# 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)[source]

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

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.

Références

[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).

Paramètres
• 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

Lève

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

Methods

 construct_circuit Construct the circuit Q^k A |0>. 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.

Renvoie

the A operator as CircuitFactory

Type renvoyé

CircuitFactory

backend

Returns backend.

Type renvoyé

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.

Type renvoyé

Optional[QuantumCircuit]

Renvoie

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.

Renvoie

the index of the objective qubit

Type renvoyé

int

objective_qubits

Get the criterion for a measurement outcome to be in a “good” state.

Type renvoyé

Optional[List[int]]

Renvoie

The criterion as list of qubit indices.

precision

Returns the target precision epsilon of the algorithm.

Type renvoyé

float

Renvoie

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

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.

Renvoie

returns the current Q factory of the algorithm

Type renvoyé

QFactory

quantum_instance

Returns quantum instance.

Type renvoyé

Optional[QuantumInstance]

random

Return a numpy random.

state_preparation

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

Type renvoyé

QuantumCircuit

Renvoie

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