# 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]

Bases: qiskit.aqua.algorithms.amplitude_estimators.ae_algorithm.AmplitudeEstimationAlgorithm

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

Methods

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

Attributes

a_factory

Get the A operator encoding the amplitude a that’s approximated, i.e.

A |0>_n |0> = sqrt{1 - a} |psi_0>_n |0> + sqrt{a} |psi_1>_n |1>

see the original Brassard paper (https://arxiv.org/abs/quant-ph/0005055) for more detail.

Returns

the A operator as CircuitFactory

Return type

CircuitFactory

backend

Returns backend.

Return type

Union[Backend, BaseBackend]

grover_operator

Get the $$\mathcal{Q}$$ operator, or Grover operator.

If the Grover operator is not set, we try to build it from the $$\mathcal{A}$$ operator and objective_qubits. This only works if objective_qubits is a list of integers.

Return type

Optional[QuantumCircuit]

Returns

The Grover operator, or None if neither the Grover operator nor the $$\mathcal{A}$$ operator is set.

i_objective

Get the index of the objective qubit. The objective qubit marks the |psi_0> state (called ‘bad states’ in https://arxiv.org/abs/quant-ph/0005055) with |0> and |psi_1> (‘good’ states) with |1>. If the A operator performs the mapping

A |0>_n |0> = sqrt{1 - a} |psi_0>_n |0> + sqrt{a} |psi_1>_n |1>

then, the objective qubit is the last one (which is either |0> or |1>).

If the objective qubit (i_objective) is not set, we check if the Q operator (q_factory) is set and return the index specified there. If the q_factory is not defined, the index equals the number of qubits of the A operator (a_factory) minus one. If also the a_factory is not set, return None.

Returns

the index of the objective qubit

Return type

int

objective_qubits

Get the criterion for a measurement outcome to be in a ‘good’ state.

Return type

Optional[List[int]]

Returns

The criterion as list of qubit indices.

q_factory

Get the Q operator, or Grover-operator for the Amplitude Estimation algorithm, i.e.

$\mathcal{Q} = \mathcal{A} \mathcal{S}_0 \mathcal{A}^\dagger \mathcal{S}_f,$

where $$\mathcal{S}_0$$ reflects about the |0>_n state and S_psi0 reflects about $$|\Psi_0\rangle_n$$. See https://arxiv.org/abs/quant-ph/0005055 for more detail.

If the Q operator is not set, we try to build it from the A operator. If neither the A operator is set, None is returned.

Returns

returns the current Q factory of the algorithm

Return type

QFactory

quantum_instance

Returns quantum instance.

Return type

Optional[QuantumInstance]

random

Return a numpy random.

state_preparation

Get the $$\mathcal{A}$$ operator encoding the amplitude $$a$$.

Return type

QuantumCircuit

Returns

The $$\mathcal{A}$$ operator as QuantumCircuit.