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)[source]¶

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__(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)[source]¶

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 \(a\).

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='likelihood_ratio')[source]¶

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(measurement=False)[source]¶

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.

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.