qiskit.aqua.algorithms.IterativeAmplitudeEstimation

class IterativeAmplitudeEstimation(epsilon, alpha, confint_method='beta', min_ratio=2, state_preparation=None, grover_operator=None, objective_qubits=None, post_processing=None, a_factory=None, q_factory=None, i_objective=None, initial_state=None, quantum_instance=None)

The Iterative Amplitude Estimation algorithm.

This class implements the Iterative Quantum Amplitude Estimation (IQAE) algorithm, proposed in [1]. The output of the algorithm is an estimate that, with at least probability \(1 - \alpha\), differs by epsilon to the target value, where both alpha and epsilon can be specified.

It differs from the original QAE algorithm proposed by Brassard [2] in that it does not rely on Quantum Phase Estimation, but is only based on Grover’s algorithm. IQAE iteratively applies carefully selected Grover iterations to find an estimate for the target amplitude.

References

[1]: Grinko, D., Gacon, J., Zoufal, C., & Woerner, S. (2019).

Iterative Quantum Amplitude Estimation. arXiv:1912.05559.

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

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

The output of the algorithm is an estimate for the amplitude a, that with at least probability 1 - alpha has an error of epsilon. The number of A operator calls scales linearly in 1/epsilon (up to a logarithmic factor).

Parameters

• epsilon (float) – Target precision for estimation target a, has values between 0 and 0.5

• alpha (float) – Confidence level, the target probability is 1 - alpha, has values between 0 and 1

• confint_method (str) – Statistical method used to estimate the confidence intervals in each iteration, can be ‘chernoff’ for the Chernoff intervals or ‘beta’ for the Clopper-Pearson intervals (default)

• min_ratio (float) – Minimal q-ratio (\(K_{i+1} / K_i\)) for FindNextK

• 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 A operator, specifying the QAE problem

• q_factory (Optional[CircuitFactory]) – The Q operator (Grover operator), constructed from the A operator

• i_objective (Optional[int]) – Index of the objective qubit, that marks the ‘good/bad’ states

• initial_state (Optional[QuantumCircuit]) – A state to prepend to the constructed circuits.

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

Raises

AquaError – if the method to compute the confidence intervals is not supported

__init__(epsilon, alpha, confint_method='beta', min_ratio=2, state_preparation=None, grover_operator=None, objective_qubits=None, post_processing=None, a_factory=None, q_factory=None, i_objective=None, initial_state=None, quantum_instance=None)

The output of the algorithm is an estimate for the amplitude a, that with at least probability 1 - alpha has an error of epsilon. The number of A operator calls scales linearly in 1/epsilon (up to a logarithmic factor).

Parameters

• epsilon (float) – Target precision for estimation target a, has values between 0 and 0.5

• alpha (float) – Confidence level, the target probability is 1 - alpha, has values between 0 and 1

• confint_method (str) – Statistical method used to estimate the confidence intervals in each iteration, can be ‘chernoff’ for the Chernoff intervals or ‘beta’ for the Clopper-Pearson intervals (default)

• min_ratio (float) – Minimal q-ratio (\(K_{i+1} / K_i\)) for FindNextK

• 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 A operator, specifying the QAE problem

• q_factory (Optional[CircuitFactory]) – The Q operator (Grover operator), constructed from the A operator

• i_objective (Optional[int]) – Index of the objective qubit, that marks the ‘good/bad’ states

• initial_state (Optional[QuantumCircuit]) – A state to prepend to the constructed circuits.

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

Raises

AquaError – if the method to compute the confidence intervals is not supported

Methods

__init__(epsilon, alpha[, confint_method, …])	The output of the algorithm is an estimate for the amplitude a, that with at least probability 1 - alpha has an error of epsilon.
construct_circuit(k[, measurement])	Construct the circuit Q^k A |0>.
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.
precision	Returns the target precision epsilon of the algorithm.
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]

construct_circuit(k, measurement=False)

Construct the circuit Q^k A |0>.

The A operator is the unitary specifying the QAE problem and Q the associated Grover operator.

Parameters

• k (int) – The power of the Q operator.

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

Return type

QuantumCircuit

Returns

The circuit Q^k A |0>.

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 precision

Returns the target precision epsilon of the algorithm.

Return type

float

Returns

The target precision (which is half the width of the confidence 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.