qiskit.aqua.algorithms.AmplitudeEstimation

class AmplitudeEstimation(num_eval_qubits, state_preparation=None, grover_operator=None, objective_qubits=None, post_processing=None, phase_estimation_circuit=None, iqft=None, quantum_instance=None, a_factory=None, q_factory=None, i_objective=None)

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The Quantum Phase Estimation-based Amplitude Estimation algorithm.

This class implements the original Quantum Amplitude Estimation (QAE) algorithm, introduced by [1]. This canonical version uses quantum phase estimation along with a set of $m$ additional evaluation qubits to find an estimate $\tilde{a}$ , that is restricted to the grid

\tilde{a} \in \{\sin^2(\pi y / 2^m) : y = 0, ..., 2^{m-1}\}

More evaluation qubits produce a finer sampling grid, therefore the accuracy of the algorithm increases with $m$ .

Using a maximum likelihood post processing, this grid constraint can be circumvented. This improved estimator is implemented as well, see [2] Appendix A for more detail.

References

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

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

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

Iterative Quantum Amplitude Estimation. arXiv:1912.05559.

Parameters

num_eval_qubits (int) – The number of evaluation qubits.
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 to specify the oracle in the Grover operator, if the Grover operator is not supplied. 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 result of the algorithm $0 \leq a \leq 1$ , usually used to map the estimate to a target interval.
phase_estimation_circuit (Optional[QuantumCircuit]) – The phase estimation circuit used to run the algorithm. Defaults to the standard phase estimation circuit from the circuit library, qiskit.circuit.library.PhaseEstimation.
iqft (Optional[QuantumCircuit]) – The inverse quantum Fourier transform component, defaults to using a standard implementation from qiskit.circuit.library.QFT when None.
quantum_instance (Union[QuantumInstance, Backend, BaseBackend, None]) – The backend (or QuantumInstance) to execute the circuits on.
a_factory (Optional[CircuitFactory]) – Deprecated, use state_preparation. The CircuitFactory subclass object representing the problem unitary.
q_factory (Optional[CircuitFactory]) – Deprecated, use grover_operator. The CircuitFactory subclass object representing an amplitude estimation sample (based on a_factory).
i_objective (Optional[int]) – Deprecated, use objective_qubits. The index of the objective qubit, i.e. the qubit marking ‘good’ solutions with the state $|1\rangle$ and ‘bad’ solutions with the state $0\rangle$ .

init

__init__(num_eval_qubits, state_preparation=None, grover_operator=None, objective_qubits=None, post_processing=None, phase_estimation_circuit=None, iqft=None, quantum_instance=None, a_factory=None, q_factory=None, i_objective=None)

Parameters

num_eval_qubits (int) – The number of evaluation qubits.
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 to specify the oracle in the Grover operator, if the Grover operator is not supplied. 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 result of the algorithm $0 \leq a \leq 1$ , usually used to map the estimate to a target interval.
phase_estimation_circuit (Optional[QuantumCircuit]) – The phase estimation circuit used to run the algorithm. Defaults to the standard phase estimation circuit from the circuit library, qiskit.circuit.library.PhaseEstimation.
iqft (Optional[QuantumCircuit]) – The inverse quantum Fourier transform component, defaults to using a standard implementation from qiskit.circuit.library.QFT when None.
quantum_instance (Union[QuantumInstance, Backend, BaseBackend, None]) – The backend (or QuantumInstance) to execute the circuits on.
a_factory (Optional[CircuitFactory]) – Deprecated, use state_preparation. The CircuitFactory subclass object representing the problem unitary.
q_factory (Optional[CircuitFactory]) – Deprecated, use grover_operator. The CircuitFactory subclass object representing an amplitude estimation sample (based on a_factory).
i_objective (Optional[int]) – Deprecated, use objective_qubits. The index of the objective qubit, i.e. the qubit marking ‘good’ solutions with the state $|1\rangle$ and ‘bad’ solutions with the state $0\rangle$ .

Methods


`__init__`(num_eval_qubits[, …])	type num_eval_qubits`int`
`confidence_interval`(alpha[, kind])	Compute the (1 - alpha) confidence interval.
`construct_circuit`([measurement])	Construct the Amplitude Estimation quantum circuit.
`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 $a$ .

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]

confidence_interval

confidence_interval(alpha, kind='likelihood_ratio')

Compute the (1 - alpha) confidence interval.

Parameters

alpha (float) – Confidence level: compute the (1 - alpha) confidence interval.
kind (str) – The method to compute the confidence interval, can be ‘fisher’, ‘observed_fisher’ or ‘likelihood_ratio’ (default)

Return type

List[float]

Returns

The (1 - alpha) confidence interval of the specified kind.

Raises

AquaError – If ‘mle’ is not in self._ret.keys() (i.e. run was not called yet).
NotImplementedError – If the confidence interval method kind is not implemented.

construct_circuit

construct_circuit(measurement=False)

Construct the Amplitude Estimation quantum circuit.

Parameters

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

Return type

QuantumCircuit

Returns

The QuantumCircuit object for the constructed circuit.

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

is_good_state

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.

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

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.

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.

run

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

set_backend(backend, **kwargs)

Sets backend with configuration.

Return type

None

state_preparation

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

Return type

QuantumCircuit

Returns

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

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