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)

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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

__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 $a$ .

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='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

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.

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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