qiskit.aqua.algorithms.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)

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

__init__(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)

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

__init__(num_oracle_circuits[, …])	type num_oracle_circuits int
confidence_interval(alpha[, kind])	Compute the alpha confidence interval using the method kind.
construct_circuits([measurement])	Construct the Amplitude Estimation w/o QPE quantum circuits.
is_good_state(measurement)	Determine whether a given state is a good state.
post_processing(value)	Post processing of the raw amplitude estimation output \(0 \leq a \leq 1\).
run([quantum_instance])	Execute the algorithm with selected backend.
set_backend(backend, **kwargs)	Sets backend with configuration.

Attributes

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

property 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

property backend

Returns backend.

Return type

Union[Backend, BaseBackend]

confidence_interval(alpha, kind='fisher')

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.

Parameters

• alpha (float) – The confidence level.

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

Return type

List[float]

Returns

The specified confidence interval.

Raises

• AquaError – If run() hasn’t been called yet.

• NotImplementedError – If the method kind is not supported.

construct_circuits(measurement=False)

Construct the Amplitude Estimation w/o QPE quantum circuits.

Parameters

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

Return type

List[QuantumCircuit]

Returns

A list with the QuantumCircuit objects for the algorithm.

property 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.

property 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

is_good_state(measurement)

Determine whether a given state is a good state.

Parameters

measurement (str) – A measurement as bitstring, e.g. ‘01100’.

Return type

bool

Returns

True if the measurement corresponds to a good state, False otherwise.

Raises

ValueError – If self.objective_qubits is not set.

property 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.

post_processing(value)

Post processing of the raw amplitude estimation output \(0 \leq a \leq 1\).

Parameters

value (float) – The estimation value \(a\).

Return type

float

Returns

The value after post processing, usually mapping the interval \([0, 1]\) to the target interval.

property 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

property quantum_instance

Returns quantum instance.

Return type

Optional[QuantumInstance]

property random

Return a numpy random.

run(quantum_instance=None, **kwargs)

Execute the algorithm with selected backend.

Parameters

• quantum_instance (Union[QuantumInstance, Backend, BaseBackend, None]) – the experimental setting.

• kwargs (dict) – kwargs

Returns

results of an algorithm.

Return type

dict

Raises

AquaError – If a quantum instance or backend has not been provided

set_backend(backend, **kwargs)

Sets backend with configuration.

Return type

None

property state_preparation

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

Return type

QuantumCircuit

Returns

The \(\mathcal{A}\) operator as QuantumCircuit.