otherwise it is classified as ‘bad’.

• post_processing (Optional[Callable[[float], float]]) – A mapping applied to the estimate of $$0 \leq a \leq 1$$, usually used to map the estimate to a target interval.

• a_factory (Optional[CircuitFactory]) – The CircuitFactory subclass object representing the problem unitary.

• q_factory (Optional[CircuitFactory]) – The CircuitFactory subclass object representing. an amplitude estimation sample (based on a_factory)

• i_objective (Optional[int]) – The index of the objective qubit, i.e. the qubit marking ‘good’ solutions with the state |1> and ‘bad’ solutions with the state |0>

• likelihood_evals (Optional[int]) – The number of gridpoints for the maximum search of the likelihood function

• quantum_instance (Union[QuantumInstance, Backend, BaseBackend, None]) – Quantum Instance or Backend

Attributes

 MaximumLikelihoodAmplitudeEstimation.a_factory Get the A operator encoding the amplitude a that’s approximated, i.e. MaximumLikelihoodAmplitudeEstimation.backend Returns backend. MaximumLikelihoodAmplitudeEstimation.grover_operator Get the $$\mathcal{Q}$$ operator, or Grover operator. MaximumLikelihoodAmplitudeEstimation.i_objective Get the index of the objective qubit. MaximumLikelihoodAmplitudeEstimation.objective_qubits Get the criterion for a measurement outcome to be in a ‘good’ state. MaximumLikelihoodAmplitudeEstimation.q_factory Get the Q operator, or Grover-operator for the Amplitude Estimation algorithm, i.e. MaximumLikelihoodAmplitudeEstimation.quantum_instance Returns quantum instance. MaximumLikelihoodAmplitudeEstimation.random Return a numpy random. MaximumLikelihoodAmplitudeEstimation.state_preparation Get the $$\mathcal{A}$$ operator encoding the amplitude $$a$$.

Methods

 Compute the alpha confidence interval using the method kind. Construct the Amplitude Estimation w/o QPE quantum circuits. Determine whether a given state is a good state. Post processing of the raw amplitude estimation output $$0 \leq a \leq 1$$. Execute the algorithm with selected backend. Sets backend with configuration.